# Testing GR and alternative theories with planetary ephemerides

Agnès Fienga, Olivier Minazzoli

► To cite this version:

Agnès Fienga, Olivier Minazzoli. Testing GR and alternative theories with planetary ephemerides. Living Rev.Rel., 2024, 27 (1), pp.1. 10.1007/s41114-023-00047-0 . hal-04039028

HAL Id: hal-04039028

https://hal.science/hal-04039028

Submitted on 6 Feb 2024

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<figure><img src="image_1.png" /></figure>

# Testing theories of gravity with planetary ephemerides

Agnès Fienga<fnref n="1" />,<fnref n="2" /> ID · Olivier Minazzoli<fnref n="3" />,<fnref n="4" /> ID

Received: 11 January 2023 / Accepted: 10 November 2023

© The Author(s) 2024

## Abstract

We describe here how planetary ephemerides are built in the framework of General Relativity and how they can be used to test alternative theories. We focus on the definition of the reference frame (space and time) in which the planetary ephemeris is described, the equations of motion that govern the orbits of solar system bodies and electromagnetic waves. After a review on the existing planetary and lunar ephemerides, we summarize the results obtained considering full modifications of the ephemeris framework with direct comparisons with the observations of planetary systems, with a specific attention for the PPN formalism. We then discuss other formalisms such as Einstein-dilaton theories, the massless graviton and MOND. The paper finally concludes on some comments and recommendations regarding misinterpreted measurements of the advance of perihelia.

**Keywords** Gravitation · Alternative gravity theories · Experimental studies of gravity · Solar system

## Contents

1 Introduction .................................................................................................................... 3

2 Basic concepts behind planetary ephemerides .................................................................... 5

2.1 Prerequisites: a brief primer on general relativity .................................................... 5

2.1.1 Newton's theory ........................................................................................................ 6

2.1.2 Proper time in special relativity ............................................................................ 7

2.1.3 Free fall and proper time in general relativity .................................................... 10

2.1.4 Einstein equation and the Newtonian approximation ............................................ 11

2.1.5 Invariance of the laws through a change of coordinate system ................................. 15

2.1.6 The action of general relativity ............................................................................ 19

<figure><img src="image_2.png" /></figure>

<footnote>Agnès Fienga
agnes.fienga@oca.eu</footnote>

<footnote n="1">¹ Geoazur, Observatoire de la Côte d'Azur, Av. A. Einstein, Sophia-Antipolis 06560, France</footnote>

<footnote n="2">² IMCCE, Observatoire de Paris, Av. Denfert-Rocheau, Paris 75014, France</footnote>

<footnote n="3">³ ARTEMIS, Observatoire de la Côte d'Azur, Av. Observatoire, Nice 06560, France</footnote>

<footnote n="4">⁴ Bureau des Affaires Spatiales, 2 rue du Gabian, Monaco 98000, Monaco</footnote>
<figure><img src="image_1.png" /></figure>

# Testing theories of gravity with planetary ephemerides

Agnès Fienga<fnref n="1" />,<fnref n="2" /> ID · Olivier Minazzoli<fnref n="3" />,<fnref n="4" /> ID

Received: 11 January 2023 / Accepted: 10 November 2023

© The Author(s) 2024

## Abstract

We describe here how planetary ephemerides are built in the framework of General Relativity and how they can be used to test alternative theories. We focus on the definition of the reference frame (space and time) in which the planetary ephemeris is described, the equations of motion that govern the orbits of solar system bodies and electromagnetic waves. After a review on the existing planetary and lunar ephemerides, we summarize the results obtained considering full modifications of the ephemeris framework with direct comparisons with the observations of planetary systems, with a specific attention for the PPN formalism. We then discuss other formalisms such as Einstein-dilaton theories, the massless graviton and MOND. The paper finally concludes on some comments and recommendations regarding misinterpreted measurements of the advance of perihelia.

**Keywords** Gravitation · Alternative gravity theories · Experimental studies of gravity · Solar system

## Contents

<table><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>2</td><td>Basic concepts behind planetary ephemerides</td><td>5</td></tr><tr><td>2.1</td><td>Prerequisites: a brief primer on general relativity</td><td>5</td></tr><tr><td>2.1.1</td><td>Newton's theory</td><td>6</td></tr><tr><td>2.1.2</td><td>Proper time in special relativity</td><td>7</td></tr><tr><td>2.1.3</td><td>Free fall and proper time in general relativity</td><td>10</td></tr><tr><td>2.1.4</td><td>Einstein equation and the Newtonian approximation</td><td>11</td></tr><tr><td>2.1.5</td><td>Invariance of the laws through a change of coordinate system</td><td>15</td></tr><tr><td>2.1.6</td><td>The action of general relativity</td><td>19</td></tr></table>

<figure><img src="image_2.png" /></figure>

<footnote>Agnès Fienga
agnes.fienga@oca.eu</footnote>

<footnote n="1">¹ Geoazur, Observatoire de la Côte d'Azur, Av. A. Einstein, Sophia-Antipolis 06560, France</footnote>

<footnote n="2">² IMCCE, Observatoire de Paris, Av. Denfert-Rocheau, Paris 75014, France</footnote>

<footnote n="3">³ ARTEMIS, Observatoire de la Côte d'Azur, Av. Observatoire, Nice 06560, France</footnote>

<footnote n="4">⁴ Bureau des Affaires Spatiales, 2 rue du Gabian, Monaco 98000, Monaco</footnote>
<table>
  <tbody>
    <tr>
      <td>2.1.7</td>
      <td>The post-Newtonian approximation of general relativity and gauge invariance</td>
      <td>20</td>
    </tr>
    <tr>
      <td>2.1.8</td>
      <td>The Einstein-Infeld-Hoffman-Droste-Lorentz equation of motion</td>
      <td>22</td>
    </tr>
    <tr>
      <td>2.1.9</td>
      <td>The propagation of light in general relativity</td>
      <td>24</td>
    </tr>
    <tr>
      <td>2.1.10</td>
      <td>Alternative gravitational theories</td>
      <td>27</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>Basics on ephemeris</td>
      <td>27</td>
    </tr>
    <tr>
      <td>3</td>
      <td>Planetary and lunar ephemerides in general relativity</td>
      <td>30</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>State-of-the-art for planetary and lunar ephemerides</td>
      <td>30</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Reference frame theory in general relativity</td>
      <td>31</td>
    </tr>
    <tr>
      <td>3.2.1</td>
      <td>General relativity barycentric metric</td>
      <td>33</td>
    </tr>
    <tr>
      <td>3.2.2</td>
      <td>Time-scales in the solar system</td>
      <td>34</td>
    </tr>
    <tr>
      <td>3.2.3</td>
      <td>Inertial frame</td>
      <td>35</td>
    </tr>
    <tr>
      <td>3.2.4</td>
      <td>Definition of the solar system barycenter (SSB)</td>
      <td>36</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>The dynamical model</td>
      <td>37</td>
    </tr>
    <tr>
      <td>3.3.1</td>
      <td>Point-mass interactions</td>
      <td>37</td>
    </tr>
    <tr>
      <td>3.3.2</td>
      <td>Asteroid perturbations</td>
      <td>38</td>
    </tr>
    <tr>
      <td>3.3.3</td>
      <td>Lense-Thirring acceleration</td>
      <td>39</td>
    </tr>
    <tr>
      <td>3.3.4</td>
      <td>Shape of the sun</td>
      <td>39</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Planetary datasets</td>
      <td>41</td>
    </tr>
    <tr>
      <td>3.4.1</td>
      <td>Time delay of the propagation of light</td>
      <td>42</td>
    </tr>
    <tr>
      <td>3.4.2</td>
      <td>Radio science and navigation data</td>
      <td>44</td>
    </tr>
    <tr>
      <td>3.4.3</td>
      <td>Angular ground-based observations</td>
      <td>46</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Fitting procedures</td>
      <td>46</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Tests of alternative theoretical frameworks with planetary ephemeris</td>
      <td>47</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Parametrized post-Newtonian framework</td>
      <td>48</td>
    </tr>
    <tr>
      <td>4.1.1</td>
      <td>Equations of motion, Lense-Thirring and Shapiro delay</td>
      <td>50</td>
    </tr>
    <tr>
      <td>4.1.2</td>
      <td>Time-scales</td>
      <td>52</td>
    </tr>
    <tr>
      <td>4.1.3</td>
      <td>Constraints and correlations</td>
      <td>52</td>
    </tr>
    <tr>
      <td>4.1.4</td>
      <td>The extended PPN formalism</td>
      <td>56</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Equivalence principle</td>
      <td>59</td>
    </tr>
    <tr>
      <td>4.2.1</td>
      <td>The case of metric theories</td>
      <td>61</td>
    </tr>
    <tr>
      <td>4.2.2</td>
      <td>The case of non-metric theories</td>
      <td>61</td>
    </tr>
    <tr>
      <td>4.2.3</td>
      <td>The equivalence principle in a planet-satellite system</td>
      <td>62</td>
    </tr>
    <tr>
      <td>4.2.4</td>
      <td>The equivalence principle at planetary scale</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.2.5</td>
      <td>Comment on the mass in the Shapiro delay equation</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.2.6</td>
      <td>The issue of time in non-metric theories</td>
      <td>64</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Variation of the gravitational constant G</td>
      <td>65</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Massive gravity</td>
      <td>68</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Yukawa potential and fifth force</td>
      <td>71</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>Einstein-dilaton theories</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.6.1</td>
      <td>Equations of motion and Shapiro delay</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.6.2</td>
      <td>Conserved quantities and the definition of the SSB</td>
      <td>75</td>
    </tr>
    <tr>
      <td>4.6.3</td>
      <td>Results</td>
      <td>76</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>MOND</td>
      <td>76</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Inconsistent tests with ephemeris outputs</td>
      <td>78</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Definitions</td>
      <td>78</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>The case of the advance of the perihelia and nodes</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>What can often be found in the literature</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Why consistency matters</td>
      <td>82</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Future directions</td>
      <td>84</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Theory</td>
      <td>84</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Observation</td>
      <td>86</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Summary</td>
      <td>86</td>
    </tr>
    <tr>
      <td colspan="2">Appendix A: Derivation of the Shapiro delay at the c<sup>4</sup> level, and related issues</td>
      <td>87</td>
    </tr>
    <tr>
      <td colspan="2">References</td>
      <td>88</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>2.1.7</td>
      <td>The post-Newtonian approximation of general relativity and gauge invariance</td>
      <td>20</td>
    </tr>
    <tr>
      <td>2.1.8</td>
      <td>The Einstein-Infeld-Hoffman-Droste-Lorentz equation of motion</td>
      <td>22</td>
    </tr>
    <tr>
      <td>2.1.9</td>
      <td>The propagation of light in general relativity</td>
      <td>24</td>
    </tr>
    <tr>
      <td>2.1.10</td>
      <td>Alternative gravitational theories</td>
      <td>27</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>Basics on ephemeris</td>
      <td>27</td>
    </tr>
    <tr>
      <td>3</td>
      <td>Planetary and lunar ephemerides in general relativity</td>
      <td>30</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>State-of-the-art for planetary and lunar ephemerides</td>
      <td>30</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Reference frame theory in general relativity</td>
      <td>31</td>
    </tr>
    <tr>
      <td>3.2.1</td>
      <td>General relativity barycentric metric</td>
      <td>33</td>
    </tr>
    <tr>
      <td>3.2.2</td>
      <td>Time-scales in the solar system</td>
      <td>34</td>
    </tr>
    <tr>
      <td>3.2.3</td>
      <td>Inertial frame</td>
      <td>35</td>
    </tr>
    <tr>
      <td>3.2.4</td>
      <td>Definition of the solar system barycenter (SSB)</td>
      <td>36</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>The dynamical model</td>
      <td>37</td>
    </tr>
    <tr>
      <td>3.3.1</td>
      <td>Point-mass interactions</td>
      <td>37</td>
    </tr>
    <tr>
      <td>3.3.2</td>
      <td>Asteroid perturbations</td>
      <td>38</td>
    </tr>
    <tr>
      <td>3.3.3</td>
      <td>Lense-Thirring acceleration</td>
      <td>39</td>
    </tr>
    <tr>
      <td>3.3.4</td>
      <td>Shape of the sun</td>
      <td>39</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Planetary datasets</td>
      <td>41</td>
    </tr>
    <tr>
      <td>3.4.1</td>
      <td>Time delay of the propagation of light</td>
      <td>42</td>
    </tr>
    <tr>
      <td>3.4.2</td>
      <td>Radio science and navigation data</td>
      <td>44</td>
    </tr>
    <tr>
      <td>3.4.3</td>
      <td>Angular ground-based observations</td>
      <td>46</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Fitting procedures</td>
      <td>46</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Tests of alternative theoretical frameworks with planetary ephemeris</td>
      <td>47</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Parametrized post-Newtonian framework</td>
      <td>48</td>
    </tr>
    <tr>
      <td>4.1.1</td>
      <td>Equations of motion, Lense-Thirring and Shapiro delay</td>
      <td>50</td>
    </tr>
    <tr>
      <td>4.1.2</td>
      <td>Time-scales</td>
      <td>52</td>
    </tr>
    <tr>
      <td>4.1.3</td>
      <td>Constraints and correlations</td>
      <td>52</td>
    </tr>
    <tr>
      <td>4.1.4</td>
      <td>The extended PPN formalism</td>
      <td>56</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Equivalence principle</td>
      <td>59</td>
    </tr>
    <tr>
      <td>4.2.1</td>
      <td>The case of metric theories</td>
      <td>61</td>
    </tr>
    <tr>
      <td>4.2.2</td>
      <td>The case of non-metric theories</td>
      <td>61</td>
    </tr>
    <tr>
      <td>4.2.3</td>
      <td>The equivalence principle in a planet-satellite system</td>
      <td>62</td>
    </tr>
    <tr>
      <td>4.2.4</td>
      <td>The equivalence principle at planetary scale</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.2.5</td>
      <td>Comment on the mass in the Shapiro delay equation</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.2.6</td>
      <td>The issue of time in non-metric theories</td>
      <td>64</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Variation of the gravitational constant G</td>
      <td>65</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Massive gravity</td>
      <td>68</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Yukawa potential and fifth force</td>
      <td>71</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>Einstein-dilaton theories</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.6.1</td>
      <td>Equations of motion and Shapiro delay</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.6.2</td>
      <td>Conserved quantities and the definition of the SSB</td>
      <td>75</td>
    </tr>
    <tr>
      <td>4.6.3</td>
      <td>Results</td>
      <td>76</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>MOND</td>
      <td>76</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Inconsistent tests with ephemeris outputs</td>
      <td>78</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Definitions</td>
      <td>78</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>The case of the advance of the perihelia and nodes</td>
      <td>79</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>What can often be found in the literature</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Why consistency matters</td>
      <td>82</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Future directions</td>
      <td>84</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Theory</td>
      <td>84</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Observation</td>
      <td>86</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Summary</td>
      <td>86</td>
    </tr>
    <tr>
      <td colspan="2">Appendix A: Derivation of the Shapiro delay at the c<sup>4</sup> level, and related issues</td>
      <td>87</td>
    </tr>
    <tr>
      <td colspan="2">References</td>
      <td>88</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_3.png" /></figure>
# 1 Introduction

The current definition of what is an ephemeris is a table giving the future positions of a planet, comet, or satellite. By extension, it also includes the dynamical framework from which the planetary positions and velocities are estimated. This framework includes not only the dynamical modeling and the reference system with which the motion is described, but also the sample of observations used for adjusting the constants of the model. In this review, we will mainly focused on ephemerides of planets, but there are also ephemerides of natural and artificial satellites, small bodies (comets, asteroids) and pulsars.

With the 19th century and the golden age of the big refractors (Verdet 1990), the astrometry of planets has known a significant improvement, leading to an increased accuracy of the dynamical theories describing their motions. Together with these improvements, came the inconsistencies between very accurate observed positions of planets, such as Uranus, and the classical Newtonian modeling of gravity. In the case of Uranus, a significant—even at this epoch—difference of several seconds of arc between its observed positions and the positions proposed by the models, based on Newton's laws, opens the door to a major controversy of the century. Why the Uranus orbit does not match with the observations when, for instances, comet ephemerides are able to predict accurately their return in the visible sky? Since 1821, the explanation of an unknown planet perturbing the orbit of Uranus has been proposed. A hunt for the unseen planet started with an intensification in 1844 and the hidden celestial object, called Neptune, was finally discovered in 1846 by Galle in Berlin at the location that was predicted by Le Verrier and Addams. This episode signs the paramount of classical celestial mechanics of the 19th century. However, an other problem remains: the orbit of Mercury. At the beginning of the 20th century, catalogs of planetary observations have been built with a worldwide effort and provided angular positions for almost one century with accuracies below the level of few seconds of arc. During this period, when no direct measure of the planetary distance was possible, the constraints on the size of the planetary orbits were obtained thanks to the long interval of observations and precise determination of the mean motions, and consequently, with the third Kepler law, of the semi-major axis.

In the case of Mercury, the observed variation of the Mercury orbit was far too important to be left unexplained in comparison to the observational accuracy of the end of the 19th century and the beginning of the 20th century. Following the example of Neptune and Uranus, people proposed the existence of another hidden object, called Vulcain, perturbing the orbit of Mercury the same way that Neptune perturbs the one of Uranus. The problem was that no one was able to observe Vulcain. Some proposed that the planet could be always located on the other side of the sun relative to the earth, making it impossible to be seen from earth. Others proposed a modification of Newton's potential equation. However, it was Einstein who ultimately explained Mercury orbit, positing that gravity was not a force, but a result of the spacetime curvature caused by the mass of celestial bodies (Einstein 1915).
# 1 Introduction

The current definition of what is an ephemeris is a table giving the future positions of a planet, comet, or satellite. By extension, it also includes the dynamical framework from which the planetary positions and velocities are estimated. This framework includes not only the dynamical modeling and the reference system with which the motion is described, but also the sample of observations used for adjusting the constants of the model. In this review, we will mainly focused on ephemerides of planets, but there are also ephemerides of natural and artificial satellites, small bodies (comets, asteroids) and pulsars.

With the 19th century and the golden age of the big refractors (Verdet 1990), the astrometry of planets has known a significant improvement, leading to an increased accuracy of the dynamical theories describing their motions. Together with these improvements, came the inconsistencies between very accurate observed positions of planets, such as Uranus, and the classical Newtonian modeling of gravity. In the case of Uranus, a significant—even at this epoch—difference of several seconds of arc between its observed positions and the positions proposed by the models, based on Newton's laws, opens the door to a major controversy of the century. Why the Uranus orbit does not match with the observations when, for instances, comet ephemerides are able to predict accurately their return in the visible sky? Since 1821, the explanation of an unknown planet perturbing the orbit of Uranus has been proposed. A hunt for the unseen planet started with an intensification in 1844 and the hidden celestial object, called Neptune, was finally discovered in 1846 by Galle in Berlin at the location that was predicted by Le Verrier and Addams. This episode signs the paramount of classical celestial mechanics of the 19th century. However, an other problem remains: the orbit of Mercury. At the beginning of the 20th century, catalogs of planetary observations have been built with a worldwide effort and provided angular positions for almost one century with accuracies below the level of few seconds of arc. During this period, when no direct measure of the planetary distance was possible, the constraints on the size of the planetary orbits were obtained thanks to the long interval of observations and precise determination of the mean motions, and consequently, with the third Kepler law, of the semi-major axis.

In the case of Mercury, the observed variation of the Mercury orbit was far too important to be left unexplained in comparison to the observational accuracy of the end of the 19th century and the beginning of the 20th century. Following the example of Neptune and Uranus, people proposed the existence of another hidden object, called Vulcain, perturbing the orbit of Mercury the same way that Neptune perturbs the one of Uranus. The problem was that no one was able to observe Vulcain. Some proposed that the planet could be always located on the other side of the sun relative to the earth, making it impossible to be seen from earth. Others proposed a modification of Newton's potential equation. However, it was Einstein who ultimately explained Mercury orbit, positing that gravity was not a force, but a result of the spacetime curvature caused by the mass of celestial bodies (Einstein 1915).
<table><caption>**Table 1** 4 generations of planetary ephemerides: accuracies of the observations used for the construction of the ephemerides either with angular measurements (columns labeled *angle*) or with direct distance radar and spacecraft tracking observations (Columns labeled *distance Earth*). These values give an idea of the expected precision obtained by the ephemerides during the period of observations indicated in the second row of the table. It does not preclude from a degradation of the quality of the ephemerides out from the period. The last column gives the estimation of the advance of perihelium $\Delta\dot{\tau}$ as computed with general relativity. For the Earth, $\Delta\dot{\tau}$ is of about 0.1".yr$^{-1}$</caption><thead><tr><th rowspan="3">Ephemerides</th><th colspan="2">Gaillot (Gaillot and Le Verrier 1913)</th><th colspan="2">DE102 (Standish 1983)</th><th colspan="2">DE440/INPOP19a (Park et al. 2021; Fienga et al. 2019)</th><th rowspan="3">GR</th></tr><tr><th colspan="2">1800–1913</th><th colspan="2">1913–1983</th><th colspan="2">1924–2021</th></tr><tr><th>Angle "</th><th>Distance Earth-km</th><th>Angle "</th><th>Distance Earth-km</th><th>Angle "</th><th>Distance Earth-km</th><th>Δω".yr<sup>-1</sup></th></tr></thead><tbody><tr><td>Mercury</td><td>1</td><td>450</td><td>0.050</td><td>5</td><td>0.002</td><td>0.004</td><td>0.43</td></tr><tr><td>Venus</td><td>0.5</td><td>100</td><td>0.050</td><td>2</td><td>0.002</td><td>0.006</td><td>0.14</td></tr><tr><td>Mars</td><td>0.5</td><td>150</td><td>0.050</td><td>0.050</td><td>0.001</td><td>0.0015</td><td>0.065</td></tr><tr><td>Jupiter</td><td>0.5</td><td>1400</td><td>0.1</td><td>10</td><td>0.010</td><td>0.020</td><td>0.019</td></tr><tr><td>Saturn</td><td>0.5</td><td>3000</td><td>0.1</td><td>600</td><td>0.001</td><td>0.020</td><td>0.010</td></tr><tr><td>Uranus</td><td>1</td><td>12,700</td><td>0.2</td><td>2540</td><td>0.050</td><td>10</td><td>0.005</td></tr><tr><td>Neptune</td><td>1</td><td>22,000</td><td>0.2</td><td>4400</td><td>0.050</td><td>50</td><td>0.0033</td></tr><tr><td>Pluto</td><td>1</td><td>24,000</td><td>0.2</td><td>4800</td><td>0.050</td><td>2400</td><td>0.0027</td></tr></tbody></table>

Decades later, thanks to the remote exploration of the solar system, planetary ephemerides have evolved with high accurate astrometric observations obtained for planet and natural satellites thanks to the navigation tracking of spacecraft (s/c) orbiting these systems. Table 1 gives a good comparison between three generations of planetary ephemerides developed at key stages of their evolution: from the end of the classical Newtonian celestial mechanics (Gaillot and Le Verrier 1913), to the beginning of the space regular exploration (Standish 1983) and up to the present time (Park et al. 2021; Fienga et al. 2019).

