Our derivation considered a macroscopic quantum object (the superfluid), but we did not discuss the self interaction of a single particle. The Schrödinger-Newton equation is described as follow in literature: "we shall consider a class of systems that can be formally described by the Schrödinger-Poisson (SP) equation [also known as the Schrödinger-Newton equation]. The latter results from the combination of the Schrödinger equation and the Poisson equation describing the self-potential. Historically speaking, the SP equation was first advocated in the context of gravitation by Diosi and Penrose as a simple quasi-classical model to introduce quantum effects in gravitational problems. It describes quantum matter confined by gravitational fields and, as such, finds many applications in astrophysics; it describes (hypothesized) boson stars (which could be a source of 'exotic' Laser Interferometer Gravitational-Wave Observatory (LIGO) detections) and it is a central ingredient of scalar field dark matter models. Interestingly, the same equation applies equally well to a variety of other systems, such as quantum plasmas and atomic gases in magneto-optical traps (MOTs). A bridge with Bose-Einstein condensates can also be established [...]. These similarities may inspire future laboratory experiments using gravity analogs" [193]. The validity of the Schrödinger-Newton equation is a debated subject, especially when single particles are considered (the standard interpretation of quantum mechanics is based on the Schrödinger equation, not on the Schrödinger-Newton equation).

In the standard model of particle physics, particles are classified by eigenvalues of mass and spin [62]: massless spin-1 particles obey Maxwell's equations while massive spin-0 particles obey the Klein-Gordon equation. The two fields should be treated on equal footing because a massive spin-0 particle can decay into two massless spin-1 particles (we mentioned the example $\pi^0 \rightarrow \gamma\gamma$ several times). When general relativity is added, the unification of Maxwell's equations and Einstein's gravity is provided by the Einstein-Maxwell equations (7.23) and (7.25). Similarly, the unification of the Klein-Gordon theory and Einstein's gravity is provided by equations (7.8) and (7.9) which produces the Schrödinger-Newton equation (8.2) in the non-relativistic limit.

Generally covariant quantum mechanics exhibits that Klein-Gordon fields (massive spin-0 particles) and Maxwell's fields (massless spin-1 particles) can transform into each other such that, if the two kinds of fields are treated on equal footing, the experimental validity of the Schrödinger-Newton equation in quantum mechanics is strongly related to the experimental validity of the Einstein-Maxwell's equations in general relativity. Generally covariant quantum mechanics even suggests that the equations of the new Kaluza-Klein theory introduced in [Section 4] could replace the Einstein-Maxwell equations and modify the unification of the Klein-Gordon theory and Einstein's gravity if the massive spin-0 particles have a non vanishing electric charge (the two other fundamental interactions are not discussed).

## 8.5 Classical tests of general relativity and quantum considerations

We are now interested by two well known tests of general relativity: the bending of light by the sun and the gravitational redshift. We first reproduce the computation of each of them, according to general relativity. Then we examine the new insights provided by generally covariant quantum mechanics.

**Bending of light by the sun:**

We start from the Schwarzschild metric in a plane. We use equation (7.42) with $\theta = \frac{\pi}{2}$.

$$c^2 d\tau^2 = \left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 - \frac{1}{1 - \frac{2GM}{c^2 r}} dr^2 - r^2 d\varphi^2 \Leftrightarrow 1 = \left(1 - \frac{2GM}{c^2 r}\right) \dot{t}^2 - \frac{1/c^2}{1 - \frac{2GM}{c^2 r}} \dot{r}^2 - \frac{r^2}{c^2} \dot{\varphi}^2 \quad (8.3)$$

The mechanical Lagrangian is $L = \int mc^2 d\tau$. Since the metric is independent of the time and angular coordinates, we define the conserved energy per unit mass-energy $\frac{E}{mc^2} = (1 - \frac{2GM}{c^2 r})\dot{t}$ and the angular momentum per unit mass-energy $\frac{L}{mc^2} = \frac{r^2}{c^2}\dot{\varphi}$ such that equation (8.3) becomes:

$$1 = \frac{1}{1 - \frac{2GM}{c^2 r}} \left(\frac{E}{mc^2}\right)^2 - \frac{1/c^2}{1 - \frac{2GM}{c^2 r}} \dot{r}^2 - \frac{L^2}{m^2 r^2 c^2} \quad (8.4)$$

