# GPS observables in general relativity

Carlo Rovelli

► To cite this version:

Carlo Rovelli. GPS observables in general relativity. ACTA Futura, 2013, 7 (4), pp.97-101. 10.2420/AF07.2013.97. hal-00957714

HAL Id: hal-00957714

https://hal.science/hal-00957714v1

Submitted on 29 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
# GPS observables in general relativity

CARLO ROVELLI*

Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France

Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France

**Abstract.** I review the motivations and the main ideas that have lead to the introduction of relativistic positioning systems in the form of the so called “GPS” variables. These variables allow us to define a *gauge invariant* metric, and a gauge-invariant notion of length, curvature... in the context of general relativity. I show explicitly how the metric can be measured locally.

## 1 GPS coordinates

It is common to read that the metric is not an observable in general relativity (GR), because it is not gauge invariant, or, equivalently, in not invariant under coordinate transformations. But this is not always true. When the coordinates are defined in terms of physical degrees of freedom, the quantities that depend on these physical coordinates are gauge-invariant, and observable. The GPS coordinates [11] are particularly simple physical coordinates, that are easy to implement concretely and realistically. In fact, they are already implicitly implemented concretely by the global positioning systems, such as the Global Navigation Systems, or by pulsar signal receivers. To define these coordinate theoretically, indeed, it suffices to have four objects in spacetime, and a way to track light signals.

Furthermore, the GPS coordinates of a spacetime point depend only on (the closure of) the past of that

point. As a consequence, evolution equations in these coordinates are *local*. A remarkable property that is absent in most other manners to define physical coordinates.

Previous attempts to define a complete class of observables abound in the literature. For instance, in the presence of matter we can localize things with respect to the matter [15, 10, 4]. Other (earlier) attempts to write physical coordinates in GR is to use curvature scalars [3]. The result is however mathematically very intricate and physically extremely unrealistic.

GPS coordinates, introduced in [11], are much simpler: consider a general covariant system including the gravitational field as well as four small bodies. These are taken to have negligible mass; they will be considered as point particles for simplicity, and called “satellites”. They follow timelike worldliness, and we assume that a preferred point $O_\alpha$, $\alpha = 1, 2, 3, 4$ is marked in each of them. Then (there is a region $\mathcal{R}$ of spacetime for which) we can uniquely associate four numbers $s^\alpha$ to each spacetime point $p$ as follows. The past lightcone of $p$ intersect the four geodesics in four points $p_\alpha$. The numbers $s^\alpha$ are defined as the time lapsed between $O_\alpha$ and $p_\alpha$ along the $\alpha$ world line. We can use the $s^\alpha$'s as physically defined coordinates.

The components $g_{\alpha\beta}(s)$ of the metric tensor in these coordinates are observable quantities. They are invariant under four-dimensional diffeomorphisms (because, of course, these deform the metric as well as the satel-

<footnote>*E-mail: rovelli@cpt.univ-mrs.fr</footnote>
<figure><img src="image_2.png" /><figcaption>FIGURE 1. s₁ and s₂ are the GPS coordinates of the point p. Σ is a Cauchy surface with p in its future domain of dependence.</figcaption></figure>

lites' worldlines). They define a complete set of observables for the region $\mathcal{R}$.

The physical picture is simple, and its realism is transparent. Imagine that the four “satellites” are in fact satellites, each carrying a clock that measures the proper time along its trajectory, starting at $O_\alpha$. Imagine also that each satellite broadcasts its local time with a radio signal. At the point p, a receiver can receives the four signals and displays the four readings. See Figure 1. These four numbers are precisely the four physical coordinates $s^\alpha$ defined above. Current technology permits to perform these measurements with accuracy well within the relativistic regime [2, 5, 6, 7, 8, 12, 13, 14]. A rod and a clock measuring physical distances between $s^\alpha$ coordinates points, measure the components of the metric tensor in the physical coordinate system. (In the terminology of Ref. [10], the $s^\alpha$'s are partial observables, while $g_{\alpha\beta}(s)$ are complete observables.)