Since the beginning of the space exploration, general relativity has been continuously tested with planetary orbits, for instance, using the first direct radar measurements on the telluric planet surfaces (Shapiro 1964). These measurements were the first direct estimations of the Earth–Venus, Mercury or Mars distances used as direct constraints in the construction of planetary ephemerides (Ash et al. 1967; Standish et al. 1976). With the Apollo and Lunokhod missions on the Moon and the installation of light reflecting corner cubes, a new step is reached as centimetric measures of the Earth–Moon distances are obtained with Lunar Laser Ranging (LLR) since then, with regular improvement of the measurement accuracy (Murphy et al. 2008; Turyshev et al. 2013b; Courde et al. 20170. In 1983, the first tracking data from the Viking landers on Mars have been included in the numerical integration of the Mars orbit at JPL (Newhall et al. 1983). With these measurements and those of the first Pioneer and Voyager flybys of the Jovian and Saturnian systems, the accuracy of the planetary ephemerides enters into the age of kilometer accuracies for
<table><caption>**Table 1** 4 generations of planetary ephemerides: accuracies of the observations used for the construction of the ephemerides either with angular measurements (columns labeled *angle*) or with direct distance radar and spacecraft tracking observations (Columns labeled *distance Earth*). These values give an idea of the expected precision obtained by the ephemerides during the period of observations indicated in the second row of the table. It does not preclude from a degradation of the quality of the ephemerides out from the period. The last column gives the estimation of the advance of perihelium $\Delta\dot{\tau}$ as computed with general relativity. For the Earth, $\Delta\dot{\tau}$ is of about 0.1".yr$^{-1}$</caption><thead><tr><th rowspan="3">Ephemerides</th><th colspan="2">Gaillot (Gaillot and Le Verrier 1913)</th><th colspan="2">DE102 (Standish 1983)</th><th colspan="2">DE440/INPOP19a (Park et al. 2021; Fienga et al. 2019)</th><th rowspan="3">GR</th></tr><tr><th colspan="2">1800–1913</th><th colspan="2">1913–1983</th><th colspan="2">1924–2021</th></tr><tr><th>Angle "</th><th>Distance Earth-km</th><th>Angle "</th><th>Distance Earth-km</th><th>Angle "</th><th>Distance Earth-km</th><th>Δω".yr<sup>-1</sup></th></tr></thead><tbody><tr><td>Mercury</td><td>1</td><td>450</td><td>0.050</td><td>5</td><td>0.002</td><td>0.004</td><td>0.43</td></tr><tr><td>Venus</td><td>0.5</td><td>100</td><td>0.050</td><td>2</td><td>0.002</td><td>0.006</td><td>0.14</td></tr><tr><td>Mars</td><td>0.5</td><td>150</td><td>0.050</td><td>0.050</td><td>0.001</td><td>0.0015</td><td>0.065</td></tr><tr><td>Jupiter</td><td>0.5</td><td>1400</td><td>0.1</td><td>10</td><td>0.010</td><td>0.020</td><td>0.019</td></tr><tr><td>Saturn</td><td>0.5</td><td>3000</td><td>0.1</td><td>600</td><td>0.001</td><td>0.020</td><td>0.010</td></tr><tr><td>Uranus</td><td>1</td><td>12,700</td><td>0.2</td><td>2540</td><td>0.050</td><td>10</td><td>0.005</td></tr><tr><td>Neptune</td><td>1</td><td>22,000</td><td>0.2</td><td>4400</td><td>0.050</td><td>50</td><td>0.0033</td></tr><tr><td>Pluto</td><td>1</td><td>24,000</td><td>0.2</td><td>4800</td><td>0.050</td><td>2400</td><td>0.0027</td></tr></tbody></table>

Decades later, thanks to the remote exploration of the solar system, planetary ephemerides have evolved with high accurate astrometric observations obtained for planet and natural satellites thanks to the navigation tracking of spacecraft (s/c) orbiting these systems. Table 1 gives a good comparison between three generations of planetary ephemerides developed at key stages of their evolution: from the end of the classical Newtonian celestial mechanics (Gaillot and Le Verrier 1913), to the beginning of the space regular exploration (Standish 1983) and up to the present time (Park et al. 2021; Fienga et al. 2019).

Since the beginning of the space exploration, general relativity has been continuously tested with planetary orbits, for instance, using the first direct radar measurements on the telluric planet surfaces (Shapiro 1964). These measurements were the first direct estimations of the Earth–Venus, Mercury or Mars distances used as direct constraints in the construction of planetary ephemerides (Ash et al. 1967; Standish et al. 1976). With the Apollo and Lunokhod missions on the Moon and the installation of light reflecting corner cubes, a new step is reached as centimetric measures of the Earth–Moon distances are obtained with Lunar Laser Ranging (LLR) since then, with regular improvement of the measurement accuracy (Murphy et al. 2008; Turyshev et al. 2013b; Courde et al. 20170. In 1983, the first tracking data from the Viking landers on Mars have been included in the numerical integration of the Mars orbit at JPL (Newhall et al. 1983). With these measurements and those of the first Pioneer and Voyager flybys of the Jovian and Saturnian systems, the accuracy of the planetary ephemerides enters into the age of kilometer accuracies for
the planets and few tens of centimeter for the Moon thanks to LLR. At this level of accuracy, the modelisation of the planetary dynamics was upgraded including asteroid perturbations of the 5 biggest objects, Mars rotation improvement but also figure-figure effect for the modelling of the Earth-Moon tidal deformations (Standish 2001). Since 2010, the s/c tracking measurements are still improving thanks to the installation of more efficient transpondeurs. It is now possible to study the orbit of Mars with an accuracy of less than a meter and since 2016 to have a monitoring of Jupiter and Saturn orbits with an accuracy of tens of meters.

With these accuracies, the complexity of the dynamical model increases and it is clear that the solar system becomes a more and more interesting tool for testing general relativity or alternative theories of gravity. First tests of general relativity with modern planetary ephemerides have started in 1978 (Anderson et al. 1978) with regular improvements since then.

In this review, we will focus on planetary ephemerides, setting aside the topic of gravity tests using the Earth-Moon system due to its complexity. Our incomplete understanding of the Moon internal structure significantly limits, indeed, the testing of alternative theories. But in order to introduce these limitations, a detailed presentation of the complex mechanisms between the internal structure of the Moon, its deformations, general relativity and their impacts on LLR observations (at very close frequencies) has to be done, and we think that it is out of the scope of the present work.

Here, we first explain how a planetary ephemeris is built from its native relativistic framework. We give the dynamical model and the data sets used for its fits as well as the problematics related to the fit itself. We try, at this stage, to give an introduction to some of the concepts hidden behind the construction of planetary ephemerides. We then review classic results obtained in the general relativity and post-Newtonian limits for the modern period. In a third part, we discuss state-of-the-art approaches for directly confronting alternative theories of gravity with planetary observations in the most consistent possible way. Finally we give a quick overview of other types of tests, deduced by interpreting planetary ephemeris accuracy at the light of different gravitational frameworks. We will try to convince the readers that such results have to be considered with a lot of caution because of the lack of consistency between the framework with which the planetary ephemeris is built and the one proposed by the authors.

Finally, we acknowledge that this review may have limitations. We apologize in advance for any inadvertent omissions in the references and welcome feedback from our readers.

# 2 Basic concepts behind planetary ephemerides

## 2.1 Prerequisites: a brief primer on general relativity

In classical Newtonian mechanics, space and time are treated as absolute concepts. For instance, the time span is assumed to be uniform throughout the universe. But the
the planets and few tens of centimeter for the Moon thanks to LLR. At this level of accuracy, the modelisation of the planetary dynamics was upgraded including asteroid perturbations of the 5 biggest objects, Mars rotation improvement but also figure-figure effect for the modelling of the Earth-Moon tidal deformations (Standish 2001). Since 2010, the s/c tracking measurements are still improving thanks to the installation of more efficient transpondeurs. It is now possible to study the orbit of Mars with an accuracy of less than a meter and since 2016 to have a monitoring of Jupiter and Saturn orbits with an accuracy of tens of meters.

With these accuracies, the complexity of the dynamical model increases and it is clear that the solar system becomes a more and more interesting tool for testing general relativity or alternative theories of gravity. First tests of general relativity with modern planetary ephemerides have started in 1978 (Anderson et al. 1978) with regular improvements since then.

In this review, we will focus on planetary ephemerides, setting aside the topic of gravity tests using the Earth-Moon system due to its complexity. Our incomplete understanding of the Moon internal structure significantly limits, indeed, the testing of alternative theories. But in order to introduce these limitations, a detailed presentation of the complex mechanisms between the internal structure of the Moon, its deformations, general relativity and their impacts on LLR observations (at very close frequencies) has to be done, and we think that it is out of the scope of the present work.

Here, we first explain how a planetary ephemeris is built from its native relativistic framework. We give the dynamical model and the data sets used for its fits as well as the problematics related to the fit itself. We try, at this stage, to give an introduction to some of the concepts hidden behind the construction of planetary ephemerides. We then review classic results obtained in the general relativity and post-Newtonian limits for the modern period. In a third part, we discuss state-of-the-art approaches for directly confronting alternative theories of gravity with planetary observations in the most consistent possible way. Finally we give a quick overview of other types of tests, deduced by interpreting planetary ephemeris accuracy at the light of different gravitational frameworks. We will try to convince the readers that such results have to be considered with a lot of caution because of the lack of consistency between the framework with which the planetary ephemeris is built and the one proposed by the authors.

Finally, we acknowledge that this review may have limitations. We apologize in advance for any inadvertent omissions in the references and welcome feedback from our readers.

# 2 Basic concepts behind planetary ephemerides

## 2.1 Prerequisites: a brief primer on general relativity

In classical Newtonian mechanics, space and time are treated as absolute concepts. For instance, the time span is assumed to be uniform throughout the universe. But the
main lesson from general relativity is that not only space and time are relative notions rather than absolute ones, but also that the structure formed by space and time is curved by the energy of matter. This notably means that the very notion of distances in space and time depends on the observer, and also that observation may be impacted by the curvature of spacetime. This implies a wide range of subtleties when one describes motions at a level at which Newton's theory is no longer accurate enough. In what follows, we provide a brief summary of what is required to understand those subtleties.

### 2.1.1 Newton's theory

According to the first law of motion in Newton's theory, an inertial motion is an uniform motion in a straight line. As a consequence, free fall motions are not inertial in Newton's theory, but are accelerated by the gravitational force $F$. This force acts between every massive objects and reads

$$F_{AB} = -F_{BA} = \frac{GM_{A}M_{B}}{r_{AB}^{2}} \mathbf{r}_{AB}, \qquad (1)$$

where $M$ is the equivalent to the charge in Coulomb's law, but for gravity and $G$ the constant of gravitation.<fnref n="1" /> $M$ is usually named gravitational mass, since it is related to the gravitational force. According to the second law of motion, the acceleration that follows from this force is

$$m_{A}\mathbf{a}_{A} = F_{AB}, \qquad (2)$$

where $m_A$ is the inertial mass of the body $A$. Namely, to accelerate a body $A$ with acceleration $\mathbf{a}_A$, a force $F_{AB}$ needs to be applied, regardless of the type of force in question-it could be gravitational or electrostatic etc. Just as one would not expect a relationship between the charge $q$ in the electrostatic force and the inertial mass, one should not anticipate a relationship between the gravitational “charge” $M$ and the inertial mass $m$. However, observations indicate otherwise, compelling Newton to postulate the equivalence of gravitational and inertial mass. This is known as the *equivalence principle*.

A common way to derive the equation of motion in classical mechanics is through the definition of a Lagrangian of motion—of the general form $L = K - V$, where $K$ is the kinetic energy and $V$ the potential energy. The Lagrangian of motion in the theory of Newton reads

$$L_N = \frac{mv^2}{2} + mU, \qquad (3)$$

<footnote n="1">¹ Indeed, Coulomb's law between two charges reads $|F_{12}| = k_C|q_1||q_2|/r_{12}^2$, where $k_C$ is Coulomb's constant.</footnote>
main lesson from general relativity is that not only space and time are relative notions rather than absolute ones, but also that the structure formed by space and time is curved by the energy of matter. This notably means that the very notion of distances in space and time depends on the observer, and also that observation may be impacted by the curvature of spacetime. This implies a wide range of subtleties when one describes motions at a level at which Newton's theory is no longer accurate enough. In what follows, we provide a brief summary of what is required to understand those subtleties.

### 2.1.1 Newton's theory

According to the first law of motion in Newton's theory, an inertial motion is an uniform motion in a straight line. As a consequence, free fall motions are not inertial in Newton's theory, but are accelerated by the gravitational force $F$. This force acts between every massive objects and reads

$$F_{AB} = -F_{BA} = \frac{GM_{A}M_{B}}{r_{AB}^{2}} \frac{r_{AB}}{\|r_{AB}\|}, \qquad (1)$$

where $M$ is the equivalent to the charge in Coulomb's law, but for gravity and $G$ the constant of gravitation.<fnref n="1" /> $M$ is usually named gravitational mass, since it is related to the gravitational force. According to the second law of motion, the acceleration that follows from this force is

$$m_{A}a_{A} = F_{AB}, \qquad (2)$$

where $m_A$ is the inertial mass of the body $A$. Namely, to accelerate a body $A$ with acceleration $a_A$, a force $F_{AB}$ needs to be applied, regardless of the type of force in question-it could be gravitational or electrostatic etc. Just as one would not expect a relationship between the charge $q$ in the electrostatic force and the inertial mass, one should not anticipate a relationship between the gravitational “charge” $M$ and the inertial mass $m$. However, observations indicate otherwise, compelling Newton to postulate the equivalence of gravitational and inertial mass. This is known as the equivalence principle.

A common way to derive the equation of motion in classical mechanics is through the definition of a Lagrangian of motion—of the general form $L = K - V$, where $K$ is the kinetic energy and $V$ the potential energy. The Lagrangian of motion in the theory of Newton reads

$$L_N = \frac{mv^2}{2} + mU, \qquad (3)$$

<footnote n="1">¹ Indeed, Coulomb's law between two charges reads $|F_{12}| = k_{C}|q_{1}||q_{2}|/r_{12}^{2}$, where $k_{C}$ is Coulomb's constant.</footnote>
where $v$ is the velocity of the moving object and $U$ is the Newtonian potential that satisfies the Poisson equation

$$ \Delta U = -4\pi G\rho_m, \qquad (4) $$

where $\rho_m$ is the mass density of the gravitational sources.<fnref n="2" /> For instance, for a sum of spherical bodies A,

$$ U(x^i, t) = \sum_A \frac{Gm_A}{|x^i - x_A^i(t)|}, \qquad (5) $$

is solution of Eq. (4), with $x^i$ and $x_A^i(t)$ the positions of the moving object and of the gravitational sources A, respectively, where

$$ m_A = \int_A \rho_m d^3 r. \qquad (6) $$

Indeed, applying the Euler–Lagrange equation

$$ \frac{\partial L}{\partial \mathbf{x}} - \frac{d}{dt} \frac{\partial L}{\partial \mathbf{v}} = 0, \qquad (7) $$

on $L_N$ leads to Newton's equation of motion

$$ \mathbf{a} = \nabla U, \qquad (8) $$

which reproduces Eq. (2) with Eq. (1) for $M = m$. In the Newtonian framework, the speed of light, $c$, and the constant of gravitation, $G$, are constants. Time and space are universal and space is flat.

### 2.1.2 Proper time in special relativity

At the onset of the XXth century, a group of physicists understood that time and space were not separate concepts, but instead, composed a singular entity known as spacetime. In this new understanding, time and space became relative notions, and the structure of spacetime, according to Minkowski, possessed a Lorentzian nature—which means that the variation of the proper time of an observer follows

$$ c^2 d\tau^2 = -\eta_{\alpha\beta} dx^\alpha dx^\beta = c^2 dt^2 - dx^2 - dy^2 - dz^2, \qquad (9) $$

where $\eta_{\alpha\beta}$ is named the Minkowski metric, and $\{t, \mathbf{x}\}$ inertial and non-rotating coordinate systems.<fnref n="3" /> Throughout the text, we use the metric signature $(-, +, +, +)$ and Einstein's summation notational convention—which is such that

<footnote n="2">² Note that the potential energy with this definition of the gravitational potential indeed reads $V = -mU$, such that one indeed verifies that $L_N = K - V$ in Eq. (3).</footnote>

<footnote n="3">³ Non-rotating with respect to what will be the subject of a discussion in Sect. 2.1.4.</footnote>
where $v$ is the velocity of the moving object and $U$ is the Newtonian potential that satisfies the Poisson equation

$$ \Delta U = -4\pi G\rho_m, \qquad (4) $$

where $\rho_m$ is the mass density of the gravitational sources.<fnref n="2" /> For instance, for a sum of spherical bodies A,

$$ U(x^i, t) = \sum_A \frac{Gm_A}{|x^i - x_A^i(t)|}, \qquad (5) $$

is solution of Eq. (4), with $x^i$ and $x_A^i(t)$ the positions of the moving object and of the gravitational sources A, respectively, where

$$ m_A = \int_A \rho_m d^3 r. \qquad (6) $$

Indeed, applying the Euler–Lagrange equation

$$ \frac{\partial L}{\partial \mathbf{x}} - \frac{d}{dt} \frac{\partial L}{\partial \mathbf{v}} = 0, \qquad (7) $$

on $L_N$ leads to Newton's equation of motion

$$ \mathbf{a} = \nabla U, \qquad (8) $$

which reproduces Eq. (2) with Eq. (1) for $M = m$. In the Newtonian framework, the speed of light, $c$, and the constant of gravitation, $G$, are constants. Time and space are universal and space is flat.

### 2.1.2 Proper time in special relativity

At the onset of the XXth century, a group of physicists understood that time and space were not separate concepts, but instead, composed a singular entity known as spacetime. In this new understanding, time and space became relative notions, and the structure of spacetime, according to Minkowski, possessed a Lorentzian nature—which means that the variation of the proper time of an observer follows

$$ c^2 d\tau^2 = -\eta_{\alpha\beta} dx^\alpha dx^\beta = c^2 dt^2 - dx^2 - dy^2 - dz^2, \qquad (9) $$

where $\eta_{\alpha\beta}$ is named the Minkowski metric, and $\{t, \mathbf{x}\}$ inertial and non-rotating coordinate systems.<fnref n="3" /> Throughout the text, we use the metric signature $(-, +, +, +)$ and Einstein's summation notational convention—which is such that

<footnote n="2">² Note that the potential energy with this definition of the gravitational potential indeed reads $V = -mU$, such that one indeed verifies that $L_N = K - V$ in Eq. (3).</footnote>

<footnote n="3">³ Non-rotating with respect to what will be the subject of a discussion in Sect. 2.1.4.</footnote>
$A_σB^σ := \sum_σ A_σB^σ$. The variable $c$ will be identified in Sect. 2.1.9 as the speed of light, but more fundamentaly, it fixes the causal structure of the flat spacetime equation Eq. (9). Lorentz transformations of special relativity are simply the coordinate transformations that leave the structure of the metric Eq. (9) unchanged. They read

$$T = \gamma \left( t - \frac{\mathbf{v} \cdot \mathbf{x}}{c^2} \right), \quad \mathbf{X} = \mathbf{x} + \gamma \mathbf{v} t + (\gamma - 1) \frac{\mathbf{v}(\mathbf{v} \cdot \mathbf{r})}{v^2}, \qquad (10)$$

where $v^2 = \mathbf{v} \cdot \mathbf{v}$ and where $\mathbf{v}$ is the velocity between the two inertial reference frames, and with the Lorentz factor $\gamma$ defined as

$$\gamma := \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}, \qquad (11)$$

Contrary to popular wisdom, it is Eq. (9)—not the Lorentz transformations equation Eq. (10)—that is needed to derive the difference in time elapsed between two twins in Langevin's twin paradox.<fnref n="4" /> Let us assume a pair of observers that are at the same location at the spacetime events A and B, but one of the observer stays at rest while the other accelerates to leave and return to the other observer (see Fig. 1).

In terms of the reference frame of the observer at rest {$\tau^r, \mathbf{x}$}, the accelerated observer's proper time $\tau^a$ between the two events A and B reads

$$\tau_{AB}^a = \int_{C_{AB}} d\tau^a = \int_{C_{AB}} \sqrt{1 - \frac{v^2}{c^2}} d\tau^r, \qquad (12)$$

according to Eq. (9), with $\mathbf{v} = d\mathbf{x}/d\tau^r$, and where $C_{AB}$ is the accelerated trajectory of the accelerated observer in the reference frame of the observer at rest. Given a trajectory $C_{AB}$, one can derive the proper time of the accelerated observer elapsed between the events A and B. It is worth noting that Eq. (12) would not maintain the same form if one attempts to derive the proper time elapsed for the rest observer from the coordinates of the accelerated observer. This is because the Minkowski metric [Eq. (9)] does not remain invariant for an accelerated reference frame {$\tau^a, \mathbf{X}$}. In the proper reference frame of an accelerated (but non-rotating) observer, the metric Eq. (9) would instead read (Ni and Zimmermann 1978)

$$c^2 d\tau^2 = \left(1 + \frac{\mathbf{a} \cdot \mathbf{X}}{c^2}\right)^2 c^2 (d\tau^a)^2 - dX^2 - dY^2 - dZ^2, \qquad (13)$$

<footnote n="4">⁴ Langevin's twin paradox presents a scenario in which one twin travels to space at a high speed and returns to find the other twin has aged more. This highlights the relativistic effect of time dilation, as described in the theory of special relativity. The paradox's essence is rooted in a perceived contradiction: both twins should ostensibly observe the other with a similar relative motion, which would therefore result in the same 'anomalous' aging effect for both when solely using Lorentz's transformations Eq. (10) in the derivation.</footnote>
$A_σB^σ := \sum_σ A_σB^σ$. The variable $c$ will be identified in Sect. 2.1.9 as the speed of light, but more fundamentaly, it fixes the causal structure of the flat spacetime equation Eq. (9). Lorentz transformations of special relativity are simply the coordinate transformations that leave the structure of the metric Eq. (9) unchanged. They read

$$T = \gamma \left(t - \frac{\mathbf{v} \cdot \mathbf{X}}{c^2}\right), \quad \mathbf{X} = \mathbf{X} + \gamma \mathbf{v} t + (\gamma - 1) \frac{\mathbf{v}(\mathbf{v} \cdot \mathbf{r})}{v^2}, \qquad (10)$$

where $v^2 = \mathbf{v} \cdot \mathbf{v}$ and where $\mathbf{v}$ is the velocity between the two inertial reference frames, and with the Lorentz factor $\gamma$ defined as

$$\gamma := \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}, \qquad (11)$$

Contrary to popular wisdom, it is Eq. (9)—not the Lorentz transformations equation Eq. (10)—that is needed to derive the difference in time elapsed between two twins in Langevin's twin paradox.<fnref n="4" /> Let us assume a pair of observers that are at the same location at the spacetime events A and B, but one of the observer stays at rest while the other accelerates to leave and return to the other observer (see Fig. 1).

In terms of the reference frame of the observer at rest {$\tau^r, \mathbf{x}$}, the accelerated observer's proper time $\tau^a$ between the two events A and B reads

$$\tau_{AB}^a = \int_{C_{AB}} d\tau^a = \int_{C_{AB}} \sqrt{1 - \frac{v^2}{c^2}} d\tau^r, \qquad (12)$$

according to Eq. (9), with $\mathbf{v} = d\mathbf{x}/d\tau^r$, and where $C_{AB}$ is the accelerated trajectory of the accelerated observer in the reference frame of the observer at rest. Given a trajectory $C_{AB}$, one can derive the proper time of the accelerated observer elapsed between the events A and B. It is worth noting that Eq. (12) would not maintain the same form if one attempts to derive the proper time elapsed for the rest observer from the coordinates of the accelerated observer. This is because the Minkowski metric [Eq. (9)] does not remain invariant for an accelerated reference frame {$\tau^a, \mathbf{X}$}. In the proper reference frame of an accelerated (but non-rotating) observer, the metric Eq. (9) would instead read (Ni and Zimmermann 1978)

$$c^2 d\tau^2 = \left(1 + \frac{\mathbf{a} \cdot \mathbf{X}}{c^2}\right)^2 c^2 (d\tau^a)^2 - dX^2 - dY^2 - dZ^2, \qquad (13)$$

<footnote n="4">⁴ Langevin's twin paradox presents a scenario in which one twin travels to space at a high speed and returns to find the other twin has aged more. This highlights the relativistic effect of time dilation, as described in the theory of special relativity. The paradox's essence is rooted in a perceived contradiction: both twins should ostensibly observe the other with a similar relative motion, which would therefore result in the same 'anomalous' aging effect for both when solely using Lorentz's transformations Eq. (10) in the derivation.</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1 Schematic illustration of the trajectories (world-lines) of two observers, one at rest and the other accelerated. The accelerated trajectory correspond to the curve $C_{AB}$</figcaption></figure>

where $a$ is the acceleration of the proper reference frame of an accelerated observer. This fact alone dispels any paradox, as the situations of the two observers are distinct both physically and mathematically. From Eq. (12), one can infer that the elapsed proper time of an observer who has been accelerated between two arbitrary spacetime events A and B is always smaller than the elapsed proper time of an observer at rest $\tau_{AB}^a < \tau_{AB}^r$. In other words, this means that for an inertial observer the following action

$$S = -mc^2 \int d\tau = -mc^2 \int \sqrt{1 - \frac{v^2}{c^2}} dt = \int L_{SR}(x^\alpha, v^\alpha) dt, \quad (14)$$

is maximal, where $L_{SR}$ is, therefore, the Lagrangian of motion of special relativity:

$$L_{SR} = -mc^2 \sqrt{1 - \frac{v^2}{c^2}}. \quad (15)$$
<figure><img src="image_4.png" /><figcaption>Fig. 1 Schematic illustration of the trajectories (world-lines) of two observers, one at rest and the other accelerated. The accelerated trajectory correspond to the curve $C_{AB}$</figcaption></figure>

where $a$ is the acceleration of the proper reference frame of an accelerated observer. This fact alone dispels any paradox, as the situations of the two observers are distinct both physically and mathematically. From Eq. (12), one can infer that the elapsed proper time of an observer who has been accelerated between two arbitrary spacetime events A and B is always smaller than the elapsed proper time of an observer at rest $\tau_{AB}^a < \tau_{AB}^r$. In other words, this means that for an inertial observer the following action

$$S = -mc^2 \int d\tau = -mc^2 \int \sqrt{1 - \frac{v^2}{c^2}} dt = \int L_{SR}(x^\alpha, v^\alpha) dt, \quad (14)$$

is maximal, where $L_{SR}$ is, therefore, the Lagrangian of motion of special relativity:

$$L_{SR} = -mc^2 \sqrt{1 - \frac{v^2}{c^2}}. \quad (15)$$
### 2.1.3 Free fall and proper time in general relativity

In general relativity, the free fall of a point mass $m$ aslo maximizes the proper time $\tau$, but which is now defined by

$$ \tau_{AB} = \int_{C_{AB}} \sqrt{-g_{\alpha\beta}dx^\alpha dx^\beta}, \qquad (16) $$

where $C_{AB}$ means that the integration is taken along a given path between the spacetime positions<fnref n="5" /> A and B of the observer.<fnref n="6" /> In cartesian coordinates for instance, one has $x^i = (x, y, z)$. $g_{\alpha\beta}$ is the spacetime metric, which depends on the position, and which represents the curvature of spacetime. As we will see below, the metric is solution of the Einstein-Hilbert equation of general relativity.

The definition of the proper time for an observer in Eq. (16) is true whether or not the motion of the observer is accelerated. It is important to keep in mind that unlike in Newton’s theory, a free falling observer in general relativity has an inertial motion —that is, it is not accelerated. In general relativity, inertial bodies follow geodesics of spacetime that correspond to free fall trajectories. This is forced upon the theory by the *equivalence principle*, which states that the inertial mass is indeed equivalent to the gravitational mass.

But since the free fall of an observer maximizes its proper time, it means that Eq. (16) has to be an extremum in the case of a free fall motion. It means that the path $C_{AB}$ between the events A and B must be such that it extremizes Eq. (16). As a consequence, it must follow the *Least Action Principle*<fnref n="7" /> of Lagrangian mechanics that demandes the extremization of an action defined upon a Lagrangian. Indeed, from Eq. (16), one can define the following action

$$ S = mc^2 \tau = \int L(x^\alpha, v^\alpha) dt, \qquad (17) $$

where $v^\alpha = c \frac{dx^\alpha}{dt} = \frac{dx^\alpha}{dt}$ is a velocity expressed in terms of the coordinates $\{x^\alpha\}$ and where the Lagrangian reads

$$ L = mc^2 \sqrt{-g_{\alpha\beta}(x^\sigma)v^\alpha v^\beta}. \qquad (18) $$

The *Least Action Principle* demands that $\delta S = 0$. Given a specific spacetime metric $g_{\alpha\beta}$, one can therefore compute the equations of motion by applying the Euler-Lagrange equation on Eq. (18). The equation can also be applied on the formal definition of equation (18) and one then gets the general form—that is, for all spacetime metrics—of the equations of motion (for massive point particles) that reads

<footnote n="5">⁵ Also named “events”.</footnote>

<footnote n="6">⁶ Let us note that we use the mostly plus convention for the signature of the metric (– + + +), and that we use the Einstein summation convention, such that $X_\sigma Y^\sigma := \sum_{\sigma=0}^3 X_\sigma Y^\sigma$. $x^\alpha = (ct, x^i) = (ct, \vec{x}) = (ct, \mathbf{x})$, where $i = 1, 2, 3$.</footnote>

<footnote n="7">⁷ The Least Action Principle might be more appropriately termed the Extremum Action Principle, since it primarily requires that the variation of the action remains null: $\delta S = 0$. This implies that the action is at an extremum.</footnote>
### 2.1.3 Free fall and proper time in general relativity

In general relativity, the free fall of a point mass $m$ aslo maximizes the proper time $\tau$, but which is now defined by

$$ \tau_{AB} = \int_{C_{AB}} \sqrt{-g_{\alpha\beta}dx^{\alpha}dx^{\beta}}, \qquad (16) $$

where $C_{AB}$ means that the integration is taken along a given path between the spacetime positions<fnref n="5" /> A and B of the observer.<fnref n="6" /> In cartesian coordinates for instance, one has $x^i = (x, y, z)$. $g_{\alpha\beta}$ is the spacetime metric, which depends on the position, and which represents the curvature of spacetime. As we will see below, the metric is solution of the Einstein-Hilbert equation of general relativity.

The definition of the proper time for an observer in Eq. (16) is true whether or not the motion of the observer is accelerated. It is important to keep in mind that unlike in Newton’s theory, a free falling observer in general relativity has an inertial motion —that is, it is not accelerated. In general relativity, inertial bodies follow geodesics of spacetime that correspond to free fall trajectories. This is forced upon the theory by the *equivalence principle*, which states that the inertial mass is indeed equivalent to the gravitational mass.

But since the free fall of an observer maximizes its proper time, it means that Eq. (16) has to be an extremum in the case of a free fall motion. It means that the path $C_{AB}$ between the events A and B must be such that it extremizes Eq. (16). As a consequence, it must follow the *Least Action Principle*<fnref n="7" /> of Lagrangian mechanics that demandes the extremization of an action defined upon a Lagrangian. Indeed, from Eq. (16), one can define the following action

$$ S = mc^2 \tau = \int L(x^\alpha, v^\alpha) dt, \qquad (17) $$

where $v^\alpha = c \frac{dx^\alpha}{dx^0} = \frac{dx^\alpha}{dt}$ is a velocity expressed in terms of the coordinates $\{x^\alpha\}$ and where the Lagrangian reads

$$ L = mc^2 \sqrt{-g_{\alpha\beta}(x^\sigma)v^\alpha v^\beta}. \qquad (18) $$

The *Least Action Principle* demands that $\delta S = 0$. Given a specific spacetime metric $g_{\alpha\beta}$, one can therefore compute the equations of motion by applying the Euler-Lagrange equation on Eq. (18). The equation can also be applied on the formal definition of equation (18) and one then gets the general form—that is, for all spacetime metrics—of the equations of motion (for massive point particles) that reads

<footnote n="5">⁵ Also named “events”.</footnote>

<footnote n="6">⁶ Let us note that we use the mostly plus convention for the signature of the metric (– + + +), and that we use the Einstein summation convention, such that $X_\sigma Y^\sigma := \sum_{\sigma=0}^3 X_\sigma Y^\sigma$. $x^\alpha = (ct, x^i) = (ct, \vec{x}) = (ct, \mathbf{x})$, where $i = 1, 2, 3$.</footnote>

<footnote n="7">⁷ The Least Action Principle might be more appropriately termed the Extremum Action Principle, since it primarily requires that the variation of the action remains null: $\delta S = 0$. This implies that the action is at an extremum.</footnote>
$$ \frac{d^2 x^\alpha}{dt^2} = \left( \Gamma^\alpha_{\sigma\epsilon} - \Gamma^0_{\sigma\epsilon} \frac{v^\alpha}{c} \right) v^\sigma v^\epsilon, \qquad (19) $$

where

$$ \Gamma^{\gamma}_{\alpha\beta} := \frac{1}{2} g^{\gamma\sigma} (\partial_{\alpha} g_{\sigma\beta} - \partial_{\beta} g_{\alpha\sigma} - \partial_{\sigma} g_{\alpha\beta}), \qquad (20) $$

is named the Christoffel connection with $g^{\gamma\sigma}$, the contravariant metric—which is such that $g_{\alpha\sigma}g^{\sigma\beta} = \delta_{\alpha\beta}$, where $\delta$ is the Kronecker symbol. Besides, an accelerated trajectory would read

$$ \frac{d^2 x^\alpha}{dt^2} - \left( \Gamma^\alpha_{\sigma\epsilon} - \Gamma^0_{\sigma\epsilon} \frac{v^\alpha}{c} \right) v^\sigma v^\epsilon = F^\alpha, \qquad (21) $$

where $F^\alpha$ is the force responsible for the acceleration.

### 2.1.4 Einstein equation and the Newtonian approximation

The spacetime metric is curved by the energy of matter according to the Einstein equation that reads

$$ R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R = \frac{8\pi G}{c^4} T_{\alpha\beta}, \qquad (22) $$

where $R_{\alpha\beta}$ is the Ricci tensor (see Sect. 2.1.5) defined as

$$ R_{\alpha\beta} = \partial_{\mu}\Gamma^{\mu}_{\alpha\beta} - \partial_{\beta}\Gamma^{\mu}_{\alpha\mu} + \Gamma^{\mu}_{\sigma\mu}\Gamma^{\sigma}_{\alpha\beta} - \Gamma^{\mu}_{\sigma\beta}\Gamma^{\sigma}_{\alpha\mu}, \qquad (23) $$

and

$$ R = g^{\sigma\epsilon} R_{\sigma\epsilon} \qquad (24) $$

is the Ricci scalar, and $T_{\alpha\beta}$ is the stress-energy tensor of the material fields—which we will define in Sect. 2.1.6. The stress-energy tensor represents the energy content of the matter fields that generate the curvature of spacetime. Otherwise, we will see below that $G$ corresponds to the constant of Newton, also known as the gravitational constant.

#### The Newtonian approximation

In the Newtonian formalism given in Sect. 2.1.1, we assume that the speed of the considered bodies is small with respect to the speed of light $v^2/c^2 \ll 1$. Let us add the constant $-mc^2$ to this Lagrangian, which does not impact the equation of motion, but which is such that one reproduces the Lagrangian of motion of special relativity equation Eq. (15) at leading order when $U = 0$, that is
$$ \frac{d^2 x^\alpha}{dt^2} = \left( \Gamma^\alpha_{\sigma\epsilon} - \Gamma^0_{\sigma\epsilon} \frac{v^\alpha}{c} \right) v^\sigma v^\epsilon, \qquad (19) $$

where

$$ \Gamma^{\gamma}_{\alpha\beta} := \frac{1}{2} g^{\gamma\sigma} (\partial_{\alpha} g_{\sigma\beta} - \partial_{\beta} g_{\alpha\sigma} - \partial_{\sigma} g_{\alpha\beta}), \qquad (20) $$

is named the Christoffel connection with $g^{\gamma\sigma}$, the contravariant metric—which is such that $g_{\alpha\sigma}g^{\sigma\beta} = \delta_{\alpha\beta}$, where $\delta$ is the Kronecker symbol. Besides, an accelerated trajectory would read

$$ \frac{d^2 x^\alpha}{dt^2} - \left( \Gamma^\alpha_{\sigma\epsilon} - \Gamma^0_{\sigma\epsilon} \frac{v^\alpha}{c} \right) v^\sigma v^\epsilon = F^\alpha, \qquad (21) $$

where $F^\alpha$ is the force responsible for the acceleration.

### 2.1.4 Einstein equation and the Newtonian approximation

The spacetime metric is curved by the energy of matter according to the Einstein equation that reads

$$ R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R = \frac{8\pi G}{c^4} T_{\alpha\beta}, \qquad (22) $$

where $R_{\alpha\beta}$ is the Ricci tensor (see Sect. 2.1.5) defined as

$$ R_{\alpha\beta} = \partial_{\mu}\Gamma^{\mu}_{\alpha\beta} - \partial_{\beta}\Gamma^{\mu}_{\alpha\mu} + \Gamma^{\mu}_{\sigma\mu}\Gamma^{\sigma}_{\alpha\beta} - \Gamma^{\mu}_{\sigma\beta}\Gamma^{\sigma}_{\alpha\mu}, \qquad (23) $$

and

$$ R = g^{\sigma\epsilon} R_{\sigma\epsilon} \qquad (24) $$

is the Ricci scalar, and $T_{\alpha\beta}$ is the stress-energy tensor of the material fields—which we will define in Sect. 2.1.6. The stress-energy tensor represents the energy content of the matter fields that generate the curvature of spacetime. Otherwise, we will see below that $G$ corresponds to the constant of Newton, also known as the gravitational constant.

#### The Newtonian approximation

In the Newtonian formalism given in Sect. 2.1.1, we assume that the speed of the considered bodies is small with respect to the speed of light $v^2/c^2 \ll 1$. Let us add the constant $-mc^2$ to this Lagrangian, which does not impact the equation of motion, but which is such that one reproduces the Lagrangian of motion of special relativity equation Eq. (15) at leading order when $U = 0$, that is
$$L_{SR} = -mc^2 \sqrt{1 - \frac{v^2}{c^2}} = -mc^2 \left(1 - \frac{v^2}{2c^2}\right) + \mathcal{O}\left(\frac{v^4}{c^4}\right). \quad (25)$$

Now, the Lagrangian of motion for the theory of Newton reads

$$L_N = -mc^2 \left( 1 - \frac{v^2}{2c^2} - \frac{U}{c^2} \right). \quad (26)$$

Let us stress that for an orbit, one has $v^2/c^2 \sim U/c^2 \ll 1$ in the theory of Newton. This approximation ought to remain valid at leading order in general relativity in the weak field limit. As a consequence, for now on, we will use the notation $\mathcal{O}(c^{-2n})$ to refer to both $\mathcal{O}(v^{2n}/c^{2n})$ and $\mathcal{O}(U^n/c^{2n})$. Since the theory of Newton is already accurate in the solar system, we expect the theory of general relativity to reproduce the trajectories of the Newtonian theory at leading order. Therefore, one expects that there exists a spacetime metric that reproduces the Lagrangian equation Eq. (26) at leading order. One can check that injecting the following metric in Eq. (18)

$$-c^2 d\tau^2 = ds^2 = -\left(1 - \frac{2U}{c^2}\right)c^2 dt^2 + dx^2 + dy^2 + dz^2, \quad (27)$$

where $ds$ is the spacetime line element defined as $ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta$, gives back the Lagrangian of motion of the theory of Newton equation Eq. (26), up to corrections of order $\mathcal{O}(c^{-4})$. Therefore, one expects the metric in Eq. (27) to be solution of general relativity at leading order for the kind of weakly gravitating sources that we have in the solar system.<fnref n="8" /> It can be indeed verified that by injecting the metric Eq. (27) in the Einstein-Hilbert equation Eq. (23), the resulting differential equation reads

$$\Delta U = -4\pi G\rho_m + \mathcal{O}(c^{-2}), \quad (28)$$

where $\rho_m$ is the mass density of the gravitational sources defined as $\rho_m = T^{00}/c^2 + \mathcal{O}(c^{-2})$. It corresponds to Newton's universal law of gravitation as long as $G$ is indeed the gravitational constant. One has therefore identified $G$ in Eq. (22) to Newton's constant from the weak-field and slowly moving limit of the theory that leads to the same field equation as Newton's theory at leading order.

Therefore, general relativity produces the same trajectories at leading order as Newton's theory. This is called the *Newtonian approximation* of general relativity.

However, let us stress that even at this level of approximation, the two theories differ drastically—in a way that can be tested at the experimental level already. Indeed, in general relativity, the variation of an observer proper time $d\tau$ [Eq. (27)] with respect to the proper time of another observer depends explicitly on their different positions in a gravitational potential $U$. This means that two observers at different locations in the gravitational potential will not agree on the evolution of

<footnote n="8">⁸ As opposed to a neutron star or a black-hole.</footnote>
$$L_{SR} = -mc^2 \sqrt{1 - \frac{v^2}{c^2}} = -mc^2 \left( 1 - \frac{v^2}{2c^2} \right) + \mathcal{O}\left( \frac{v^4}{c^4} \right). \quad (25)$$

Now, the Lagrangian of motion for the theory of Newton reads

$$L_N = -mc^2 \left( 1 - \frac{v^2}{2c^2} - \frac{U}{c^2} \right). \quad (26)$$

Let us stress that for an orbit, one has $v^2/c^2 \sim U/c^2 \ll 1$ in the theory of Newton. This approximation ought to remain valid at leading order in general relativity in the weak field limit. As a consequence, for now on, we will use the notation $\mathcal{O}(c^{-2n})$ to refer to both $\mathcal{O}(v^{2n}/c^{2n})$ and $\mathcal{O}(U^n/c^{2n})$. Since the theory of Newton is already accurate in the solar system, we expect the theory of general relativity to reproduce the trajectories of the Newtonian theory at leading order. Therefore, one expects that there exists a spacetime metric that reproduces the Lagrangian equation Eq. (26) at leading order. One can check that injecting the following metric in Eq. (18)

$$-c^2 d\tau^2 = ds^2 = -\left(1 - \frac{2U}{c^2}\right)c^2 dt^2 + dx^2 + dy^2 + dz^2, \quad (27)$$

where $ds$ is the spacetime line element defined as $ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta$, gives back the Lagrangian of motion of the theory of Newton equation Eq. (26), up to corrections of order $\mathcal{O}(c^{-4})$. Therefore, one expects the metric in Eq. (27) to be solution of general relativity at leading order for the kind of weakly gravitating sources that we have in the solar system.<fnref n="8" /> It can be indeed verified that by injecting the metric Eq. (27) in the Einstein-Hilbert equation Eq. (23), the resulting differential equation reads

$$\Delta U = -4\pi G\rho_m + \mathcal{O}(c^{-2}), \quad (28)$$

where $\rho_m$ is the mass density of the gravitational sources defined as $\rho_m = T^{00}/c^2 + \mathcal{O}(c^{-2})$. It corresponds to Newton's universal law of gravitation as long as $G$ is indeed the gravitational constant. One has therefore identified $G$ in Eq. (22) to Newton's constant from the weak-field and slowly moving limit of the theory that leads to the same field equation as Newton's theory at leading order.

Therefore, general relativity produces the same trajectories at leading order as Newton's theory. This is called the Newtonian approximation of general relativity.

However, let us stress that even at this level of approximation, the two theories differ drastically—in a way that can be tested at the experimental level already. Indeed, in general relativity, the variation of an observer proper time $d\tau$ [Eq. (27)] with respect to the proper time of another observer depends explicitly on their different positions in a gravitational potential $U$. This means that two observers at different locations in the gravitational potential will not agree on the evolution of

<footnote n="8">⁸ As opposed to a neutron star or a black-hole.</footnote>
time. This effect, although minute, can be tested if one has accurate enough clocks. In other words, had we developed atomic clocks with sufficient precision prior to our ability to observe the motions of celestial bodies in the solar system, we could have confirmed the superiority of general relativity over Newton's theory. This could have been achieved by comparing the frequencies of two clocks located at different positions relative to the geoid. However, due to our atmosphere's transparency, astronomers identified the limitations of Newton's theory through the anomalous advance of Mercury's perihelion—see Table 1 for the numerical value of this advance—before quantum physicists could create sufficiently accurate atomic clocks. We will further discuss the advance of the perihelion of Mercury in Sect. 2.1.8.

Finally, one can see in Eq. (27) that space it flat<fnref n="9" /> at leading order, such that free fall trajectories in the solar system are essentially the consequence of the curvature of the temporal dimension, not the spatial one.

### Gauge invariance of Newton's equation and the definition of coordinate times

Newton's equation Eq. (28) is notably invariant under the following change of the potential:

$$U \rightarrow U + a_k(t)x^k + b(t), \qquad (29)$$

where $a_k(t)$ and $b(t)$ are arbitrary three-vector and scalar that depend on time.<fnref n="10" /> One often says that the Eq. (28) is invariant under the change of gauge defined by Eq. (29).

The change of gauge in Eq. (29) corresponds to a change of the coordinate time being used to describe a motion. This is a major difference with respect to Newton's theory where there is a unique time. A coordinate time is simply a mathematical construction of a time that can be related to the proper time of an observer, but not necessarily. For instance, it can simply be defined according to purely mathematical criteria, such as the simplicity of the field equations.