The physical coordinates $s^\alpha$ have nice geometrical properties. First, they are characterized by

$$g^{\alpha\alpha}(s) = 0, \quad \alpha = 1, \dots, 4. \qquad (1)$$

This is because the constant $s^\alpha$ surface, for a given $\alpha$, is null, therefore the surface element has null volume, and $g^{\alpha\alpha}$ is the volume element of this surface. Second, the evolution equations for $g_{\alpha\beta}(s)$ are local, as shown in [11]. These evolution equations can be written explicitly

using the Arnowitt-Deser-Misner (ADM) variables [1] and Lapse and Shift turn out to be fixed local functions of the three metric.

The coordinates $s^\alpha$ are null coordinates, but we can easily use them to define more common space and time coordinates in a region. For instance, we can define $s^\mu$ around some point p by

$$s^\mu = W_\alpha^\mu \delta s^\alpha \qquad (2)$$

where $\delta s^\alpha$ is the difference with the respect of the coordinates of p and $W_\alpha^\mu$ is a fixed matrix that transforms null direction is space and time directions.

## 2 Measuring the metric

How to measure the spacetime metric with local measurements? Here I recall that this is possible, and how it can be done.

The setting is the following. Assume that the spacetime region R of interest is described by a metric field $g_{\mu\nu}(x)$ which is not known. The only assumption about this metric is that in the region of interest the maximum curvature scale is given by a value A, which we assume to be known. In other words, the metric is not too “rough”, below a given scale. Assume that somewhere in the universe there are four emitters that broadcast a signal, which can be received in the region R. Concretely, this can be a Navigation System-like artificial signal broadcasting the emission proper time, or the periodic signal of a pulsar. Assume that the measuring device for measuring the metric is a swarm of freely falling flying objects equipped with the following devices:

* a clock measuring proper time along the world line of each device,

* a receiving antenna that detects the signals of the four emitters and reads the emitter's proper time. In the case of the pulsar, I assume that onboard clock can resolve time by a greater precision than the millisecond pulsar period, by simply linearly interpolating between pulses.

* a laser ranging technology that allows each flying object to track the distance of its immediate neighbors in the swarm, by sending a light signal to the neighbor, and measuring the proper time between emission and reception of the reflected signal.

* a way to detect the moment at which the radar signal emitted by a neighboring device is reflected.
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Local measurements. The blue lines are the external emitters, the grid gives the null coordinates, the purple lines are the free falling objects.</figcaption></figure>

With these data, the metric can be reconstructed with arbitrary precision, if the average distance between the flying devices and the clock's accuracy are sufficiently small with respect to A.

This is how the reconstruction works. On a given object O, at some given moment, let $s^\alpha = (s^1, s^2, s^3, s^4)$ be the reading of the four signals emitted by the four external emitters (interpolated is necessary for the pulsars), that is, GPS coordinates. These readings will be used as a concrete coordinatization of the points in R. At some given moment p, let this reading be $s_p^\alpha$. Wait for a proper time T (much shorter than A) and take the new reading of the four signals. Let this be $s_T^\alpha = s_p^\alpha + \Delta s_T^\alpha$. We can use the basic definition of the metric field, which is

$$ds^2 = g_{\mu\nu}(x)dx^\mu dx^\nu \quad (3)$$

and since we have assumed that T is much shorter than A, we can treat the finite $\Delta s_T^\alpha$ intervals as infinitesimals, and get a condition for the components $g_{\alpha\beta}(s)$ of the metric in the coordinate system defined by the coordinates $s^\alpha$:

$$T = g_{\alpha\beta}(s_p) \Delta s_T^\alpha \Delta s_T^\beta. \quad (4)$$

Next, at the same moment $s_p^\alpha$, a signal was sent from O to the three of the neighboring devices on the swarm (labelled $\alpha = 1, 2, 3$), was received back on O after a time $2T_\alpha$, and the coordinates of the neighboring object at the moment of the reflection is recorded (on the object