In order to illustrate the dependence of the Newtonian potential to the coordinate time being used, let us define a coordinate time that would correspond to the proper time of an imaginary observer that is so remote from the barycenter of the solar system that it can be at rest with respect to the solar system because the potential of the solar system and its gradient at their location is negligible. Assuming that this imaginary observer is indeed at rest with respect to the solar system, this means that the coordinate time $t$ corresponds to the (fictional) proper time $\tau$ such that $dt = d\tau$ at the location of this (fictional) observer. The Newtonian potential in this gauge—or, equivalently, with this coordinate time—reads

$$U(x^i, t) = \sum_A \frac{Gm_A}{|x^i - x_A^i(t)|}, \qquad (30)$$

where $m_A$ and $x_A^i(t)$ are the masses and the positions of the bodies $A$. Alternatively, one could define a coordinate time that corresponds to a (fictional) observer that

<footnote n="9">⁹ Space is Euclidean in Eq. (27), such that it satisfies the Pythagorean theorem.</footnote>

<footnote n="10">¹⁰ Adding any term $\delta U$ that is such that $\Delta\delta U = 0$ leaves Newton's equation invariant.</footnote>
time. This effect, although minute, can be tested if one has accurate enough clocks. In other words, had we developed atomic clocks with sufficient precision prior to our ability to observe the motions of celestial bodies in the solar system, we could have confirmed the superiority of general relativity over Newton's theory. This could have been achieved by comparing the frequencies of two clocks located at different positions relative to the geoid. However, due to our atmosphere's transparency, astronomers identified the limitations of Newton's theory through the anomalous advance of Mercury's perihelion—see Table 1 for the numerical value of this advance—before quantum physicists could create sufficiently accurate atomic clocks. We will further discuss the advance of the perihelion of Mercury in Sect. 2.1.8.

Finally, one can see in Eq. (27) that space it flat<fnref n="9" /> at leading order, such that free fall trajectories in the solar system are essentially the consequence of the curvature of the temporal dimension, not the spatial one.

### Gauge invariance of Newton's equation and the definition of coordinate times

Newton's equation Eq. (28) is notably invariant under the following change of the potential:

$$U \rightarrow U + a_k(t)x^k + b(t), \qquad (29)$$

where $a_k(t)$ and $b(t)$ are arbitrary three-vector and scalar that depend on time.<fnref n="10" /> One often says that the Eq. (28) is invariant under the change of gauge defined by Eq. (29).

The change of gauge in Eq. (29) corresponds to a change of the coordinate time being used to describe a motion. This is a major difference with respect to Newton's theory where there is a unique time. A coordinate time is simply a mathematical construction of a time that can be related to the proper time of an observer, but not necessarily. For instance, it can simply be defined according to purely mathematical criteria, such as the simplicity of the field equations.

In order to illustrate the dependence of the Newtonian potential to the coordinate time being used, let us define a coordinate time that would correspond to the proper time of an imaginary observer that is so remote from the barycenter of the solar system that it can be at rest with respect to the solar system because the potential of the solar system and its gradient at their location is negligible. Assuming that this imaginary observer is indeed at rest with respect to the solar system, this means that the coordinate time $t$ corresponds to the (fictional) proper time $\tau$ such that $dt = d\tau$ at the location of this (fictional) observer. The Newtonian potential in this gauge—or, equivalently, with this coordinate time—reads

$$U(x^i, t) = \sum_A \frac{Gm_A}{|x^i - x_A^i(t)|}, \qquad (30)$$

where $m_A$ and $x_A^i(t)$ are the masses and the positions of the bodies $A$. Alternatively, one could define a coordinate time that corresponds to a (fictional) observer that

<footnote n="9">⁹ Space is Euclidean in Eq. (27), such that it satisfies the Pythagorean theorem.</footnote>

<footnote n="10">¹⁰ Adding any term $\delta U$ that is such that $\Delta\delta U = 0$ leaves Newton's equation invariant.</footnote>
would be at rest at the center of the coordinate system $x^i = 0$. In this gauge, the Newtonian potential would instead read

$$U(x^i, t) = \sum_A Gm_A \left[ \frac{1}{|x^i - x_A^i(t)|} - \frac{1}{|x_A^i(t)|} - \frac{x_A^i x_A^j(t)}{|x_A^i(t)|^3} \right], \quad (31)$$

because having $dt = d\tau$ in Eq. (27) at the center imposes $U(x^i = 0) = 0$, and being at rest imposes $\vec{\nabla}U|_{x^i=0} = 0$. The transformation from the potential expressed in the coordinate time of a remote observer to the potential expressed in the coordinate time of an observer at the center of the coordinate system takes the form of Eq. (29) with

$$a^k(t) = - \sum_A Gm_A \frac{x_A^k(t)}{|x_A^j(t)|^3} \quad \text{and} \quad b(t) = - \sum_A \frac{Gm_A}{|x_A^j(t)|} \quad (32)$$

This notably means that the leading order of the equation of general relativity is invariant under specific changes of time coordinate. As we will see in Sect. 2.1.5, this is a leftover of the invariance of general relativity through a change of coordinate system. It is really important to notice that while the field equation Eq. (28) is invariant under some changes of the coordinate time, the actual solution of the potential is not, as one can check from Eqs. (30) and (31).

In the coordinate time defined as the proper time of the fictional remote observer $t = \tau_r$, the proper time of fictional observer at the center of the coordinate system $\tau_o$ reads

$$\tau_o = t - \frac{1}{2c^2} \int \sum_A \frac{Gm_A}{|x_A^j(t)|} dt, \quad (33)$$

provided that they are indeed at rest with one another. See Sect. 3.2.2 for a discussion on the coordinate times that are used by the community, following the IAU recommendations (Soffel 2003).

### Dependence of a coordinate time to the trajectories of the gravitational bodies

Equation (33) teaches us something very important. Depending on its definition, a coordinate time can depend explicitly on the trajectories of the gravitational bodies $x_A(t)$. As a consequence, in order to define such a coordinate time in practice, it is necessary to estimate the positions of the celestial bodies over time with enough accuracy.

But why would one want to define such a coordinate time if it means that one must already have an accurate knowledge of the trajectories of the gravitational bodies in order to define it? Simply because one has to define such a coordinate time in order to compare different observations made at various locations by several observers. The proper time of each observer is indeed different and the differences between the various proper times depend on the observers' relative trajectories in the gravitational potential. Therefore, in order to compare the various observations, one first has to define a coordinate time that will be used to transform the proper time of every
would be at rest at the center of the coordinate system $x^i = 0$. In this gauge, the Newtonian potential would instead read

$$U(x^i, t) = \sum_A Gm_A \left[ \frac{1}{|x^i - x_A^i(t)|} - \frac{1}{|x_A^i(t)|} - \frac{x_A^i x_A^j(t)}{|x_A^i(t)|^3} \right], \quad (31)$$

because having $dt = d\tau$ in Eq. (27) at the center imposes $U(x^i = 0) = 0$, and being at rest imposes $\nabla U|_{x^i=0} = 0$. The transformation from the potential expressed in the coordinate time of a remote observer to the potential expressed in the coordinate time of an observer at the center of the coordinate system takes the form of Eq. (29) with

$$a^k(t) = - \sum_A Gm_A \frac{x_A^k(t)}{|x_A^j(t)|^3} \quad \text{and} \quad b(t) = - \sum_A \frac{Gm_A}{|x_A^j(t)|} \quad (32)$$

This notably means that the leading order of the equation of general relativity is invariant under specific changes of time coordinate. As we will see in Sect. 2.1.5, this is a leftover of the invariance of general relativity through a change of coordinate system. It is really important to notice that while the field equation Eq. (28) is invariant under some changes of the coordinate time, the actual solution of the potential is not, as one can check from Eqs. (30) and (31).

In the coordinate time defined as the proper time of the fictional remote observer $t = \tau_r$, the proper time of fictional observer at the center of the coordinate system $\tau_o$ reads

$$\tau_o = t - \frac{1}{2c^2} \int \sum_A \frac{Gm_A}{|x_A^j(t)|} dt, \quad (33)$$

provided that they are indeed at rest with one another. See Sect. 3.2.2 for a discussion on the coordinate times that are used by the community, following the IAU recommendations (Soffel 2003).

### Dependence of a coordinate time to the trajectories of the gravitational bodies

Equation (33) teaches us something very important. Depending on its definition, a coordinate time can depend explicitly on the trajectories of the gravitational bodies $x_A(t)$. As a consequence, in order to define such a coordinate time in practice, it is necessary to estimate the positions of the celestial bodies over time with enough accuracy.

But why would one want to define such a coordinate time if it means that one must already have an accurate knowledge of the trajectories of the gravitational bodies in order to define it? Simply because one has to define such a coordinate time in order to compare different observations made at various locations by several observers. The proper time of each observer is indeed different and the differences between the various proper times depend on the observers' relative trajectories in the gravitational potential. Therefore, in order to compare the various observations, one first has to define a coordinate time that will be used to transform the proper time of every
observer to this coordinate time, such that after the transformation, all the
observations are expressed in a common time. We will see in Sect. 2.1.7 that at
the post-Newtonian level—or, the next-to-leading order—similar considerations
must be taken into account for the definition of space.

Any definition of a coordinate time could be used in theory. However, in practice,
a coordinate time is usually defined as being the proper time of a fictional observer at
some convenient location in the solar system, such as at its barycenter or at the geoid
of the Earth. Such coordinate times are defined by the International Astronomical
Union, as we will further discuss in Sect. 3.2.2.

But since one needs to know the trajectories of the gravitational bodies in the solar
system in order to construct this coordinate time, it means that one needs to use
planetary ephemerides to construct such a time. Therefore, planetary ephemerides
actually deliver the coordinate times defined by the International Astronomical Union
to the community, as discussed more in details in Sect. 3.2.2.

Rotating reference frames

The Newtonian approximation of the spacetime metric in general relativity assumes the form of Eq. (27) only within a specific category of coordinate systems, typically referred to as inertial frames. Notably, the coordinate system must be kinematically non-rotating relative to distant celestial objects such as quasars. This suggests that the rotation (or absence thereof) of a local reference frame must be defined in relation to exceedingly distant objects. In support of his theory, Newton posited this coincidence as evidence of an absolute space, serving as the stage for dynamic events and providing a reference for defining rotation. In contrast, figures such as Leibniz and later Mach, contended that this apparent coincidence reveals that inertia is relative, depending more on the universe matter content than on an absolute space. Heavily influenced by these perspectives, Einstein proposed the *principle of relativity of inertia*, later referred to as *Mach's principle* (Einstein 1918). This principle stipulates that spacetime is wholly determined by its matter content. Considering general relativity accommodates vacuum solutions, whether Einstein's theory satisfies the principle of relativity of inertia remains debated (Barbour and Pfister 1995). However, practically speaking, the solar system asymptotic metric must be integrated into the larger spacetime metric effectively generated by distant sources. This requirement accounts for why inertial reference frames are non-rotating relative to very remote sources in general relativity—considering they can be approximated as stationary due to their minimal relative angular velocity in the sky over certain timescales.

2.1.5 Invariance of the laws through a change of coordinate system

Tensors

The tensorial nature of the Einstein–Hilbert equation Eq. (22) enforces the *covariance principle*,<fnref n="11" /> which demands that the laws of Nature do not depend on the choice of the coordinate system. Indeed, tensorial equations—such as Eq. (22)—

<footnote n="11">¹¹ Or, *principle of relativity* (Einstein 1918).</footnote>
observer to this coordinate time, such that after the transformation, all the
observations are expressed in a common time. We will see in Sect. 2.1.7 that at
the post-Newtonian level—or, the next-to-leading order—similar considerations
must be taken into account for the definition of space.

Any definition of a coordinate time could be used in theory. However, in practice,
a coordinate time is usually defined as being the proper time of a fictional observer at
some convenient location in the solar system, such as at its barycenter or at the geoid
of the Earth. Such coordinate times are defined by the International Astronomical
Union, as we will further discuss in Sect. 3.2.2.

But since one needs to know the trajectories of the gravitational bodies in the solar
system in order to construct this coordinate time, it means that one needs to use
planetary ephemerides to construct such a time. Therefore, planetary ephemerides
actually deliver the coordinate times defined by the International Astronomical Union
to the community, as discussed more in details in Sect. 3.2.2.

Rotating reference frames

The Newtonian approximation of the spacetime metric in general relativity assumes the form of Eq. (27) only within a specific category of coordinate systems, typically referred to as inertial frames. Notably, the coordinate system must be kinematically non-rotating relative to distant celestial objects such as quasars. This suggests that the rotation (or absence thereof) of a local reference frame must be defined in relation to exceedingly distant objects. In support of his theory, Newton posited this coincidence as evidence of an absolute space, serving as the stage for dynamic events and providing a reference for defining rotation. In contrast, figures such as Leibniz and later Mach, contended that this apparent coincidence reveals that inertia is relative, depending more on the universe matter content than on an absolute space. Heavily influenced by these perspectives, Einstein proposed the *principle of relativity of inertia*, later referred to as *Mach's principle* (Einstein 1918). This principle stipulates that spacetime is wholly determined by its matter content. Considering general relativity accommodates vacuum solutions, whether Einstein's theory satisfies the principle of relativity of inertia remains debated (Barbour and Pfister 1995). However, practically speaking, the solar system asymptotic metric must be integrated into the larger spacetime metric effectively generated by distant sources. This requirement accounts for why inertial reference frames are non-rotating relative to very remote sources in general relativity—considering they can be approximated as stationary due to their minimal relative angular velocity in the sky over certain timescales.

**2.1.5 Invariance of the laws through a change of coordinate system**

Tensors

The tensorial nature of the Einstein–Hilbert equation Eq. (22) enforces the *covariance principle*,<fnref n="11" /> which demands that the laws of Nature do not depend on the choice of the coordinate system. Indeed, tensorial equations—such as Eq. (22)—

<footnote n="11">¹¹ Or, *principle of relativity* (Einstein 1918).</footnote>
are invariant under change of coordinates. A tensor is defined by the way it transforms under a change of coordinates. A tensor $W$, with $n$ contravariant indices and $m$ covariant indices transform as

$$W_{\beta_1 \dots \beta_m}^{\alpha_1 \dots \alpha_n} \{x^\sigma\} = \left( \prod_{\alpha_k} \prod_{\beta_k} \frac{\partial x^{\alpha_k}}{\partial X^{\mu_k}} \frac{\partial X^{\nu_k}}{\partial x^{\beta_k}} \right) W_{\nu_1 \dots \nu_m}^{\mu_1 \dots \mu_n} \{X^\sigma\}. \quad (34)$$

Tensorial equations are manifestly invariant under coordinate change, that is:

$$A_{\alpha_1 \alpha_2 \dots \alpha_n}^{\sigma_1 \sigma_2 \dots \sigma_p} \{x^\sigma\} = B_{\alpha_1 \alpha_2 \dots \alpha_n}^{\sigma_1 \sigma_2 \dots \sigma_p} \{x^\sigma\} \Leftrightarrow A_{\beta_1 \beta_2 \dots \beta_n}^{\omega_1 \omega_2 \dots \omega_p} \{X^\sigma\} = B_{\beta_1 \beta_2 \dots \beta_n}^{\omega_1 \omega_2 \dots \omega_p} \{X^\sigma\}. \quad (35)$$

Because the metric field is a tensor, one can notably verify with Eq. (34) that the line-element $ds$ defined from the metric as follows

$$ds^2 = -c^2 d\tau^2 = g_{\alpha\beta} \{x^\sigma\} dx^\alpha dx^\beta = g_{\alpha\beta} \{X^\sigma\} dX^\alpha dX^\beta \quad (36)$$

is invariant under coordinate change when integrated along a world-line. Indeed, the proper time of an observer cannot depend on any coordinate system as it is a measured quantity. Let us note however, that according to this definition, the Christoffel connection defined in Eq. (20) is not a tensor as it transforms as follow:

$$\Gamma^\alpha_{\mu\nu} \{x^\sigma\} = \frac{\partial x^\alpha}{\partial X^\beta} \frac{\partial X^\sigma}{\partial x^\mu} \frac{\partial X^\rho}{\partial x^\nu} \Gamma^\beta_{\sigma\rho} \{X^\sigma\} - \frac{\partial^2 x^\alpha}{\partial X^\sigma \partial X^\rho} \frac{\partial X^\sigma}{\partial x^\mu} \frac{\partial X^\rho}{\partial x^\nu}. \quad (37)$$

One can also check that the difference between two connections is a tensor. What the covariance principle means in particular is that the laws of physics are independent with respect to the observer, whether the observer is inertial or not. While this is very satisfying at the fundamental level, it also implies an intrinsic ambiguity about the coordinates that one can use, as they are all equivalent in principle.

### Observables

Certain coordinate systems can result in what are known as *spurious effects*. These are essentially false effects arising solely from the coordinate system and do not reflect reality. Determining whether an effect is genuine or spurious can be challenging unless all calculations are made with respect to actual observables. Parameters referred to as *observables* are independent of the coordinate system, and therefore, can correspond to quantities that an observer can directly measure. These include values such as the proper time elapsed for an observer between two events, or, as we will discuss shortly, the angle between two light cones at the observer's location. On the contrary, it is very important to keep in mind that positions, trajectories, or velocities, are not observables, but have to be reconstructed from observations after assuming a specific coordinate system—as well, as we will see, as a model for the dynamics of bodies and light.

### Observables are scalar quantitites

According to Eq. (34), the only type of tensors that do not depend on the coordinate system are scalars, which are such that
are invariant under change of coordinates. A tensor is defined by the way it transforms under a change of coordinates. A tensor W, with n contravariant indices and m covariant indices transform as

$$W_{\beta_1 \dots \beta_m}^{\alpha_1 \dots \alpha_n} \{x^\sigma\} = \left( \prod_{\alpha_k} \prod_{\beta_k} \frac{\partial x^{\alpha_k}}{\partial X^{\mu_k}} \frac{\partial X^{\nu_k}}{\partial x^{\beta_k}} \right) W_{\nu_1 \dots \nu_m}^{\mu_1 \dots \mu_n} \{X^\sigma\}. \quad (34)$$

Tensorial equations are manifestly invariant under coordinate change, that is:

$$A_{\alpha_1 \alpha_2 \dots \alpha_n}^{\sigma_1 \sigma_2 \dots \sigma_p} \{x^\sigma\} = B_{\alpha_1 \alpha_2 \dots \alpha_n}^{\sigma_1 \sigma_2 \dots \sigma_p} \{x^\sigma\} \Leftrightarrow A_{\beta_1 \beta_2 \dots \beta_n}^{\omega_1 \omega_2 \dots \omega_p} \{X^\sigma\} = B_{\beta_1 \beta_2 \dots \beta_n}^{\omega_1 \omega_2 \dots \omega_p} \{X^\sigma\}. \quad (35)$$

Because the metric field is a tensor, one can notably verify with Eq. (34) that the line-element ds defined from the metric as follows

$$ds^2 = -c^2 d\tau^2 = g_{\alpha\beta} \{x^\sigma\} dx^\alpha dx^\beta = g_{\alpha\beta} \{X^\sigma\} dX^\alpha dX^\beta \quad (36)$$

is invariant under coordinate change when integrated along a world-line. Indeed, the proper time of an observer cannot depend on any coordinate system as it is a measured quantity. Let us note however, that according to this definition, the Christoffel connection defined in Eq. (20) is not a tensor as it transforms as follow:

$$\Gamma^\alpha_{\mu\nu} \{x^\sigma\} = \frac{\partial x^\alpha}{\partial X^\beta} \frac{\partial X^\sigma}{\partial x^\mu} \frac{\partial X^\rho}{\partial x^\nu} \Gamma^\beta_{\sigma\rho} \{X^\sigma\} - \frac{\partial^2 x^\alpha}{\partial X^\sigma \partial X^\rho} \frac{\partial X^\sigma}{\partial x^\mu} \frac{\partial X^\rho}{\partial x^\nu}. \quad (37)$$

One can also check that the difference between two connections is a tensor. What the covariance principle means in particular is that the laws of physics are independent with respect to the observer, whether the observer is inertial or not. While this is very satisfying at the fundamental level, it also implies an intrinsic ambiguity about the coordinates that one can use, as they are all equivalent in principle.

### Observables

Certain coordinate systems can result in what are known as spurious effects. These are essentially false effects arising solely from the coordinate system and do not reflect reality. Determining whether an effect is genuine or spurious can be challenging unless all calculations are made with respect to actual observables. Parameters referred to as observables are independent of the coordinate system, and therefore, can correspond to quantities that an observer can directly measure. These include values such as the proper time elapsed for an observer between two events, or, as we will discuss shortly, the angle between two light cones at the observer's location. On the contrary, it is very important to keep in mind that positions, trajectories, or velocities, are not observables, but have to be reconstructed from observations after assuming a specific coordinate system—as well, as we will see, as a model for the dynamics of bodies and light.

### Observables are scalar quantitites

According to Eq. (34), the only type of tensors that do not depend on the coordinate system are scalars, which are such that
<figure><img src="image_4.png" /><figcaption>**Fig. 2** Schematic illustration of the angle (α) determined by an observer (O) at their location, between two null-geodesics—see Sect. 2.1.9—that link the observer to two sources (Q and S) of electromagnetic waves. Each null-geodesic represents a segment of the light-cone centered on an emitter</figcaption></figure>

$$S\{x^\sigma\} = S\{X^\sigma\}. \qquad (38)$$

Tensors, if not scalars, therefore cannot represent observable quantities.

*Positions in two dimensions: projection on the celestial sphere* From an observer's perspective, the only tangible element related to the spatial distribution of any distant object is the relative angular separation in the sky between the images of two objects. This measurement can be made without reference to any specific coordinate system. This angular separation corresponds to angles between the light cones connecting the distant objects to the observer at the observer's location, as shown in Fig. 2.

However, observers typically project at their location the vector of the light cones, which join the distant objects to the observer's location, onto a coordinate-dependent representation of their celestial spheres. This process is where coordinate systems become to be used. Importantly, it must be noted that the relative position (angle) of the same distant objects as viewed by two different observers can vary. This is because in general relativity, both time and space are relative to the observer and to its gravitational environment. Hence, there is no such thing as absolute positions. In simpler terms, the celestial spheres for different observers differ in general. Although this effect is minute for observers in the solar system, it nonetheless needs to be accounted for in modern astrometry.
<figure><img src="image_5.png" /><figcaption>**Fig. 2** Schematic illustration of the angle (α) determined by an observer (O) at their location, between two null-geodesics—see Sect. 2.1.9—that link the observer to two sources (Q and S) of electromagnetic waves. Each null-geodesic represents a segment of the light-cone centered on an emitter</figcaption></figure>

$$S\{x^\sigma\} = S\{X^\sigma\}. \qquad (38)$$

Tensors, if not scalars, therefore cannot represent observable quantities.

*Positions in two dimensions: projection on the celestial sphere* From an observer's perspective, the only tangible element related to the spatial distribution of any distant object is the relative angular separation in the sky between the images of two objects. This measurement can be made without reference to any specific coordinate system. This angular separation corresponds to angles between the light cones connecting the distant objects to the observer at the observer's location, as shown in Fig. 2.

However, observers typically project at their location the vector of the light cones, which join the distant objects to the observer's location, onto a coordinate-dependent representation of their celestial spheres. This process is where coordinate systems become to be used. Importantly, it must be noted that the relative position (angle) of the same distant objects as viewed by two different observers can vary. This is because in general relativity, both time and space are relative to the observer and to its gravitational environment. Hence, there is no such thing as absolute positions. In simpler terms, the celestial spheres for different observers differ in general. Although this effect is minute for observers in the solar system, it nonetheless needs to be accounted for in modern astrometry.
In practical terms, an object location on the celestial sphere is determined by observing the angular separation in the sky between the object and reference points like quasars, from the perspective of the observer. Indeed, due to their extremely remote location, the apparent movement of these reference points is negligible at the present level of accuracy for the astrometric measurements for these objects (see Sect. 3.2.3), and they can therefore indeed serve as static reference points (or astrometric candle light).

However, there is another significant effect to consider, which is also illustrated in Fig. 2: the propagation of light is not generally linear. This means that if an observer sees a remote object at a particular location on his celestial sphere, it doesn't imply that the object is actually at that position on the sphere. This effect is referred to as gravitational lensing (or deflection of light in the solar system) and is due to the fact that light propagates on null-geodesics, see Sect. 2.1.9—which are not straight lines in general. Although its impact is relatively weak for objects within the solar system, it highlights the inherent difficulty of attributing positions on the celestial sphere in general relativity.

Hence, assigning a two-dimensional position to an object in general relativity not only necessitates the definition of a coordinate system, but also requires a model of the gravitational field through which light has propagated. Consequently, any position in general relativity is model-dependent and is essentially a reconstruction from observations.

*The third dimension: range* There are various methods to infer the distance between a remote object and an observer. For s/c, the most accurate technique is typically the measure of the time delay between the emission of a signal in the direction of the spacecraft and the reception of the returned signal by the probe (ranging). Essentially, this involves measuring the round-trip time of an electromagnetic signal emitted at a given frequency between an observer and a spacecraft. Given that the signal propagates at a constant speed in vacuum, this propagation time can be converted into a distance. However, while the signal indeed travels at the speed of light in a vacuum locally, its propagation time measured by a given observer can be affected by the gravitational field localised between the source of emission and this observer, along the trajectory of the electromagnetic signal. As a result, even when a the signal follows a straight line between two points, the propagation time would not correspond to the Euclidean distance between these points divided by the speed of light. This effect is known as the *Shapiro delay*—See Sect. 2.1.9.

Therefore, determining distances through ranging techniques not only depends on the specific choice of the coordinate system considered but also depends on the knowledge one has on the gravitational potential in the solar system. Consequently, distances are both model and theory dependent. Indeed, the potential in the solar system is reconstructed by solving the field equations of a given theory—e.g. Equation (27)—after assuming a specific model for the solar system that “sources” the field equations—e.g. the right hand side of Eq. (27).
In practical terms, an object location on the celestial sphere is determined by observing the angular separation in the sky between the object and reference points like quasars, from the perspective of the observer. Indeed, due to their extremely remote location, the apparent movement of these reference points is negligible at the present level of accuracy for the astrometric measurements for these objects (see Sect. 3.2.3), and they can therefore indeed serve as static reference points (or astrometric candle light).

However, there is another significant effect to consider, which is also illustrated in Fig. 2: the propagation of light is not generally linear. This means that if an observer sees a remote object at a particular location on his celestial sphere, it doesn't imply that the object is actually at that position on the sphere. This effect is referred to as gravitational lensing (or deflection of light in the solar system) and is due to the fact that light propagates on null-geodesics, see Sect. 2.1.9—which are not straight lines in general. Although its impact is relatively weak for objects within the solar system, it highlights the inherent difficulty of attributing positions on the celestial sphere in general relativity.

Hence, assigning a two-dimensional position to an object in general relativity not only necessitates the definition of a coordinate system, but also requires a model of the gravitational field through which light has propagated. Consequently, any position in general relativity is model-dependent and is essentially a reconstruction from observations.

*The third dimension: range* There are various methods to infer the distance between a remote object and an observer. For s/c, the most accurate technique is typically the measure of the time delay between the emission of a signal in the direction of the spacecraft and the reception of the returned signal by the probe (ranging). Essentially, this involves measuring the round-trip time of an electromagnetic signal emitted at a given frequency between an observer and a spacecraft. Given that the signal propagates at a constant speed in vacuum, this propagation time can be converted into a distance. However, while the signal indeed travels at the speed of light in a vacuum locally, its propagation time measured by a given observer can be affected by the gravitational field localised between the source of emission and this observer, along the trajectory of the electromagnetic signal. As a result, even when a the signal follows a straight line between two points, the propagation time would not correspond to the Euclidean distance between these points divided by the speed of light. This effect is known as the *Shapiro delay*—See Sect. 2.1.9.

Therefore, determining distances through ranging techniques not only depends on the specific choice of the coordinate system considered but also depends on the knowledge one has on the gravitational potential in the solar system. Consequently, distances are both model and theory dependent. Indeed, the potential in the solar system is reconstructed by solving the field equations of a given theory—e.g. Equation (27)—after assuming a specific model for the solar system that “sources” the field equations—e.g. the right hand side of Eq. (27).
Let us stress in particular that a modification of the field equations of general relativity therefore implies the need for a new analysis of the measures, leading to a new determination of the distances.

*The need of conventions for coordinate systems* Because space and time are relative, for various observers to compare their respective observations and agree on the positions of bodies, they must define a set of coordinates that allows to convert their respective observations in terms of positions in space and time. This is the role of the IAU recommendations that we will discuss in Sect. 3.2.

### 2.1.6 The action of general relativity

Just as the equations of motion can be derived from the Euler–Lagrange equation, the field equation of general relativity can be derived from a Lagrangian density that is defined based on the metric field and the various material fields. The key difference lies in the type of dynamical variables used in the Lagrangian: instead of using position and velocity as dynamical variables, one considers the fields and their derivatives as dynamical variables in the case of a Lagrangian density.<fnref n="12" /> One can define the action of general relativity as follows

$$S_{GR} = \int \sqrt{-g} d^4 x \left[ \frac{R}{2\kappa} + \mathcal{L}_m \right], \quad (39)$$

where $g$ is the metric determinant, $\mathcal{L}_m$ the Lagrangian density of matter fields, and $\kappa = 8\pi G/c^4$ the coupling constant between matter and curvature. In order to derive the Euler–Lagragian equation, $R$ is treated as a functional constructed upon the metric field and its derivative—see Eq. (24). $\mathcal{L}_m$ is then a functional constructed upon the matter fields and their derivative. For instance, for an electromagnetic field one has

$$\mathcal{L}_m = \frac{1}{4\mu_0} g^{\alpha\sigma} g^{\beta\epsilon} (\partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}) (\partial_{\sigma} A_{\epsilon} - \partial_{\epsilon} A_{\sigma}) \quad (40)$$

where $\mu_0$ is the magnetic permeability of vacuum, and $A^\alpha$ is the electromagnetic four-vector. Let us note that one has $A^\alpha = (\phi/c, A)$ where $\phi$ is the scalar potential and $A$ the vector potential of classical electromagnetism, form which one can compute the electric and magnetic fields (Jackson 1998).

Defining a theory from its action rather than from its equations is convenient because it allows one to be sure that the theory possesses conservation laws for the considered theory, which derive from Noether’s theorem that states that every symmetry of a Lagrangian implies the existence of a conservation law (Noether 1918; Wald 1984). Therefore, most modern theories are defined from an action, although there are some exceptions—such as MOND, see Sect. 4.7.

<footnote n="12">¹² Note that mathematicians typically refer to $\sqrt{-g}\mathcal{L}$ as a density because $\mathcal{L}$ is a scalar and a scalar multiplied by the square root of the determinant corresponds to a mathematical object referred to as a density of weight $-1$; however, physicists typically refer to $\mathcal{L}$ as a density, as it carries the dimension of an energy density and is frequently associated with the definitions of kinetic and potential energy densities.</footnote>
Let us stress in particular that a modification of the field equations of general relativity therefore implies the need for a new analysis of the measures, leading to a new determination of the distances.

*The need of conventions for coordinate systems* Because space and time are relative, for various observers to compare their respective observations and agree on the positions of bodies, they must define a set of coordinates that allows to convert their respective observations in terms of positions in space and time. This is the role of the IAU recommendations that we will discuss in Sect. 3.2.

### 2.1.6 The action of general relativity

Just as the equations of motion can be derived from the Euler–Lagrange equation, the field equation of general relativity can be derived from a Lagrangian density that is defined based on the metric field and the various material fields. The key difference lies in the type of dynamical variables used in the Lagrangian: instead of using position and velocity as dynamical variables, one considers the fields and their derivatives as dynamical variables in the case of a Lagrangian density.<fnref n="12" /> One can define the action of general relativity as follows

$$S_{GR} = \int \sqrt{-g} d^4 x \left[ \frac{R}{2\kappa} + \mathcal{L}_m \right], \quad (39)$$

where $g$ is the metric determinant, $\mathcal{L}_m$ the Lagrangian density of matter fields, and $\kappa = 8\pi G/c^4$ the coupling constant between matter and curvature. In order to derive the Euler–Lagragian equation, $R$ is treated as a functional constructed upon the metric field and its derivative—see Eq. (24). $\mathcal{L}_m$ is then a functional constructed upon the matter fields and their derivative. For instance, for an electromagnetic field one has

$$\mathcal{L}_m = \frac{1}{4\mu_0} g^{\alpha\sigma} g^{\beta\epsilon} (\partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}) (\partial_{\sigma} A_{\epsilon} - \partial_{\epsilon} A_{\sigma}) \quad (40)$$

where $\mu_0$ is the magnetic permeability of vacuum, and $A^\alpha$ is the electromagnetic four-vector. Let us note that one has $A^\alpha = (\phi/c, A)$ where $\phi$ is the scalar potential and $A$ the vector potential of classical electromagnetism, form which one can compute the electric and magnetic fields (Jackson 1998).

Defining a theory from its action rather than from its equations is convenient because it allows one to be sure that the theory possesses conservation laws for the considered theory, which derive from Noether's theorem that states that every symmetry of a Lagrangian implies the existence of a conservation law (Noether 1918; Wald 1984). Therefore, most modern theories are defined from an action, although there are some exceptions—such as MOND, see Sect. 4.7.

<footnote n="12">¹² Note that mathematicians typically refer to $\sqrt{-g}\mathcal{L}$ as a *density* because $\mathcal{L}$ is a scalar and a scalar multiplied by the square root of the determinant corresponds to a mathematical object referred to as a density of weight $-1$; however, physicists typically refer to $\mathcal{L}$ as a *density*, as it carries the dimension of an energy density and is frequently associated with the definitions of kinetic and potential energy densities.</footnote>
Applying the principle of least action on the action (39), one recovers the Einstein–Hilbert equation of general relativity Eq. (22), with

$$T_{\alpha\beta} = - \frac{2 \delta(\sqrt{-g} \mathcal{L}_m)}{\sqrt{-g} \delta g^{\alpha\beta}}, \quad (41)$$

where $\delta$ stands for a variational derivative. It is worth noting that from Noether’s theorem, the diffeomorphism invariance—that is, the invariance under change of coordinates—of the matter Lagrangian $\mathcal{L}_m$ implies the (covariant) conservation of the stress-energy tensor

$$\nabla_{\sigma} T^{\mu\sigma} = 0, \quad (42)$$

where $\nabla_{\sigma}$ is the covariant derivative defined as followed for a tensor $X$ with $k$ contravariant indices and $l$ covariant indices:

$$\begin{aligned} \nabla_{\sigma} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} = & \partial_{\sigma} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} \\ & + \Gamma_{\sigma\lambda}^{\mu_1} X_{\nu_1 \dots \nu_l}^{\lambda \mu_1 \dots \mu_k} + \dots + \Gamma_{\sigma\lambda}^{\mu_k} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \lambda \mu_k} \\ & - \Gamma_{\sigma\lambda}^{\lambda} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} - \dots - \Gamma_{\sigma\nu_l}^{\lambda} X_{\nu_1 \dots \lambda \mu_k}^{\mu_1 \dots \mu_k}. \end{aligned} \quad (43)$$

This is, of course, consistent with the geometrical fact that $\nabla_{\sigma} G^{\mu\sigma} = 0$, with $G^{\mu\nu} = R^{\mu\nu} - 1/2 g^{\mu\nu} R$, the Einstein tensor, where $R^{\mu\nu}$ is the contravariant Ricci tensor defined from the covariant Ricci tensor $R_{\mu\nu}$ and the contravariant metric tensor $g^{\mu\nu}$ as $R^{\mu\nu} = g^{\mu\alpha} g^{\nu\beta} R_{\alpha\beta}$.

### 2.1.7 The post-Newtonian approximation of general relativity and gauge invariance

First, let us recall that the metric of general relativity at the Newtonian level, for a reference frame that is not kinematically rotating with respect to distant celestial bodies,<fnref n="13" /> reads

$$g_{00} = -1 + \frac{2U}{c^2} + \mathcal{O}(c^{-3}), \quad (44)$$

$$g_{0i} = \mathcal{O}(c^{-2}), \quad (45)$$

$$g_{ij} = \delta_{ij} + \mathcal{O}(c^{-1}), \quad (46)$$

such that one indeed has

$$ds^2 = - \left(1 - \frac{2U}{c^2}\right) c^2 dt^2 + dx^2 + dy^2 + dz^2, \quad (47)$$

at leading order. The truncation between leading and next-to-leading orders in Eqs. (44)–(46) are made with respect to the equation of motion that derives from the

<footnote n="13">¹³ Beyond rotating coordinate systems, it’s noteworthy that one could also define and utilize coordinate systems with shear, instead of pseudo-cartesian coordinate systems. However, such systems are rarely used because they would notably complicate the calculations, and likely also hinder understanding.</footnote>
Applying the principle of least action on the action (39), one recovers the Einstein–Hilbert equation of general relativity Eq. (22), with

$$T_{\alpha\beta} = - \frac{2 \delta(\sqrt{-g} \mathcal{L}_m)}{\sqrt{-g} \delta g^{\alpha\beta}}, \qquad (41)$$

where $\delta$ stands for a variational derivative. It is worth noting that from Noether’s theorem, the diffeomorphism invariance—that is, the invariance under change of coordinates—of the matter Lagrangian $\mathcal{L}_m$ implies the (covariant) conservation of the stress-energy tensor

$$\nabla_{\sigma} T^{\mu\sigma} = 0, \qquad (42)$$

where $\nabla_{\sigma}$ is the covariant derivative defined as followed for a tensor $X$ with $k$ contravariant indices and $l$ covariant indices:

$$\begin{aligned} \nabla_{\sigma} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} = & \partial_{\sigma} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} \\ & + \Gamma_{\sigma\lambda}^{\mu_1} X_{\nu_1 \dots \nu_l}^{\lambda \mu_1 \dots \mu_k} + \dots + \Gamma_{\sigma\lambda}^{\mu_k} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \lambda \mu_k} \\ & - \Gamma_{\sigma\lambda}^{\lambda} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} - \dots - \Gamma_{\sigma\lambda}^{\lambda} X_{\nu_1 \dots \nu_l}^{\mu_1 \dots \mu_k} \end{aligned} \qquad (43)$$

This is, of course, consistent with the geometrical fact that $\nabla_{\sigma} G^{\mu\sigma} = 0$, with $G^{\mu\nu} = R^{\mu\nu} - 1/2 g^{\mu\nu} R$, the Einstein tensor, where $R^{\mu\nu}$ is the contravariant Ricci tensor defined from the covariant Ricci tensor $R_{\mu\nu}$ and the contravariant metric tensor $g^{\mu\nu}$ as $R^{\mu\nu} = g^{\mu\alpha} g^{\nu\beta} R_{\alpha\beta}$.

### 2.1.7 The post-Newtonian approximation of general relativity and gauge invariance

First, let us recall that the metric of general relativity at the Newtonian level, for a reference frame that is not kinematically rotating with respect to distant celestial bodies,<fnref n="13" /> reads

$$g_{00} = -1 + \frac{2U}{c^2} + \mathcal{O}(c^{-3}), \qquad (44)$$

$$g_{0i} = \mathcal{O}(c^{-2}), \qquad (45)$$

$$g_{ij} = \delta_{ij} + \mathcal{O}(c^{-1}), \qquad (46)$$

such that one indeed has

$$ds^2 = - \left(1 - \frac{2U}{c^2}\right) c^2 dt^2 + dx^2 + dy^2 + dz^2, \qquad (47)$$

at leading order. The truncation between leading and next-to-leading orders in Eqs. (44)–(46) are made with respect to the equation of motion that derives from the

<footnote n="13">¹³ Beyond rotating coordinate systems, it’s noteworthy that one could also define and utilize coordinate systems with shear, instead of pseudo-cartesian coordinate systems. However, such systems are rarely used because they would notably complicate the calculations, and likely also hinder understanding.</footnote>
Lagrangian of motion Eq. (18). Indeed, the Lagrangian of motion with Eqs. (44)–(46) reads

$$L \propto 1 + \frac{1}{c^2} \left( \frac{v^2}{2} - U \right) + \mathcal{O}(c^{-3}). \quad (48)$$

One can show that a set of coordinate systems exists in general relativity that are such that, at next-to-leading order, the metric reads (Damour et al. 1991)

$$g_{00} = -1 + \frac{2w}{c^2} - \frac{2w^2}{c^4} + \mathcal{O}(c^{-5}), \quad (49)$$

$$g_{0i} = - \frac{4w^i}{c^3} \mathcal{O}(c^{-4}), \quad (50)$$

$$g_{ij} = \delta_{ij} \left( 1 + \frac{2w}{c^2} \right) + \mathcal{O}(c^{-3}), \quad (51)$$

where $w$ and $w^i$ are gravitational potentials. This is the so-called post-Newtonian metric of general relativity, in conformally cartesian coordinates.<fnref n="14" /> Assuming this metric, the field equation of general relativity reduces to

$$\Delta w + \frac{3}{c^2} \partial_{tt}^2 w + \frac{4}{c^2} \partial_{ij}^2 w_j = -4\pi G\sigma + \mathcal{O}(c^{-3}), \quad (52)$$

$$\Delta w_i - \partial_{ij}^2 w_j - \partial_{tt}^2 w = -4\pi G \sigma^i + \mathcal{O}(c^{-2}), \quad (53)$$

where $\sigma = c^{-2}(T^{00} + T^{kk})$ and $\sigma^i = c^{-1}T^{0i}$, such that $\sigma$ and $\sigma^i$ are zeroth order quantities—because $T^{\mu\nu} = \mathcal{O}(c^2, c, c^0)$ in the weak-field and slowly moving approximation.<fnref n="15" />

## Post-Newtonian gauge invariance

One can check that the equations Eqs. (52)–(53) are invariant under the following transformations

$$w' = w - \frac{1}{c^2} \partial_t \lambda, \quad (54)$$

$$w_i' = w_i + \frac{1}{4} \partial_i \lambda, \quad (55)$$

where $\lambda$ is an arbitrary differential function. Indeed, while the form of the metric in Eqs. (49)–(51) completely fixes the type of spatial coordinates being considered, it

<footnote n="14">¹⁴ The name conformally cartesian stems from the fact that one has $g_{00}g_{ij} = -\delta_{ij} + \mathcal{O}(c^{-4})$—that is, the space-space component of metric $g_{ij}$ multiplied by a conformal factor (here, the time-time component of the metric $g_{00}$) is flat up to $c^{-4}$ corrections. It's worth noting that such coordinate systems generally do not exist in general in theories other than general relativity—at least when the metric considered still defines the proper time $d\tau^2 = -g_{\alpha\beta}dx^\alpha dx^\beta$ and not a conformal frame.</footnote>

<footnote n="15">¹⁵ One can verify that with the stress energy tensor of dust, which reads $T^{\alpha\beta} = -\rho_m u^\alpha u^\beta$, where $\rho_m$ is the rest mass density and $u^\alpha = dx^\alpha/d\tau$, such that $u^0 \sim c$.</footnote>
Lagrangian of motion Eq. (18). Indeed, the Lagrangian of motion with Eqs. (44)–(46) reads

$$L \propto 1 + \frac{1}{c^2} \left( \frac{v^2}{2} - U \right) + \mathcal{O}(c^{-3}). \quad (48)$$

One can show that a set of coordinate systems exists in general relativity that are such that, at next-to-leading order, the metric reads (Damour et al. 1991)

$$g_{00} = -1 + \frac{2w}{c^2} - \frac{2w^2}{c^4} + \mathcal{O}(c^{-5}), \quad (49)$$

$$g_{0i} = - \frac{4w^i}{c^3} \mathcal{O}(c^{-4}), \quad (50)$$

$$g_{ij} = \delta_{ij} \left( 1 + \frac{2w}{c^2} \right) + \mathcal{O}(c^{-3}), \quad (51)$$

where $w$ and $w^i$ are gravitational potentials. This is the so-called post-Newtonian metric of general relativity, in conformally cartesian coordinates.<fnref n="14" /> Assuming this metric, the field equation of general relativity reduces to

$$\Delta w + \frac{3}{c^2} \partial_{tt}^2 w + \frac{4}{c^2} \partial_{ij}^2 w_j = -4\pi G\sigma + \mathcal{O}(c^{-3}), \quad (52)$$

$$\Delta w_i - \partial_{ij}^2 w_j - \partial_{ti}^2 w = -4\pi G\sigma^i + \mathcal{O}(c^{-2}), \quad (53)$$

where $\sigma = c^{-2}(T^{00} + T^{kk})$ and $\sigma^i = c^{-1}T^{0i}$, such that $\sigma$ and $\sigma^i$ are zeroth order quantities—because $T^{\mu\nu} = \mathcal{O}(c^2, c, c^0)$ in the weak-field and slowly moving approximation.<fnref n="15" />

## Post-Newtonian gauge invariance

One can check that the equations Eqs. (52)–(53) are invariant under the following transformations

$$w' = w - \frac{1}{c^2} \partial_t \lambda, \quad (54)$$

$$w'_i = w_i + \frac{1}{4} \partial_i \lambda, \quad (55)$$

where $\lambda$ is an arbitrary differential function. Indeed, while the form of the metric in Eqs. (49)–(51) completely fixes the type of spatial coordinates being considered, it

<footnote n="14">¹⁴ The name conformally cartesian stems from the fact that one has $g_{00}g_{ij} = -\delta_{ij} + \mathcal{O}(c^{-4})$—that is, the space-space component of metric $g_{ij}$ multiplied by a conformal factor (here, the time-time component of the metric $g_{00}$) is flat up to $c^{-4}$ corrections. It's worth noting that such coordinate systems generally do not exist in general in theories other than general relativity—at least when the metric considered still defines the proper time $d\tau^2 = -g_{\alpha\beta}dx^\alpha dx^\beta$ and not a conformal frame.</footnote>

<footnote n="15">¹⁵ One can verify that with the stress energy tensor of dust, which reads $T^{\alpha\beta} = -\rho_m u^\alpha u^\beta$, where $\rho_m$ is the rest mass density and $u^\alpha = dx^\alpha/d\tau$, such that $u^0 \sim c$.</footnote>
leaves a freedom at the level of the time coordinate. This gauge invariance indeed corresponds to a shift of the time variable

$$ \delta t = c^{-4} \lambda(x^{\alpha}). \qquad (56) $$

After having imposed a non-kinematically-rotating frame with respect to distant celestial objects (such as quasars), a central world-line and the use of conformally cartesian coordinates, Eq. (56) is a remaining freedom of our coordinate system.