$\alpha$, and later communicated) and called $s_\alpha^\alpha = s_p^\alpha + \Delta s_\alpha^\alpha$. Then, as before, this implies

$$cT_\alpha = -g_{\alpha\beta}(s_p)\Delta s_\alpha^\alpha \Delta s_\beta^\alpha. \quad (5)$$

Next, with the same technology, let $cT_{ab}$ be the distance between the object $\alpha$ and the object $b$ at the reception time. Since these distances are much smaller than A, we can consider spacetime locally flat and from the three lengths of the three sides of the triangle formed by the three objects O, $\alpha$ and $b$, we can compute the angle $\phi_{ab}$ at of this triangle at O (which is the angle under which $\alpha$ and $b$ at seen by O). This is given by Hero formula

$$T_\alpha T_\beta \cos \phi_{ab} = T_\alpha^2 + T_\beta^2 - T_{ab}^2, \quad (6)$$

So we have immediately one more condition on the metric

$$c(T_\alpha^2 + T_\beta^2 - T_{ab}^2) = -g_{\alpha\beta}(s_p) \Delta s_\alpha^\alpha \Delta s_\beta^\beta. \quad (7)$$

Finally, let's observe that the surfaces where one coordinate is constant are null surfaces, therefore the norm of their covariant normal vanishes, as this is given by the diagonal matrix elements of the controvariant metric, we have (no summation)

$$0 = g^{\alpha\alpha}(s_p). \quad (8)$$

Now in these equations, all the quantities $T$, $T_\alpha$, $T_{ab}$, $\Delta s_T^\alpha$, $\Delta s_\alpha^\alpha$, $s_p^\alpha$ are concretely measurable. The $s^\alpha$ and $\Delta s^\alpha$ quantities are readings form the four distant emitters (or given by counting pulsate pulses plus interpolation), while the times $T$, $T_\alpha$, $T_{ab}$ are directly read by onboard clocks. Therefore the equations can be solved for $g_{\alpha\beta}(s_p)$ at the given point $s_p^\alpha$. This can be done on a grid of points. If the grid is fine enough, the metric can be reconstructed with arbitrary precision. Notice that the individual components $g_{\alpha\beta}(s)$ of the metric in the coordinates $s^\alpha$ are gauge-invariant quantities in general relativity: they are uniquely determined physically by the matter and the gravitational field. Therefore they observable quantities, in the sense of gauge theories. Here I have shown how they can be concretely measured.

Where is the curvature? Within each cell defined by the grid, there cannot be curvature, by assumption, of course: if in reality there is curvature, it is averaged away by the measuring procedure. But at the immediately higher scale, curvature is immediately detected. To see this, imagine for a moment that we focus for simplicity on spacial curvature, which is easier to have intuition
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Curvature is given by the difference between 2π and $\varphi_1 + \varphi_2 + \varphi_3$.</figcaption></figure>

about. Consider two objects, say a and b. And a ring of three other objects labelled as 1, 2, 3 forming a triangle that surrounds the line between a and b. The four devices a, b, 1, 2 form a tetrahedron. In a tetrahedron, the dihedral angle $\varphi_{ab}$ at the side (a, b) can be computed explicitly from the lengths of the sides. It is given by the formula

$$ \cos \varphi_{ab} = \frac{\cos \varphi_{a12} - \cos \varphi_{ab1} \cos \varphi_{ab2}}{\sin \varphi_{ab1} \sin \varphi_{ab2}}, \quad (9) $$

where $\varphi_{abc}$ is the angle at a in the abc triangle. Let $\varphi_1$ be the dihedral angle at the side ab of the tetrahedron a, b, 2, 3 and so on cyclically. Then clearly if space is flat the three dihedral angles must sum up to $2\pi$. If they don't, this means that there is curvature at the side ab. Therefore, following Regge, we can define the deficit angle

$$ \delta_{ab} = 2\pi - \sum_{i,1}^{3} \varphi_i. \quad (10) $$

The deficit angle is the discrete version of the curvature. (It is given by the components of the Riemann tensor that rotate in the plane normal to the side, and integrated in a plane normal to the side: $\delta_{ab} \sim R_{\beta\gamma\theta}^{\alpha} \Delta s_{1\alpha} \Delta s_{2\beta} \Delta s_{1\gamma} \Delta s_{2\theta}$) The dihedral angle is an explicit function of the measured quantities. Hence these measures detect curvature systematically. The same construction (which I do not give here for simplicity) works in four dimensions.