### Harmonic gauge

The International Astronomical Union recommends (Soffel 2003) the use of harmonic coordinates that are such that

$$ g^{\alpha\beta} \Gamma_{\alpha\beta}^{\sigma} = \mathcal{O}(c^{-5}, c^{-4}). \qquad (57) $$

However, because of the use of conformally cartesian coordinates, the space component of this condition $g^{\alpha\beta}\Gamma_{\alpha\beta}^{i} = \mathcal{O}(c^{-4})$ is already satisfied, and one is left with the time component of this condition that, considering the metric Eqs. (49)–(51), reduces to

$$ \partial_t w + \partial_j w_j = \mathcal{O}(c^{-2}). \qquad (58) $$

With this condition, Eqs. (52)–(53) reduce to

$$ \Box_m w = -4\pi G\sigma + \mathcal{O}(c^{-3}), \qquad (59) $$

$$ \Delta w_i = -4\pi G\sigma^i + \mathcal{O}(c^{-2}), \qquad (60) $$

where $\Box_m$ and $\Delta w_i$ are respectively the d’Alembertian and Laplacian of the usual flat Minkowski spacetime:

$$ \Box_m = -c^{-2}\partial_t^2 + \Delta, \qquad (61) $$

where for instance, $\Delta = \partial_{xx}^2 + \partial_{yy}^2 + \partial_{zz}^2$ in Cartesian coordinates.

### 2.1.8 The Einstein–Infeld–Hoffman–Droste–Lorentz equation of motion

Up to corrections that can be taken into account at a later stage, celestial bodies in the solar system can be approximated as being non-rotating point particles. This approximation has been explored for the first time by Lorentz and Droste (1917a, 1917b)—translated in Lorentz and Droste (1937)—and re-derived later by Einstein et al. (1938). The stress-energy tensor for point particles is simply the stress-energy tensor of a dust fluid

$$ T^{\mu\nu} = \rho_m u^{\mu} u^{\nu}, \qquad (62) $$

where $u^{\alpha} = dx^{\alpha}/d\tau$ is the proper four-velocity of the fluid. In terms of conserved mass along the fluid geodesics $dm_A/d\tau = 0$, it reads
leaves a freedom at the level of the time coordinate. This gauge invariance indeed corresponds to a shift of the time variable

$$ \delta t = c^{-4} \lambda(x^{\alpha}). \qquad (56) $$

After having imposed a non-kinematically-rotating frame with respect to distant celestial objects (such as quasars), a central world-line and the use of conformally cartesian coordinates, Eq. (56) is a remaining freedom of our coordinate system.

### Harmonic gauge

The International Astronomical Union recommends (Soffel 2003) the use of harmonic coordinates that are such that

$$ g^{\alpha\beta} \Gamma_{\alpha\beta}^{\sigma} = \mathcal{O}(c^{-5}, c^{-4}). \qquad (57) $$

However, because of the use of conformally cartesian coordinates, the space component of this condition $g^{\alpha\beta}\Gamma_{\alpha\beta}^{i} = \mathcal{O}(c^{-4})$ is already satisfied, and one is left with the time component of this condition that, considering the metric Eqs. (49)–(51), reduces to

$$ \partial_t w + \partial_j w_j = \mathcal{O}(c^{-2}). \qquad (58) $$

With this condition, Eqs. (52)–(53) reduce to

$$ \Box_m w = -4\pi G\sigma + \mathcal{O}(c^{-3}), \qquad (59) $$

$$ \Delta w_i = -4\pi G\sigma^i + \mathcal{O}(c^{-2}), \qquad (60) $$

where $\Box_m$ and $\Delta w_i$ are respectively the d’Alembertian and Laplacian of the usual flat Minkowski spacetime:

$$ \Box_m = -c^{-2}\partial_t^2 + \Delta, \qquad (61) $$

where for instance, $\Delta = \partial_{xx}^2 + \partial_{yy}^2 + \partial_{zz}^2$ in Cartesian coordinates.

### 2.1.8 The Einstein–Infeld–Hoffman–Droste–Lorentz equation of motion

Up to corrections that can be taken into account at a later stage, celestial bodies in the solar system can be approximated as being non-rotating point particles. This approximation has been explored for the first time by Lorentz and Droste (1917a, 1917b)—translated in Lorentz and Droste (1937)—and re-derived later by Einstein et al. (1938). The stress-energy tensor for point particles is simply the stress-energy tensor of a dust fluid

$$ T^{\mu\nu} = \rho_m u^{\mu} u^{\nu}, \qquad (62) $$

where $u^{\alpha} = dx^{\alpha}/d\tau$ is the proper four-velocity of the fluid. In terms of conserved mass along the fluid geodesics $dm_A/d\tau = 0$, it reads
$$T^{\mu\nu} = \sum_A \frac{m_A u^0/c}{\sqrt{-g}} v_A^\mu v_A^\nu \delta^{(3)}(\mathbf{x} - \mathbf{x}_A(t)), \quad (63)$$

where $\delta^{(3)}$ is the 3-dimensional delta function, and $v^\alpha = dx^\alpha/dt$ the coordinate four-velocity of the fluid, such that $u^\alpha = u^0 v^\alpha/c$. Equation (63) used the fact that $u_\mu \nabla_\sigma T^{\mu\sigma} = 0$ —which follows from Eq. (42)—such that one has the usual Newtonian conservation of the mass density $\partial_\sigma(\rho^* v^\sigma) = \partial_t \rho^* + \partial_i (\rho^* v^i) = 0$ for the density $\rho^* := \sqrt{-g} \rho_m u^0/c$. Solving the Einstein–Hilbert equation (22) with this approximation, and in the harmonic gauge Eq. (58), leads to the metric in Eqs. (49)–(51) with

$$w = w_0 - \frac{1}{c^2} \Delta + O(c^{-4}), \quad (64)$$

$$w^j = \sum_A \frac{Gm_A}{r_A} v_A^j + O(c^{-2}), \quad (65)$$

where

$$w_0 = \sum_A \frac{Gm_A}{r_A}, \quad (66)$$

where $\mathbf{r}_A = \mathbf{x} - \mathbf{x}_A(t)$, $r_A = |\mathbf{r}_A|$, and

$$\Delta = \sum_A \frac{Gm_A}{r_A} \left\{ -2v_a^2 + \sum_{B \neq A} \frac{Gm_B}{r_{BA}} + \frac{1}{2} \left[ \frac{(\mathbf{r}_A \mathbf{v}_A)^2}{r_A^2} + \mathbf{r}_A \mathbf{a}_A \right] \right\}, \quad (67)$$

with $r_{AB} = |\mathbf{x}_B - \mathbf{x}_A|$. We should note that this corresponds to the metric recommended by the International Astronomical Union (Soffel 2003), subject to corrections accounting for the fact that celestial bodies are not point-like but extended objects, and that these bodies possess angular momentum relative to the frame that is fixed with respect to distant objects, such as quasars. Fortunately, these corrections are numerically small, such that they can safely be added a posteriori at the leading order, without impacting the calculation of the next-to-leading order.

Now, injecting this metric into the definition of the Lagrangian of motion Eq. (18), one gets for a test particle B the following Lagrangian of motion

$$L_B = -m_B c^2 + m_B \frac{v_B^2}{2} + \sum_{A \neq B} \frac{Gm_A m_B}{r_{AB}} + \frac{m_B v_B^4}{8c^2} \\ + \frac{1}{c^2} \sum_{A \neq B} \frac{Gm_A m_B}{r_{AB}} \left[ -4\mathbf{v}_A \cdot \mathbf{v}_B + \frac{3}{2}v_B^2 + 2v_A^2 - \frac{(\mathbf{v}_A \cdot \mathbf{r}_{AB})^2}{2r_{AB}^2} - \frac{\mathbf{r}_{AB} \cdot \mathbf{a}_A}{2} \right] \\ - \frac{1}{c^2} \sum_{A \neq B} \frac{Gm_A m_B}{r_{AB}} \left[ \sum_{D \neq A} \frac{Gm_D}{r_{AD}} + \sum_{D \neq B} \frac{1}{2} \frac{Gm_D}{r_{DB}} \right] + O(c^{-4}).$$

Injecting this Lagrangian into the Euler–Lagrange equation Eq. (7), one finally gets
$$T^{\mu\nu} = \sum_A \frac{m_A u^0/c}{\sqrt{-g}} v_A^\mu v_A^\nu \delta^{(3)}(\mathbf{x} - \mathbf{x}_A(t)), \qquad (63)$$

where $\delta^{(3)}$ is the 3-dimensional delta function, and $v^\alpha = dx^\alpha/dt$ the coordinate four-velocity of the fluid, such that $u^\alpha = u^0 v^\alpha/c$. Equation (63) used the fact that $u_\mu \nabla_\sigma T^{\mu\sigma} = 0$ —which follows from Eq. (42)—such that one has the usual Newtonian conservation of the mass density $\partial_\sigma(\rho^* v^\sigma) = \partial_t \rho^* + \partial_i (\rho^* v^i) = 0$ for the density $\rho^* := \sqrt{-g} \rho_m u^0/c$. Solving the Einstein–Hilbert equation (22) with this approximation, and in the harmonic gauge Eq. (58), leads to the metric in Eqs. (49)–(51) with

$$w = w_0 - \frac{1}{c^2} \Delta + O(c^{-4}), \qquad (64)$$

$$w^j = \sum_A \frac{Gm_A}{r_A} v_A^j + O(c^{-2}), \qquad (65)$$

where

$$w_0 = \sum_A \frac{Gm_A}{r_A}, \qquad (66)$$

where $\mathbf{r}_A = \mathbf{x} - \mathbf{x}_A(t)$, $r_A = |\mathbf{r}_A|$, and

$$\Delta = \sum_A \frac{Gm_A}{r_A} \left\{ -2v_a^2 + \sum_{B \neq A} \frac{Gm_B}{r_{BA}} + \frac{1}{2} \left[ \frac{(\mathbf{r}_A \mathbf{v}_A)^2}{r_A^2} + \mathbf{r}_A \mathbf{a}_A \right] \right\}, \qquad (67)$$

with $r_{AB} = |\mathbf{x}_B - \mathbf{x}_A|$. We should note that this corresponds to the metric recommended by the International Astronomical Union (Soffel 2003), subject to corrections accounting for the fact that celestial bodies are not point-like but extended objects, and that these bodies possess angular momentum relative to the frame that is fixed with respect to distant objects, such as quasars. Fortunately, these corrections are numerically small, such that they can safely be added a posteriori at the leading order, without impacting the calculation of the next-to-leading order.

Now, injecting this metric into the definition of the Lagrangian of motion Eq. (18), one gets for a test particle B the following Lagrangian of motion

$$L_B = -m_B c^2 + m_B \frac{v_B^2}{2} + \sum_{A \neq B} \frac{Gm_A m_B}{r_{AB}} + \frac{m_B v_B^4}{8c^2} \\ + \frac{1}{c^2} \sum_{A \neq B} \frac{Gm_A m_B}{r_{AB}} \left[ -4\mathbf{v}_A \cdot \mathbf{v}_B + \frac{3}{2}v_B^2 + 2v_A^2 - \frac{(\mathbf{v}_A \cdot \mathbf{r}_{AB})^2}{2r_{AB}^2} - \frac{\mathbf{r}_{AB} \cdot \mathbf{a}_A}{2} \right] \\ - \frac{1}{c^2} \sum_{A \neq B} \frac{Gm_A m_B}{r_{AB}} \left[ \sum_{D \neq A} \frac{Gm_D}{r_{AD}} + \sum_{D \neq B} \frac{1}{2} \frac{Gm_D}{r_{DB}} \right] + O(c^{-4}).$$

Injecting this Lagrangian into the Euler–Lagrange equation Eq. (7), one finally gets
the Einstein-Infeld-Hoffman-Droste-Lorentz (EIHDL) equation of motion given in Eq. (90) in Sect. 3.3.

### The advance of perihelion of Mercury

When focusing solely on the two-body problem, one can examine the secular changes of the orbital elements by treating the $c^{-4}$ term in the equation of motion Eq. (90) as perturbations to Keplerian orbits. For instance, the secular advance of a perihelion in the two-body problem can be expressed as follows

$$ \Delta\dot{\sigma} = 6\pi \frac{G(m_1 + m_2)}{a(1 - e^2)c^2}, \qquad (68) $$

where $a$ is the Keplerian semi-major axis, $e$ the eccentricity and $m_i$ the mass of the body $i$.

## 2.1.9 The propagation of light in general relativity

From the Lagrangian of a free electromagnetic field Eq. (40), one derives the following equation for the free electromagnetic field in a curved spacetime

$$ \nabla_{\sigma} F^{\alpha\sigma} = \partial_{\sigma} F^{\alpha\sigma} + \Gamma^{\alpha}_{\sigma\epsilon} F^{\epsilon\sigma} + \Gamma^{\sigma}_{\sigma\epsilon} F^{\alpha\epsilon} = 0, \qquad (69) $$

where $\nabla_{\sigma}$ defines the covariant derivatives associated to the Christoffel connection Eq. (43). Using the definition of the electromagnetic tensor in Sect. 2.1.6, and translating the 4-vector potential in terms of electric and magnetic fields (Jackson 1998), one recovers the equations of Maxwell for free electromagnetic fields if spacetime is flat—that is $\Box_m \vec{E} = 0$ and $\Box_m \vec{B} = 0$, where $\Box_m$ is the usual D’Alembertian of flat Minkowski spacetimes defined in Eq. (61). Considering the Lorenz gauge $\nabla_{\sigma} A^{\sigma} = 0$, Eq. (69) reduces to

$$ \Box A^{\alpha} - g^{\alpha\sigma} R_{\sigma\epsilon} A^{\epsilon} = 0, \qquad (70) $$

where one defines the covariant D’Alembertian as $\Box = g^{\sigma\epsilon}\nabla_{\sigma}\nabla_{\epsilon}$. Now, let us expand the 4-vector potential as follows (Misner et al. 1973)

$$ A^{\mu} = \Re \left\{ \left( a^{\mu} + \epsilon b^{\mu} + O(\epsilon^2) \right) \exp^{i\theta/\epsilon} \right\}, \qquad (71) $$

where $\Re\{X\}$ means the real part of $X$. The leading order in the $\epsilon$ expansion corresponds to the geometric optics approximation. It induces that

$$ k^{\sigma} \nabla_{\sigma} k^{\alpha} = 0, \qquad (72) $$

where the wave-vector is defined as $k^{\alpha} := g^{\alpha\sigma}\partial_{\sigma}\theta$ and

$$ k_{\sigma} k^{\sigma} = g_{\sigma\epsilon} k^{\epsilon} k^{\sigma} = 0. \qquad (73) $$

Equation (72) means that in the geometric optics limit, electromagnetic waves follow spacetime geodesics
the Einstein–Infeld–Hoffman–Droste–Lorentz (EIHDL) equation of motion given in Eq. (90) in Sect. 3.3.

### The advance of perihelion of Mercury

When focusing solely on the two-body problem, one can examine the secular changes of the orbital elements by treating the $c^{-4}$ term in the equation of motion Eq. (90) as perturbations to Keplerian orbits. For instance, the secular advance of a perihelion in the two-body problem can be expressed as follows

$$ \Delta\dot{\sigma} = 6\pi \frac{G(m_1 + m_2)}{a(1 - e^2)c^2}, \qquad (68) $$

where $a$ is the Keplerian semi-major axis, $e$ the eccentricity and $m_i$ the mass of the body $i$.

## 2.1.9 The propagation of light in general relativity

From the Lagrangian of a free electromagnetic field Eq. (40), one derives the following equation for the free electromagnetic field in a curved spacetime

$$ \nabla_{\sigma} F^{\alpha\sigma} = \partial_{\sigma} F^{\alpha\sigma} + \Gamma^{\alpha}_{\sigma\epsilon} F^{\epsilon\sigma} + \Gamma^{\sigma}_{\sigma\epsilon} F^{\alpha\epsilon} = 0, \qquad (69) $$

where $\nabla_{\sigma}$ defines the covariant derivatives associated to the Christoffel connection Eq. (43). Using the definition of the electromagnetic tensor in Sect. 2.1.6, and translating the 4-vector potential in terms of electric and magnetic fields (Jackson 1998), one recovers the equations of Maxwell for free electromagnetic fields if spacetime is flat—that is $\Box_m \vec{E} = 0$ and $\Box_m \vec{B} = 0$, where $\Box_m$ is the usual D’Alembertian of flat Minkowski spacetimes defined in Eq. (61). Considering the Lorenz gauge $\nabla_{\sigma} A^{\sigma} = 0$, Eq. (69) reduces to

$$ \Box A^{\alpha} - g^{\alpha\sigma} R_{\sigma\epsilon} A^{\epsilon} = 0, \qquad (70) $$

where one defines the covariant D’Alembertian as $\Box = g^{\sigma\epsilon}\nabla_{\sigma}\nabla_{\epsilon}$. Now, let us expand the 4-vector potential as follows (Misner et al. 1973)

$$ A^{\mu} = \Re \left\{ \left( a^{\mu} + \epsilon b^{\mu} + O(\epsilon^2) \right) \exp^{i\theta/\epsilon} \right\}, \qquad (71) $$

where $\Re\{X\}$ means the real part of $X$. The leading order in the $\epsilon$ expansion corresponds to the geometric optics approximation. It induces that

$$ k^{\sigma} \nabla_{\sigma} k^{\alpha} = 0, \qquad (72) $$

where the wave-vector is defined as $k^{\alpha} := g^{\alpha\sigma} \partial_{\sigma} \theta$ and

$$ k_{\sigma} k^{\sigma} = g_{\sigma\epsilon} k^{\epsilon} k^{\sigma} = 0. \qquad (73) $$

Equation (72) means that in the geometric optics limit, electromagnetic waves follow spacetime geodesics
$$ \frac{dk^\alpha}{d\lambda} + \Gamma^\alpha_{\mu\nu} k^\mu k^\nu = 0, \qquad (74) $$

where $\lambda$ is an affine parameter of the geodesics, such that $k^\alpha = dx^\alpha/d\lambda$. Equation (73) means that those geodesics are such that the line element is null ($ds^2 = 0$) along the trajectories of the electromagnetic waves in the geometric optics approximation. One therefore generically says that light follows null-geodesics, although this is in fact correct only in the geometric optics approximation. The fact that $ds^2 = 0$ along the geodesics of light implies that the trajectory lies on the null spacetime cones, which define the causal structure of spacetime. In other words, the speed of light is indeed equivalent to the speed $c$ appearing in the definition of the line element Eq. (36). But it also means that there is no notion of proper time for light since the line element $ds^2 = -c^2 d\tau^2$ is null along their geodesics.

## Null-geodesics and astrometric observables

Because electromagnetic waves follow null-geodesics of a curved spacetime in the geometric optic approximation, the trajectories of electromagnetic waves are curved in general, notably leading to the deflection mentioned in Sect. 2.1.5 and represented in Fig. 2.

One side of astrometry is about determining the projection of the positions of celestial bodies on the celestial sphere as they are seen by an observer, based on their angular measurements. Indeed, as detailed in Sect. 2.1.5, what measures an observer are the angles between the null-cones that link them to the sources of the observed electromagnetic waves.

As explained notably in Hees et al. (2014a), one way to get a covariant definition<fnref n="16" /> of the position of the electromagnetic sources in the celestial sphere as it appears for an observer, is to use the tetrad formalism (Misner et al. 1973; Klioner and Kopeikin 1992), by giving the direction of observation of an incoming electromagnetic wave in a tetrad $E$ comoving with the observer $O_B$, as one can see in Fig. 3. We write $E^\mu_{(\alpha)}$ the components of this tetrad, where $\langle\alpha\rangle$ corresponds to the tetrad index and $\mu$ is a normal tensor index that can be lowered and raised by use of the metric. The tetrad is assumed to be orthonormal so that

$$ g_{\mu\nu} E^\mu_{(\alpha)} E^\nu_{(\beta)} = \eta_{(\alpha)(\beta)}, \qquad (75) $$

$\eta$ being the flat Minkowski metric, and the vector $E^\mu_{(0)}$ is chosen to be timelike, such that $E^\mu_{(i)}$ is spacelike. Then, the wave-vector $k_\mu = g_{\mu\nu}k^\nu$ becomes in the tetrad frame associated to the observer

$$ k_{(\alpha)} = E^\mu_{(\alpha)} k_\mu. \qquad (76) $$

The incident direction of the wave in the tetrad frame is given by the following normalization

<footnote n="16">¹⁶ That is, a definition that is invariant under change of coordinate systems.</footnote>
$$ \frac{dk^\alpha}{d\lambda} + \Gamma^\alpha_{\mu\nu} k^\mu k^\nu = 0, \qquad (74) $$

where $\lambda$ is an affine parameter of the geodesics, such that $k^\alpha = dx^\alpha/d\lambda$. Equation (73) means that those geodesics are such that the line element is null ($ds^2 = 0$) along the trajectories of the electromagnetic waves in the geometric optics approximation. One therefore generically says that light follows null-geodesics, although this is in fact correct only in the geometric optics approximation. The fact that $ds^2 = 0$ along the geodesics of light implies that the trajectory lies on the null spacetime cones, which define the causal structure of spacetime. In other words, the speed of light is indeed equivalent to the speed $c$ appearing in the definition of the line element Eq. (36). But it also means that there is no notion of proper time for light since the line element $ds^2 = -c^2d\tau^2$ is null along their geodesics.

## Null-geodesics and astrometric observables

Because electromagnetic waves follow null-geodesics of a curved spacetime in the geometric optic approximation, the trajectories of electromagnetic waves are curved in general, notably leading to the deflection mentioned in Sect. 2.1.5 and represented in Fig. 2.

One side of astrometry is about determining the projection of the positions of celestial bodies on the celestial sphere as they are seen by an observer, based on their angular measurements. Indeed, as detailed in Sect. 2.1.5, what measures an observer are the angles between the null-cones that link them to the sources of the observed electromagnetic waves.

As explained notably in Hees et al. (2014a), one way to get a covariant definition<fnref n="16" /> of the position of the electromagnetic sources in the celestial sphere as it appears for an observer, is to use the tetrad formalism (Misner et al. 1973; Klioner and Kopeikin 1992), by giving the direction of observation of an incoming electromagnetic wave in a tetrad $E$ comoving with the observer $O_B$, as one can see in Fig. 3. We write $E^\mu_{(\alpha)}$ the components of this tetrad, where $\langle\alpha\rangle$ corresponds to the tetrad index and $\mu$ is a normal tensor index that can be lowered and raised by use of the metric. The tetrad is assumed to be orthonormal so that

$$ g_{\mu\nu} E^\mu_{(\alpha)} E^\nu_{(\beta)} = \eta_{(\alpha)(\beta)}, \qquad (75) $$

$\eta$ being the flat Minkowski metric, and the vector $E^\mu_{(0)}$ is chosen to be timelike, such that $E^\mu_{(i)}$ is spacelike. Then, the wave-vector $k_\mu = g_{\mu\nu}k^\nu$ becomes in the tetrad frame associated to the observer

$$ k_{(\alpha)} = E^\mu_{(\alpha)} k_\mu. \qquad (76) $$

The incident direction of the wave in the tetrad frame is given by the following normalization

<footnote n="16">¹⁶ That is, a definition that is invariant under change of coordinate systems.</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Illustration of an electromagnetic wave with frequency $v_A$ emitted by $O_A$ with a wave four-vector of components $k_A^\mu$ and received by $O_B$ at a frequency $v_B$ and with a wave four-vector of components $k_B^\mu$. Image reproduced with permission from Hees et al. (2014a), copyright by APS</figcaption></figure>

$$n^{(i)} = \frac{k^{(i)}}{\sqrt{\delta_{jk} k^{(i)} k^{(j)}}} = \frac{k^{(i)}}{k^{(0)}} = -\frac{k_{(i)}}{k_{(0)}}. \qquad (77)$$

This quantity is the actual astrometric observable at the location of the observer. For more details, we refer the reader to Misner et al. (1973), Klioner and Kopeikin (1992), and Hees et al. (2014a).

The other side of astrometry is about determining the distance of remote objects, which leads us to the concept of the Shapiro delay.

### Shapiro delay

In the solar system, most of the time, the trajectory of light can be approximated as being straight lines at leading order.<fnref n="17" /> That is, one has $x^i(t) = n^i c(t - t_e) + x_e^j + O(c^{-2})$—where $e$ stands for the emission, and $n^i$ is a constant normalized vector. One can use this information in order to compute the coordinate time elapsed between the emission and the reception of light, without the need to actually solve the geodesic equation (74). From the null condition Eq. (73), one has, indeed, that $ds^2 = 0$ between the emission and the reception of the electromagnetic wave. This means that $g_{\mu\nu}dx^\mu dx^\nu = 0$, where $g_{\mu\nu}$ is given by Eqs. (49)–(51). However, since light travels at the speed of light, $v^i/c$ is not a negligible quantity, and it is necessary to maintain the same order of development in terms of $c^{-n}$ in the metric to account for all the relevant terms at a given order for the propagation of light. As a result, when considering deviations from the trajectory of light relative to special relativity up to the order $O(c^{-3})$, the relevant metric is

<footnote n="17">¹⁷ This approximation holds true when one is sufficiently distant from the gravitational lensing regime, but starts to fail near this regime, resulting in the emergence of the so-called enhanced terms during the derivation of light trajectory in geometric configurations where this approximation is only marginally accurate (Ashby and Bertotti 2010; Linet and Teyssandier 2016). For a brief discussion on this matter, we refer to Sect. 3.4.1.</footnote>
<figure><img src="image_6.png" /><figcaption>**Fig. 3** Illustration of an electromagnetic wave with frequency $v_A$ emitted by $O_A$ with a wave four-vector of components $k_A^\mu$ and received by $O_B$ at a frequency $v_B$ and with a wave four-vector of components $k_B^\mu$. Image reproduced with permission from Hees et al. (2014a), copyright by APS</figcaption></figure>

$$n^{(i)} = \frac{k^{(i)}}{\sqrt{\delta_{jk} k^{(i)} k^{(j)}}} = \frac{k^{(i)}}{k^{(0)}} = -\frac{k_{(i)}}{k_{(0)}}. \qquad (77)$$

This quantity is the actual astrometric observable at the location of the observer. For more details, we refer the reader to Misner et al. (1973), Klioner and Kopeikin (1992), and Hees et al. (2014a).

The other side of astrometry is about determining the distance of remote objects, which leads us to the concept of the Shapiro delay.

### Shapiro delay

In the solar system, most of the time, the trajectory of light can be approximated as being straight lines at leading order.<fnref n="17" /> That is, one has $x^i(t) = n^i c(t - t_e) + x_e^j + O(c^{-2})$—where $e$ stands for the emission, and $n^i$ is a constant normalized vector. One can use this information in order to compute the coordinate time elapsed between the emission and the reception of light, without the need to actually solve the geodesic equation (74). From the null condition Eq. (73), one has, indeed, that $ds^2 = 0$ between the emission and the reception of the electromagnetic wave. This means that $g_{\mu\nu}dx^\mu dx^\nu = 0$, where $g_{\mu\nu}$ is given by Eqs. (49)–(51). However, since light travels at the speed of light, $v^i/c$ is not a negligible quantity, and it is necessary to maintain the same order of development in terms of $c^{-n}$ in the metric to account for all the relevant terms at a given order for the propagation of light. As a result, when considering deviations from the trajectory of light relative to special relativity up to the order $O(c^{-3})$, the relevant metric is

<footnote n="17">¹⁷ This approximation holds true when one is sufficiently distant from the gravitational lensing regime, but starts to fail near this regime, resulting in the emergence of the so-called enhanced terms during the derivation of light trajectory in geometric configurations where this approximation is only marginally accurate (Ashby and Bertotti 2010; Linet and Teyssandier 2016). For a brief discussion on this matter, we refer to Sect. 3.4.1.</footnote>
$$ds^2 = -\left(1 - \frac{2U}{c^2}\right)c^2 dt^2 + \left(1 + \frac{2U}{c^2}\right)(dx^2 + dy^2 + dz^2) + \mathcal{O}(c^{-3}) = 0, \quad (78)$$

where $U = w + \mathcal{O}(c^{-2})$—see Eqs. (28) and (52). Integrating over this equation, one gets the coordinate time elapsed between the emission and the reception $T_{er} = t_r - t_e$

$$c T_{er} = |x_r^i - x_e^i| + \frac{2}{c^2} \int_e^r U dl + \mathcal{O}(c^{-3}), \quad (79)$$

where the integration is taken along the straight line $x^i(t) = n^i(t - t_e) + x_e^i$ that connects the emission and the reception events. It appears as if light experiences a delay due to the presence of a gravitational field. This delay is known as the *Shapiro delay*, named after Irwin Shapiro who was the first to predict this effect (Shapiro 1964). More on this delay in Sect. 3.4.1.

Hence, the Shapiro delay has to be taken into account in order to recover the distance—in terms of a given coordinate system—from the measured round-trip propagation time. This means that distances constructed from observations not only depend on the coordinate system being used, but also depend on the model for the gravitational potential $U$ along the trajectory of the electromagnetic wave.

It is crucial to have in mind that what is typically probed by solar system experiments—such as in Bertotti et al. (2003b)—is not so much the delay in Eq. (79), but rather its variation as the observed electromagnetic signal traverses different sections of the gravitational potential $U$.<fnref n="18" /> More specifically, these experiments usually probe the delay variation with respect to the the minimum distance between the electromagnetic signal trajectory and the gravitating body (also called impact parameter). For a comprehensive discussion on this topic, we direct readers to Chapter 6.3 in Wald (1984).

### 2.1.10 Alternative gravitational theories

All the aspects of the aforementioned content may be subject to modifications in an alternative theory to general relativity—see Sect. 4. Hence, one must exercise caution when considering alternative theories of gravity, as the introduced modifications can impact the entire modeling process—from the definition of the coordinate system, to the equations of motion of light and massive bodies—and consequently the analysis of the observations.

## 2.2 Basics on ephemeris

An ephemeris is a table of positions and velocities given at different time steps. One can compute an ephemeris for artificial satellites, planetary bodies (planets, natural satellites, asteroids, comets...) but also pulsars. In order to provide to the user

<footnote n="18">¹⁸ The mathematical expression of the delay as given in Eq. (79) is gauge-dependent, and thus does not represent an observable quantity by itself. For detailed discussions on this, we refer readers to Gao and Wald (2000) and Minazzoli et al. (2019).</footnote>
$$ds^2 = -\left(1 - \frac{2U}{c^2}\right)c^2 dt^2 + \left(1 + \frac{2U}{c^2}\right)(dx^2 + dy^2 + dz^2) + \mathcal{O}(c^{-3}) = 0, \quad (78)$$

where $U = w + \mathcal{O}(c^{-2})$—see Eqs. (28) and (52). Integrating over this equation, one gets the coordinate time elapsed between the emission and the reception $T_{er} = t_r - t_e$

$$c T_{er} = |x_r^i - x_e^i| + \frac{2}{c^2} \int_e^r U dl + \mathcal{O}(c^{-3}), \quad (79)$$

where the integration is taken along the straight line $x^i(t) = n^i(t - t_e) + x_e^i$ that connects the emission and the reception events. It appears as if light experiences a delay due to the presence of a gravitational field. This delay is known as the *Shapiro delay*, named after Irwin Shapiro who was the first to predict this effect (Shapiro 1964). More on this delay in Sect. 3.4.1.

Hence, the Shapiro delay has to be taken into account in order to recover the distance—in terms of a given coordinate system—from the measured round-trip propagation time. This means that distances constructed from observations not only depend on the coordinate system being used, but also depend on the model for the gravitational potential $U$ along the trajectory of the electromagnetic wave.

It is crucial to have in mind that what is typically probed by solar system experiments—such as in Bertotti et al. (2003b)—is not so much the delay in Eq. (79), but rather its variation as the observed electromagnetic signal traverses different sections of the gravitational potential $U$.<fnref n="18" /> More specifically, these experiments usually probe the delay variation with respect to the the minimum distance between the electromagnetic signal trajectory and the gravitating body (also called impact parameter). For a comprehensive discussion on this topic, we direct readers to Chapter 6.3 in Wald (1984).

### 2.1.10 Alternative gravitational theories

All the aspects of the aforementioned content may be subject to modifications in an alternative theory to general relativity—see Sect. 4. Hence, one must exercise caution when considering alternative theories of gravity, as the introduced modifications can impact the entire modeling process—from the definition of the coordinate system, to the equations of motion of light and massive bodies—and consequently the analysis of the observations.

## 2.2 Basics on ephemeris

An ephemeris is a table of positions and velocities given at different time steps. One can compute an ephemeris for artificial satellites, planetary bodies (planets, natural satellites, asteroids, comets...) but also pulsars. In order to provide to the user

<footnote n="18">¹⁸ The mathematical expression of the delay as given in Eq. (79) is gauge-dependent, and thus does not represent an observable quantity by itself. For detailed discussions on this, we refer readers to Gao and Wald (2000) and Minazzoli et al. (2019).</footnote>
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Schematic representation of the construction of ephemeris applied to the planetary case. SSB stands for solar system barycenter (see Sect. 3.2.4), TDB (UTC) stands for Barycentric Dynamical Time (Universal Time Coordinate, respectively) (see Sect. 3.2.2). The description of the dynamical model is the case of planetary ephemerides is done in Sect. 3.3, and the presentation of the observational data sets in Sect. 3.4</figcaption></figure>

accurate estimations of the dynamical states of the considered body, several ingredients are necessary.

First, one needs to agree upon a set of both time and space coordinates given in a properly defined frame. This frame will be preferably inertial and the coordinate system will give an easily understandable representation of the body motion. This coordinate system will also be used to confront the dynamical modeling of the motion to the observations (see in step 3). The selection of the frame—in which the motion and the observations will be described—and its characterisation in space and time (metrics) will constitute the step 1 of the ephemeris construction (red boxes in Fig. 4).

Second, we shall identify and develop the appropriate dynamical model for describing the most accurately the motion of the bodies. This step (black box in Fig. 4) requires the writing of an equation of motion according to the frame defined in the step 1. In the dynamical modeling, one must include all the gravitational perturbations expected for the considered system. A numerical integration of the equations of motion is also usually performed in order to provide to the users positions and velocities for discrete times. Analytical resolutions were also proposed up to the beginning of the 21st century but stopped when the planetary observations became too accurate in regards to the size of the analytical series (see for example Fienga and Simon 2005).

At this step, one may consider that the ephemeris is built. However, in particular in the solar system, the localisation of the object of interest has been monitored by
<figure><img src="image_7.png" /><figcaption>**Fig. 4** Schematic representation of the construction of ephemeris applied to the planetary case. SSB stands for solar system barycenter (see Sect. 3.2.4), TDB (UTC) stands for Barycentric Dynamical Time (Universal Time Coordinate, respectively) (see Sect. 3.2.2). The description of the dynamical model is the case of planetary ephemerides is done in Sect. 3.3, and the presentation of the observational data sets in Sect. 3.4</figcaption></figure>

accurate estimations of the dynamical states of the considered body, several ingredients are necessary.

First, one needs to agree upon a set of both time and space coordinates given in a properly defined frame. This frame will be preferably inertial and the coordinate system will give an easily understandable representation of the body motion. This coordinate system will also be used to confront the dynamical modeling of the motion to the observations (see in step 3). The selection of the frame—in which the motion and the observations will be described—and its characterisation in space and time (metrics) will constitute the step 1 of the ephemeris construction (red boxes in Fig. 4).

Second, we shall identify and develop the appropriate dynamical model for describing the most accurately the motion of the bodies. This step (black box in Fig. 4) requires the writing of an equation of motion according to the frame defined in the step 1. In the dynamical modeling, one must include all the gravitational perturbations expected for the considered system. A numerical integration of the equations of motion is also usually performed in order to provide to the users positions and velocities for discrete times. Analytical resolutions were also proposed up to the beginning of the 21st century but stopped when the planetary observations became too accurate in regards to the size of the analytical series (see for example Fienga and Simon 2005).

At this step, one may consider that the ephemeris is built. However, in particular in the solar system, the localisation of the object of interest has been monitored by
observers and a comparison between its modeled and observed dynamical states (positions and velocities) is used for improving the model and then, to continue the process of construction of the ephemeris.

It is important to stress that the transformation from computed positions and velocities to observed quantities (direct radar range to the planetary surface, s/c navigation range and frequency shift, angular positions, pulsar time of arrival) requires some hypothesis on the space and time coordinate system in play and on how the observations have been obtained (see for example the Shapiro delay in Sect. 4.2.5 or Sect. 3.5). The data used for building planetary ephemerides are then not independent from the framework used for describing gravity.

In a third step, we consider the information on the positions and/or velocities obtained by the observers. As explained previously, and as it will be described in details in Sect. 3.4 in the case of planetary ephemerides, these informations are deduced from observations that can be ranges, frequency shifts, angular positions relative to reference stellar or quasi-stellar objects (part of astrometric catalogues or not) in different wavelengths (from optical to radio), in most of the cases centered on Earth or on specific locations at its surface. In general, at this step (blue boxes in Fig. 4), the observations are closely analysed in order to remove outliers, and to identify and correct potential biases and systematics.

The fourth step (orange box in Fig. 4) is the confrontation between the observed quantities analysed at step 3 and the same quantities estimated with the dynamical model at step 2. From this comparison, are obtained the residuals, which are the differences between model and observations. In order to have the most accurate modelisation of the observed positions, one looks for minimizing the residuals by considering different causes:

* a mismodeling in the motion of the body. In this case, the step 2 has to be reconsidered and the dynamical modeling modified for example by adding more perturbations.

* parameters used in the model that are not close enough from their real values. The model is then corrected by updating the values of the parameters.

* some systematics or bias in the observations were not accounted for properly and still remain in the residuals. A modification of step 4 is then necessary.

It is of course not easy to isolate the different causes of the residuals. But, nevertheless, in order to reduce the differences obtained at step 4, one systematically starts with correcting the parameters of the model by using either a classic least squares method or a Bayesian approach (see for example the discussion in Sect. 3.5). Once the model has its parameters updated considering the current set of observations, if residuals still present some signatures different from white noise, one can investigate the correctness of the dynamical modeling or the data analysis procedure. This very generic procedure—that can be used for a wide set of natural or artificial bodies—can be sketched by Fig. 4. This figure shows that the definition of the frame in which will be described both the motion and the observations is a crucial
observers and a comparison between its modeled and observed dynamical states (positions and velocities) is used for improving the model and then, to continue the process of construction of the ephemeris.

It is important to stress that the transformation from computed positions and velocities to observed quantities (direct radar range to the planetary surface, s/c navigation range and frequency shift, angular positions, pulsar time of arrival) requires some hypothesis on the space and time coordinate system in play and on how the observations have been obtained (see for example the Shapiro delay in Sect. 4.2.5 or Sect. 3.5). The data used for building planetary ephemerides are then not independent from the framework used for describing gravity.

In a third step, we consider the information on the positions and/or velocities obtained by the observers. As explained previously, and as it will be described in details in Sect. 3.4 in the case of planetary ephemerides, these informations are deduced from observations that can be ranges, frequency shifts, angular positions relative to reference stellar or quasi-stellar objects (part of astrometric catalogues or not) in different wavelengths (from optical to radio), in most of the cases centered on Earth or on specific locations at its surface. In general, at this step (blue boxes in Fig. 4), the observations are closely analysed in order to remove outliers, and to identify and correct potential biases and systematics.

The fourth step (orange box in Fig. 4) is the confrontation between the observed quantities analysed at step 3 and the same quantities estimated with the dynamical model at step 2. From this comparison, are obtained the residuals, which are the differences between model and observations. In order to have the most accurate modelisation of the observed positions, one looks for minimizing the residuals by considering different causes:

* a mismodeling in the motion of the body. In this case, the step 2 has to be reconsidered and the dynamical modeling modified for example by adding more perturbations.

* parameters used in the model that are not close enough from their real values. The model is then corrected by updating the values of the parameters.

* some systematics or bias in the observations were not accounted for properly and still remain in the residuals. A modification of step 4 is then necessary.

It is of course not easy to isolate the different causes of the residuals. But, nevertheless, in order to reduce the differences obtained at step 4, one systematically starts with correcting the parameters of the model by using either a classic least squares method or a Bayesian approach (see for example the discussion in Sect. 3.5). Once the model has its parameters updated considering the current set of observations, if residuals still present some signatures different from white noise, one can investigate the correctness of the dynamical modeling or the data analysis procedure. This very generic procedure—that can be used for a wide set of natural or artificial bodies—can be sketched by Fig. 4. This figure shows that the definition of the frame in which will be described both the motion and the observations is a crucial
step for the prediction of the dynamical status of whatever object, from artificial satellites to quasi-stellar objects.

With the improvement of the measurement accuracy on planetary positions and velocities, the Newtonian paradigm of an universe with flat Cartesian coordinate systems and straight photon path failed to explain the observations. As discussed in introduction, the first evidence of the Newtonian failure was the case of the advance of perihelia of Mercury, explained by Einstein (1915). On Table 1, are compared the accuracies reached by three generations of planetary ephemerides and the advance of perihelia as predicted by general relativity. It is immediately visible that even with the Gaillot ephemerides in the late 19th century (Gaillot 1888; Gaillot and Le Verrier 1913), after three years of observations, the accumulated advance of perihelia for Mercury (1.29 s of arc) is greater than the observational accuracy of this epoch (1 s of arc). This leads to the choice of general relativity as a preferred framework over Newton's laws.

However, other frameworks can also be proposed for describing the most accurate astrometric observations of planets in our solar system. In most cases, these alternative theories tend asymptotically to general relativity in the context of our weak field solar system.

In Sect. 4, we review the alternative gravity models for which dedicated planetary ephemerides have been constructed and published in existing literature. We prioritize these models because we believe that only a fully developed ephemeris, as described in Fig. 4, can conclusively constrain, or even rule out, an alternative theory of gravitation. For a detailed discussion with examples, we refer to Sect. 5.

# 3 Planetary and lunar ephemerides in general relativity

## 3.1 State-of-the-art for planetary and lunar ephemerides

The motion of the planets and asteroids in our solar system can be solved directly by the numerical integration of their equations of motion, or with analytical approximations of their orbits. As it has been shown in Fienga and Simon (2005), the analytical models for the main planet orbits are not accurate enough (due to the limited number of terms in the series) in comparison with the meter level uncertainties reached by the modern observations of planets. Therefore, in the following, we will only consider the planetary ephemerides in their numerical form.

Based on the first preliminary versions of the numerical integration of planetary motions (Devine and Dunham 1966; Ash et al. 1967), the DE96 JPL ephemerides (Standish et al. 1976) was first of the known and widely distributed accurate numerical ephemerides fitted to observations developed by JPL. These were followed by DE102 (Standish 1983), DE200 (Standish 1990), DE403 (Standish 1995) and DE405 (Standish 2001). All these ephemerides are numerically integrated with a variable step-size, variable-order, Adams method. Their dynamical model includes point-mass interactions between the eight planets and Pluto, the Sun and a diverse number of asteroids, relativistic PPN effects (Moyer 1971, 2000) and lunar librations (Newhall et al. 1983). Since DE96, regular improvements have been added to the DE
step for the prediction of the dynamical status of whatever object, from artificial satellites to quasi-stellar objects.

With the improvement of the measurement accuracy on planetary positions and velocities, the Newtonian paradigm of an universe with flat Cartesian coordinate systems and straight photon path failed to explain the observations. As discussed in introduction, the first evidence of the Newtonian failure was the case of the advance of perihelia of Mercury, explained by Einstein (1915). On Table 1, are compared the accuracies reached by three generations of planetary ephemerides and the advance of perihelia as predicted by general relativity. It is immediately visible that even with the Gaillot ephemerides in the late 19th century (Gaillot 1888; Gaillot and Le Verrier 1913), after three years of observations, the accumulated advance of perihelia for Mercury (1.29 s of arc) is greater than the observational accuracy of this epoch (1 s of arc). This leads to the choice of general relativity as a preferred framework over Newton's laws.

However, other frameworks can also be proposed for describing the most accurate astrometric observations of planets in our solar system. In most cases, these alternative theories tend asymptotically to general relativity in the context of our weak field solar system.

In Sect. 4, we review the alternative gravity models for which dedicated planetary ephemerides have been constructed and published in existing literature. We prioritize these models because we believe that only a fully developed ephemeris, as described in Fig. 4, can conclusively constrain, or even rule out, an alternative theory of gravitation. For a detailed discussion with examples, we refer to Sect. 5.

# 3 Planetary and lunar ephemerides in general relativity

## 3.1 State-of-the-art for planetary and lunar ephemerides

The motion of the planets and asteroids in our solar system can be solved directly by the numerical integration of their equations of motion, or with analytical approximations of their orbits. As it has been shown in Fienga and Simon (2005), the analytical models for the main planet orbits are not accurate enough (due to the limited number of terms in the series) in comparison with the meter level uncertainties reached by the modern observations of planets. Therefore, in the following, we will only consider the planetary ephemerides in their numerical form.

Based on the first preliminary versions of the numerical integration of planetary motions (Devine and Dunham 1966; Ash et al. 1967), the DE96 JPL ephemerides (Standish et al. 1976) was first of the known and widely distributed accurate numerical ephemerides fitted to observations developed by JPL. These were followed by DE102 (Standish 1983), DE200 (Standish 1990), DE403 (Standish 1995) and DE405 (Standish 2001). All these ephemerides are numerically integrated with a variable step-size, variable-order, Adams method. Their dynamical model includes point-mass interactions between the eight planets and Pluto, the Sun and a diverse number of asteroids, relativistic PPN effects (Moyer 1971, 2000) and lunar librations (Newhall et al. 1983). Since DE96, regular improvements have been added to the DE
ephemerides. Ephemerides such as DE421 (Folkner et al. 2008), DE430 (Folkner et al. 2014) and DE440 (Park et al. 2021) have been constructed and fitted with increasingly dense sets of space mission tracking data. Numerical ephemerides have also been developed at the Institute of Applied Astronomy of the Russian Academy of Sciences (EPM) and at the Observatory of Paris and the Côte d'Azur Observatory (INPOP). They are based on a dynamical model similar to the JPL one but with specific characteristics, in particular regarding the interactions between the main planets and the asteroids. Several possible additional contributions have been included in the EPM ephemerides such as the interactions of Trans-Neptunian Objects (TNOs) by the mean of one or several rings (Pitjeva and Pitjev 2018) and the influence with the Jupiter Trojens (Pitjeva and Pitjev 2020). The EPM ephemerides, are fitted to optical, radar and space tracking data and have an accuracy comparable to the JPL ephemerides (Krasinsky et al. 1988; Krasinskii et al. 1993; Pitjeva 2001, 2005a; Pitjeva and Pitjev 2014). They have been intensively used for estimating PPN parameters and the hypothetical secular variation of the gravitational constant (Pitjeva 1993, 2005b; Pitjeva et al. 2021). Since 2003, INPOP planetary ephemerides are developed, integrating numerically the Einstein-Infeld-Hoffman-Droste-Lorentz (EIHDL) equation, as proposed by Moyer (1971, 2000) (see Sect. 3), and fitting the parameters of the dynamical model to the most accurate planetary observations. The main INPOP releases are INPOP08 (Fienga et al. 2009), INPOP10a (Fienga et al. 2011a), INPOP17a (Viswanathan et al. 2017), and INPOP19a (Fienga et al. 2019). For this family of ephemerides, a specific care has been brought on the consistency between the GR framework defined by the IAU and the actual equations of motion and time-scale used in the ephemerides. In particular in 2009, INPOP08 (Fienga et al. 2009) was the first ephemeris built with consistent planetary orbits and time-scales (see Sect. 3.2.2). In 2010, INPOP10a (Fienga et al. 2011a) was also the first to fit the gravitational mass of the Sun instead of the astronomical unit for consistency reasons.