In conclusion, given GPS coordinates and a swarm of objects equipped with clocks and capable of tracking the relative distances from their immediate neighbors via laser tracking, the spacetime geometry of a region can be fully reconstructed down to the scale of the average objects' distance.

It is may be useful to recall here the distinction between partial and complete observables [10]. A partial observable is a quantity to which a measuring procedure can be associated. A complete observable is an observable quantity that can predicted by the theory, or, equivalently, whose knowledge provides information on the state of the system. The GPS coordinates are partial observables: we can associate them a measuring procedure (this is what has been done in this paper), but we can of course not "predict" $s^1$. The complete observables, or true observables, are the quantities $g_{\mu\nu}(s)$, for any given value of the coordinates $s^1$. These quantities are diffeomorphism invariant, are uniquely determined by the initial data and in a canonical formulation are represented by functions on the phase space that commute with all constraints.

These observables are straightforward generalization of Einstein's "point coincidences"<fnref n="1" />. In a sense, they are *precisely* Einstein's point coincidences. Einstein's "material points" are just replaced by photons (light pulses): the spacetime point $s^\alpha$ is characterized as the meeting point of four photons designated by the fact of carrying the radio signals $s^\alpha$.

## References

[1] R. L. Arnowitt, S. Deser, and C. W. Misner. *Gravitation: an introduction to current research*, chapter The dynamics of general relativity, pages 227–265. John Wiley & Sons, 1962.

[2] T. B. Bahder. Fermi coordinates of an observer moving in a circle in Minkowski space: Apparent behavior of clocks. Technical Report ARL-TR-2211, Army research laboratory, 1998.

[3] P. G. Bergmann. Observables in general relativity. *Rev. Modern Phys.*, 33:510–514, 1961.

<footnote n="1">¹"All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points" [9].</footnote>
[4] J. D. Brown and D. Marolf. Relativistic material reference systems. *Physical Review D*, 53(4):1835, 1996.

[5] F. De Felice, M. Lattanzi, A. Vecchiato, and P. Bernacca. General relativistic satellite astrometry. i. a non-perturbative approach to data reduction. *Astronomy and Astrophysics*, 332:1133-1141, 1998.

[6] B. Guinot. Application of general relativity to metrology. *Metrologia*, 34(3):261, 1997.

[7] B. Hofmann-Wellenhof, H. Lichtenegger, and J. Collins. *Global Positioning System. Theory and practice*. Wien (Austria): Springer, 1993.

[8] E. D. Kaplan and C. J. Hegarty. *Understand-ing GPS: principles and applications*. Artech House Publishers, 2006.

[9] W. Perrett and G. Jeffery. *The principle of relativity*. Dover, New York, 1952.

[10] C. Rovelli. What is observable in classical and quantum gravity? *Classical and Quantum Gravity*, 8(2):297, 1991.

[11] C. Rovelli. Gps observables in general relativity. *Phys. Rev. D*, 65:044017, Jan 2002.

[12] J. Spilker. *GPS Theory and Applications, Vol. I, Progress in Astronautics and Aeronautics*, chapter Tropospheric Effects on GPS, pages 517-546. Amererica Institute of Aeronautics, 1996.

[13] A. R. Thompson, J. M. Moran, and G. W. Swen-son Jr. *Interferometry and synthesis in radio astron-omy*. Wiley-Vch, 2008.

[14] P. Visser. Gravity field determination with GOCE and GRACE. *Advances in Space Research*, 23(4):771-776, 1999.

[15] L. Witten. *Gravitation: an introduction to current research*, volume 1. New York: Wiley, 1962.