These three families of planetary ephemerides differ in the dynamical model (see Sect. 3.3.2) such as the number of point-mass objects (more or less main-belt asteroids, TNOs) to consider in the EIHDL point-mass interaction, additional accelerations to implement (Lense-Thirring acceleration, TNOs rings, Trojan rings, etc.) as well as in the size of the planetary datasets used for the adjustments of the models (see Sect. 3.4) and by the way this adjustment is performed (see Sect. 3.5). Table 2 summarises these distinctions. However, it is important to stress that, despite their specific characteristics, the accuracy of these models are very close to each other and involve towards an even closer consistency.

## 3.2 Reference frame theory in general relativity

The general relativistic framework of the planetary ephemerides since 2006 is the one summarized by the IAU2000 and IAU2006 conventions (Soffel 2003; Petit and Luzum 2010).

Because general relativity is a covariant theory, an infinite set of coordinate systems could be used in principle in order to describe space-time events—see Sect. 2.1. The International Astronomical Union (IAU) has therefore set the standard
ephemerides. Ephemerides such as DE421 (Folkner et al. 2008), DE430 (Folkner et al. 2014) and DE440 (Park et al. 2021) have been constructed and fitted with increasingly dense sets of space mission tracking data. Numerical ephemerides have also been developed at the Institute of Applied Astronomy of the Russian Academy of Sciences (EPM) and at the Observatory of Paris and the Côte d'Azur Observatory (INPOP). They are based on a dynamical model similar to the JPL one but with specific characteristics, in particular regarding the interactions between the main planets and the asteroids. Several possible additional contributions have been included in the EPM ephemerides such as the interactions of Trans-Neptunian Objects (TNOs) by the mean of one or several rings (Pitjeva and Pitjev 2018) and the influence with the Jupiter Trojens (Pitjeva and Pitjev 2020). The EPM ephemerides, are fitted to optical, radar and space tracking data and have an accuracy comparable to the JPL ephemerides (Krasinsky et al. 1988; Krasinskii et al. 1993; Pitjeva 2001, 2005a; Pitjeva and Pitjev 2014). They have been intensively used for estimating PPN parameters and the hypothetical secular variation of the gravitational constant (Pitjeva 1993, 2005b; Pitjeva et al. 2021). Since 2003, INPOP planetary ephemerides are developed, integrating numerically the Einstein-Infeld-Hoffman-Droste-Lorentz (EIHDL) equation, as proposed by Moyer (1971, 2000) (see Sect. 3), and fitting the parameters of the dynamical model to the most accurate planetary observations. The main INPOP releases are INPOP08 (Fienga et al. 2009), INPOP10a (Fienga et al. 2011a), INPOP17a (Viswanathan et al. 2017), and INPOP19a (Fienga et al. 2019). For this family of ephemerides, a specific care has been brought on the consistency between the GR framework defined by the IAU and the actual equations of motion and time-scale used in the ephemerides. In particular in 2009, INPOP08 (Fienga et al. 2009) was the first ephemeris built with consistent planetary orbits and time-scales (see Sect. 3.2.2). In 2010, INPOP10a (Fienga et al. 2011a) was also the first to fit the gravitational mass of the Sun instead of the astronomical unit for consistency reasons.

These three families of planetary ephemerides differ in the dynamical model (see Sect. 3.3.2) such as the number of point-mass objects (more or less main-belt asteroids, TNOs) to consider in the EIHDL point-mass interaction, additional accelerations to implement (Lense-Thirring acceleration, TNOs rings, Trojan rings, etc.) as well as in the size of the planetary datasets used for the adjustments of the models (see Sect. 3.4) and by the way this adjustment is performed (see Sect. 3.5). Table 2 summarises these distinctions. However, it is important to stress that, despite their specific characteristics, the accuracy of these models are very close to each other and involve towards an even closer consistency.

## 3.2 Reference frame theory in general relativity

The general relativistic framework of the planetary ephemerides since 2006 is the one summarized by the IAU2000 and IAU2006 conventions (Soffel 2003; Petit and Luzum 2010).

Because general relativity is a covariant theory, an infinite set of coordinate systems could be used in principle in order to describe space-time events—see Sect. 2.1. The International Astronomical Union (IAU) has therefore set the standard
<table><caption>**Table 2** Solve-for parameters for recent ephemerides. S01 (Standish 2001), F08 (Folkner et al. 2008), F14 (Folkner et al. 2014), P17 (Park et al. 2017), P05 (Pitjeva 2005a), P14 (Pitjeva and Pitjev 2014), P18 (Pitjeva and Pitjev 2018), P21 (Pitjeva et al. 2021), F09 stands for Fienga et al. (2009), F11 for Fienga et al. (2011a), F19 (Fienga et al. 2019), F21 (Fienga et al. 2021a). Column 3 and 4 indicate the model used for describing the perturbations induced by the main belt asteroids as well as the number of corresponding parameters where Columns 5 and 6 give the same informations but for Trans-Neptunian objects (Sect. 3.3.2). Column 7 indicates which ephemeris accounts for the Lense-Thirring acceleration (see Sect. 3.3.3) and Column 8 if the ephemeris was built with the astronomical unit (AU) fitted or with the gravitational mass of the sun fitted (GM<sub>☉</sub>). It also gives if the integration of the TT-TDB equation was done simultaneously with the equations of motion (see Sect. 3.2.2). Finally, Column 9 gives the time interval covered by the observations used for the fit</caption><thead><tr><th rowspan="2"></th><th rowspan="2">References</th><th colspan="2">Main belt asteroids</th><th colspan="2">TNO</th><th rowspan="2">Others</th><th rowspan="2">Period</th></tr><tr><th>Model</th><th>Fit</th><th>Model</th><th>Fit</th><th>LT</th></tr></thead><tbody><tr><td colspan="8"><strong>DE</strong></td></tr><tr><td>DE405</td><td>S01</td><td>300</td><td>3 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU</td><td>1924:1998</td></tr><tr><td>DE421</td><td>F08</td><td>343</td><td>11 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU</td><td>1924:2007</td></tr><tr><td>DE430</td><td>F14</td><td>343</td><td>343 GM</td><td>N</td><td>N</td><td>N</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2018</td></tr><tr><td>DE440</td><td>P17</td><td>343</td><td>343 GM</td><td>36 + 1 ring</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2020</td></tr><tr><td colspan="8"><strong>EPM</strong></td></tr><tr><td>EPM2004</td><td>P05</td><td>301 + 1 ring</td><td>6 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU</td><td>1913: 2004</td></tr><tr><td>EPM2011</td><td>P14</td><td>301 + 1 ring</td><td>21 GM +<br/>3 densities</td><td>21 + 1 ring</td><td>1</td><td>N</td><td>AU, TDB</td><td>1913:2011</td></tr><tr><td>EPM2017</td><td>P18</td><td>301 + 1 ring</td><td>21 GM +<br/>3 densities</td><td>30 + 3<br/>rings</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1913:2016</td></tr><tr><td>EPM2019</td><td>P21</td><td>301 + 1 ring<br/>+ 2 Trojan<br/>groups</td><td>24 GM +<br/>3 densities</td><td>30 + 3<br/>rings</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1913:2017</td></tr><tr><td colspan="8"><strong>INPOP</strong></td></tr><tr><td>INPOP08a</td><td>F09</td><td>300 + 1 ring</td><td>34 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU, TDB</td><td>1913:2007</td></tr><tr><td>INPOP10a</td><td>F11</td><td>289 + 1 ring</td><td>145 GM</td><td>N</td><td>N</td><td>N</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1913:2010</td></tr><tr><td>INPOP19a</td><td>F19</td><td>343</td><td>343 GM</td><td>9 + 3 rings</td><td>1</td><td>N</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2019</td></tr><tr><td>INPOP21a</td><td>F21</td><td>343</td><td>343 GM</td><td>509</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2020</td></tr></tbody></table>
<table><caption>**Table 2** Solve-for parameters for recent ephemerides. S01 (Standish 2001), F08 (Folkner et al. 2008), F14 (Folkner et al. 2014), P17 (Park et al. 2017), P05 (Pitjeva 2005a), P14 (Pitjeva and Pitjev 2014), P18 (Pitjeva and Pitjev 2018), P21 (Pitjeva et al. 2021), F09 stands for Fienga et al. (2009), F11 for Fienga et al. (2011a), F19 (Fienga et al. 2019), F21 (Fienga et al. 2021a). Column 3 and 4 indicate the model used for describing the perturbations induced by the main belt asteroids as well as the number of corresponding parameters where Columns 5 and 6 give the same informations but for Trans-Neptunian objects (Sect. 3.3.2). Column 7 indicates which ephemeris accounts for the Lense-Thirring acceleration (see Sect. 3.3.3) and Column 8 if the ephemeris was built with the astronomical unit (AU) fitted or with the gravitational mass of the sun fitted (GM<sub>☉</sub>). It also gives if the integration of the TT-TDB equation was done simultaneously with the equations of motion (see Sect. 3.2.2). Finally, Column 9 gives the time interval covered by the observations used for the fit</caption><thead><tr><th rowspan="2"></th><th rowspan="2">References</th><th colspan="2">Main belt asteroids</th><th colspan="2">TNO</th><th colspan="2">Others</th><th rowspan="2">Period</th></tr><tr><th>Model</th><th>Fit</th><th>Model</th><th>Fit</th><th>LT</th><th>LT</th></tr></thead><tbody><tr><td colspan="9"><strong>DE</strong></td></tr><tr><td>DE405</td><td>S01</td><td>300</td><td>3 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU</td><td>1924:1998</td></tr><tr><td>DE421</td><td>F08</td><td>343</td><td>11 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU</td><td>1924:2007</td></tr><tr><td>DE430</td><td>F14</td><td>343</td><td>343 GM</td><td>N</td><td>N</td><td>N</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2018</td></tr><tr><td>DE440</td><td>P17</td><td>343</td><td>343 GM</td><td>36 + 1 ring</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2020</td></tr><tr><td colspan="9"><strong>EPM</strong></td></tr><tr><td>EPM2004</td><td>P05</td><td>301 + 1 ring</td><td>6 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU</td><td>1913: 2004</td></tr><tr><td>EPM2011</td><td>P14</td><td>301 + 1 ring</td><td>21 GM +<br/>3 densities</td><td>21 + 1 ring</td><td>1</td><td>N</td><td>AU, TDB</td><td>1913:2011</td></tr><tr><td>EPM2017</td><td>P18</td><td>301 + 1 ring</td><td>21 GM +<br/>3 densities</td><td>30 + 3<br/>rings</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1913:2016</td></tr><tr><td>EPM2019</td><td>P21</td><td>301 + 1 ring<br/>+ 2 Trojan<br/>groups</td><td>24 GM +<br/>3 densities</td><td>30 + 3<br/>rings</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1913:2017</td></tr><tr><td colspan="9"><strong>INPOP</strong></td></tr><tr><td>INPOP08a</td><td>F09</td><td>300 + 1 ring</td><td>34 GM +<br/>3 densities</td><td>N</td><td>N</td><td>N</td><td>AU, TDB</td><td>1913:2007</td></tr><tr><td>INPOP10a</td><td>F11</td><td>289 + 1 ring</td><td>145 GM</td><td>N</td><td>N</td><td>N</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1913:2010</td></tr><tr><td>INPOP19a</td><td>F19</td><td>343</td><td>343 GM</td><td>9 + 3 rings</td><td>1</td><td>N</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2019</td></tr><tr><td>INPOP21a</td><td>F21</td><td>343</td><td>343 GM</td><td>509</td><td>1</td><td>Y</td><td>GM<sub>☉</sub>,<br/>TDB</td><td>1924:2020</td></tr></tbody></table>
coordinate systems that people are recommended to use, notably through the IAU2000 recommendations (Soffel 2003). Two main reference systems have been defined, as well as the transformation between one another: the barycentric celestial reference system (BCRS) and the geocentric celestial reference system (GCRS). Both reference systems are defined at the post-Newtonian level and use the harmonic gauge. Beyond the harmonic gauge condition, the freedom in choosing the coordinate systems is further reduced by fixing the form and properties of the metric and the gravitational potentials.

Planetary ephemerides are integrated in the BCRS and are linked to the realization of ICRS, by VLBI observations of s/c orbiting planets (see Sect. 3.2.3).

### 3.2.1 General relativity barycentric metric

The BCRS is defined with the coordinates $(ct, x^i)$, where $t=\text{TCB}$ (see Sect. 3.2.2). The metric is taken to be kinematically fixed with respect to distant quasi stellar objects (QSO). The catalog gathering QSO astrometric positions and velocities used as fixed standards for the definition of the kinematically fixed BCRS is the International Celestial Reference Frame (ICRF) (Ma et al. 1998; Fey et al. 2015; Charlot et al. 2020). The general form of the BCRS metric is taken to be the following Soffel (2003) (see Sect. 2.1.7 for details on its derivation)

$$ 
\begin{aligned}
g_{00} &= -1 + \frac{2w}{c^2} - \frac{2w^2}{c^4} + \mathcal{O}(c^{-5}) \\
g_{0i} &= -\frac{4w_i}{c^3} + \mathcal{O}(c^{-4}) \\
g_{ij} &= \delta_{ij} \left(1 + \frac{2w}{c^2}\right) + \mathcal{O}(c^{-3}),
\end{aligned} 
\qquad (80) $$

where $w$ and $w_i$ are respectively a scalar gravitational potential and a vector potential, $c$ being the speed of light. The harmonic gauge conditions then imply that the potentials $w$ and $w_i$ satisfy the following equations:

$$ 
\begin{aligned}
\left(-\frac{1}{c^2} \frac{\partial^2}{\partial t^2} + \nabla^2\right) w &= -4\pi G\sigma + \mathcal{O}(c^{-4}), \\
\nabla^2 w^i &= -4\pi G\sigma^i + \mathcal{O}(c^{-2}),
\end{aligned} 
\qquad (81) $$

where $\sigma$ and $\sigma^i$ are the gravitational mass and mass current defined upon the stress-energy tensor:

$$ \sigma = \frac{1}{c^2} (T^{00} + T^{ss}), \quad \sigma^i = \frac{1}{c} T^{0i}. \qquad (83) $$

In this definition, the gravitational perturbations induced by other bodies in the vicinity of the solar system (stars, galaxies, dark matter, dark energy) are ignored. The solar system is considered as an isolated system—which is possible in general relativity, thanks to the equivalence principle and the resulting “effacement of
coordinate systems that people are recommended to use, notably through the IAU2000 recommendations (Soffel 2003). Two main reference systems have been defined, as well as the transformation between one another: the barycentric celestial reference system (BCRS) and the geocentric celestial reference system (GCRS). Both reference systems are defined at the post-Newtonian level and use the harmonic gauge. Beyond the harmonic gauge condition, the freedom in choosing the coordinate systems is further reduced by fixing the form and properties of the metric and the gravitational potentials.

Planetary ephemerides are integrated in the BCRS and are linked to the realization of ICRS, by VLBI observations of s/c orbiting planets (see Sect. 3.2.3).

### 3.2.1 General relativity barycentric metric

The BCRS is defined with the coordinates $(ct, x^i)$, where $t=\text{TCB}$ (see Sect. 3.2.2). The metric is taken to be kinematically fixed with respect to distant quasi stellar objects (QSO). The catalog gathering QSO astrometric positions and velocities used as fixed standards for the definition of the kinematically fixed BCRS is the International Celestial Reference Frame (ICRF) (Ma et al. 1998; Fey et al. 2015; Charlot et al. 2020). The general form of the BCRS metric is taken to be the following Soffel (2003) (see Sect. 2.1.7 for details on its derivation)

$$ 
\begin{aligned}
g_{00} &= -1 + \frac{2w}{c^2} - \frac{2w^2}{c^4} + \mathcal{O}(c^{-5}) \\
g_{0i} &= -\frac{4w_i}{c^3} + \mathcal{O}(c^{-4}) \\
g_{ij} &= \delta_{ij} \left(1 + \frac{2w}{c^2}\right) + \mathcal{O}(c^{-3}),
\end{aligned} 
\quad (80) $$

where $w$ and $w_i$ are respectively a scalar gravitational potential and a vector potential, $c$ being the speed of light. The harmonic gauge conditions then imply that the potentials $w$ and $w_i$ satisfy the following equations:

$$ 
\begin{aligned}
\left(-\frac{1}{c^2} \frac{\partial^2}{\partial t^2} + \nabla^2\right) w &= -4\pi G\sigma + \mathcal{O}(c^{-4}), \\
\nabla^2 w^i &= -4\pi G\sigma^i + \mathcal{O}(c^{-2}),
\end{aligned} 
\quad (81) $$

where $\sigma$ and $\sigma^i$ are the gravitational mass and mass current defined upon the stress-energy tensor:

$$ \sigma = \frac{1}{c^2} (T^{00} + T^{ss}), \quad \sigma^i = \frac{1}{c} T^{0i}. \quad (83) $$

In this definition, the gravitational perturbations induced by other bodies in the vicinity of the solar system (stars, galaxies, dark matter, dark energy) are ignored. The solar system is considered as an isolated system—which is possible in general relativity, thanks to the equivalence principle and the resulting “effacement of”
<figure><img src="image_7.png" /><figcaption>Fig. 5 Various relativistic time-scales and their relations. Each of the coordinate time-scales TCB, TCG, TT and TDB can be related to the proper time $\tau_A$ of an observer A, provided that the trajectory of the observer in the BCRS and/or GCRS is known (Petit and Luzum 2010). Dashed lines represent transformations with fixed rates, whereas full lines represent transformations that depend on the metric potentials, following to the IAU recommendations (Soffel 2003)</figcaption></figure>

internal degrees of freedom in the global problem and of the external world in the local system” (Damour 1989; Klioner and Soffel 2000), but not in general for alternative theories of gravity—e.g. not in MOND (Milgrom 2009, 2014), see Sect. 4.7. For a solar system composed by non-rotating point-mass objects, the previous barycentric potentials are $w = \sum_A w_A$ and $w^i = \sum_A w_A^i$ with

$$w_A = \frac{\mu_A}{r_A} \left[ 1 + 2 \frac{v_A^2}{c^2} - \frac{1}{c^2} \sum_{B \neq A} \frac{\mu_B}{r_{BA}} - \frac{1}{c^2} \left( \frac{(\mathbf{r}_A \cdot \mathbf{v}_A)^2}{r_A^2} + \mathbf{r}_A \cdot \mathbf{a}_A \right) \right], \quad (84)$$

$$w_A^i = \frac{\mu_A}{r_A} v_A^i,$$

where $\mu_A := Gm_A$ is the gravitational parameter of the body A, $r_{AT}$ is the relative position of body T with respect to A, $r_{AT} = |\mathbf{r}_{AT}|$ and $v_A$ is the coordinate velocity of body A while $\mathbf{a}_A$ is its coordinate acceleration in the BCRS.

The same type of framework can be defined for a reference system centered on the Earth center of mass and leads to the definition of the Geocentric Celestial Reference System (GCRS). The GCRS is suitable in practice for the modeling of processes in the vicinity of the Earth, whereas the metrics of Eqs. (80) and (84) will be used for modeling the light propagation and motion of celestial objects in the solar system in the BCRS.

The coordinate transformations between the BCRS and the GCRS involve a complicated set of functions that are defined in the resolution B1.3 of the IAU2000 resolution (Soffel 2003).

### 3.2.2 Time-scales in the solar system

The time-scales in the BCRS and GCRS are denoted by TCB and TCG, respectively (Soffel 2003). The relation between TCB and TCG are given in Soffel (2003). At the Earth level, the Terrestrial Time (TT) has been defined in order to remain close to the realized atomic time (TAI). It differs from the TCG by a constant rate (Soffel 2003).
<figure><img src="image_8.png" /><figcaption>Fig. 5 Various relativistic time-scales and their relations. Each of the coordinate time-scales TCB, TCG, TT and TDB can be related to the proper time $\tau_A$ of an observer A, provided that the trajectory of the observer in the BCRS and/or GCRS is known (Petit and Luzum 2010). Dashed lines represent transformations with fixed rates, whereas full lines represent transformations that depend on the metric potentials, following to the IAU recommendations (Soffel 2003)</figcaption></figure>

internal degrees of freedom in the global problem and of the external world in the local system” (Damour 1989; Klioner and Soffel 2000), but not in general for alternative theories of gravity—e.g. not in MOND (Milgrom 2009, 2014), see Sect. 4.7. For a solar system composed by non-rotating point-mass objects, the previous barycentric potentials are $w = \sum_A w_A$ and $w^i = \sum_A w_A^i$ with

$$w_A = \frac{\mu_A}{r_A} \left[ 1 + 2 \frac{v_A^2}{c^2} - \frac{1}{c^2} \sum_{B \neq A} \frac{\mu_B}{r_{BA}} - \frac{1}{c^2} \left( \frac{(\mathbf{r}_A \cdot \mathbf{v}_A)^2}{r_A^2} + \mathbf{r}_A \cdot \mathbf{a}_A \right) \right], \quad (84)$$

$$w_A^i = \frac{\mu_A}{r_A} v_A^i,$$

where $\mu_A := Gm_A$ is the gravitational parameter of the body A, $r_{AT}$ is the relative position of body T with respect to A, $r_{AT} = |\mathbf{r}_{AT}|$ and $v_A$ is the coordinate velocity of body A while $\mathbf{a}_A$ is its coordinate acceleration in the BCRS.

The same type of framework can be defined for a reference system centered on the Earth center of mass and leads to the definition of the Geocentric Celestial Reference System (GCRS). The GCRS is suitable in practice for the modeling of processes in the vicinity of the Earth, whereas the metrics of Eqs. (80) and (84) will be used for modeling the light propagation and motion of celestial objects in the solar system in the BCRS.

The coordinate transformations between the BCRS and the GCRS involve a complicated set of functions that are defined in the resolution B1.3 of the IAU2000 resolution (Soffel 2003).

### 3.2.2 Time-scales in the solar system

The time-scales in the BCRS and GCRS are denoted by TCB and TCG, respectively (Soffel 2003). The relation between TCB and TCG are given in Soffel (2003). At the Earth level, the Terrestrial Time (TT) has been defined in order to remain close to the realized atomic time (TAI). It differs from the TCG by a constant rate (Soffel 2003).