Using $\frac{L}{mc^2} = \frac{r^2}{c^2}\dot{\varphi} \Leftrightarrow d\tau = mr^2 d\varphi/L$ we can rewrite $\dot{r} = \frac{dr}{d\tau}$ as $\dot{r} = \frac{L}{m}(\frac{dr}{r^2 d\varphi}) = \frac{L}{m}(-\frac{du}{d\varphi})$ where $u = \frac{1}{r}$ such that equation (8.4) takes the form:

$$\begin{aligned} 1 &= \frac{1}{1 - \frac{2GM}{c^2 r}} \left(\frac{E}{mc^2}\right)^2 - \frac{\frac{L^2}{m^2 c^2}}{1 - \frac{2GM}{c^2 r}} \left(\frac{dr}{r^2 d\varphi}\right)^2 - \frac{L^2}{m^2 r^2 c^2} \\ 1 &= \frac{1}{1 - \frac{2GMu}{c^2}} \left(\frac{E}{mc^2}\right)^2 - \frac{\frac{L^2}{m^2 c^2}}{1 - \frac{2GMu}{c^2}} \left(\frac{du}{d\varphi}\right)^2 - \frac{L^2 u^2}{m^2 c^2} \end{aligned} \quad (8.5)$$
$$ 
\begin{aligned}
1 - \frac{2GMu}{c^2} &= \left(\frac{E}{mc^2}\right)^2 - \frac{L^2}{m^2c^2}\left(\frac{du}{d\varphi}\right)^2 - \left(1 - \frac{2GMu}{c^2}\right)\frac{L^2u^2}{m^2c^2} \\
\frac{m^2c^2}{L^2}\left(1 - \frac{2GMu}{c^2}\right) &= \left(\frac{E}{Lc}\right)^2 - \left[\left(\frac{du}{d\varphi}\right)^2 + u^2\right] + \frac{2GMu^3}{c^2} \\
\left(\frac{du}{d\varphi}\right)^2 + u^2 &= -\frac{m^2c^2}{L^2}\left(1 - \frac{2GMu}{c^2}\right) + \left(\frac{E}{Lc}\right)^2 + \frac{2GMu^3}{c^2}
\end{aligned}
\quad (8.6) $$

Now we take the derivative of the above equation with respect to $\varphi$ and we impose the condition $m = 0$ to describe massless photons :

$$ \frac{d^2 u}{d\varphi^2} + u = \frac{3GMu^2}{c^2} \quad (8.7) $$

This equation is well known in general relativity : it allows to directly compute the bending of light by the sun ($\Delta\varphi = \frac{4GM}{c^2R_s}$ where $R_s$ is the radius of the sun). The reamining computation can be found in standard textbooks and lecture notes on general relativity [27][194][195].

**Gravitational redshift :**

Here we follow the lecture notes written by M. Blau. "First of all, let a light ray be described by the wave vector $k^\mu$. In special relativity, we would parametrise this as $k^\mu = (\omega, \vec{k})$ with $\omega = 2\pi\nu$ and $\nu$ the frequency. This is the frequency observed by an inertial observer at rest, with 4-velocity $u^\mu = (1, 0, 0, 0)$. A lorentz-invariant, and thus in our context now coordinate independent, notion of the frequency measured by an oberserver with velocity $u^\mu$ is thus $\omega = u^\mu k_\mu$. This includes as special cases the relativistic Doppler effect (where one compares $\omega$ with $\bar{\omega} = \bar{u}^\mu k_\mu$, $\bar{u}^\mu$ the tangent to the world line of a boosted observer), as well as the gravitational redshift we want to discuss here. A static observer in the spherically-symmetric and static gravitational field of equation (8.3) is described by the 4-velocity

$$ u^\mu = (u^0, 0, 0, 0) \quad ; \quad g_{\mu\nu}u^\mu u^\nu = g_{00}(u^0)^2 = 1 \quad (8.8) $$

Thus for the static observer at $r = r_A$, say, one has :

$$ u^\mu = u_A^0 = (g_{00}(r_A))^{-\frac{1}{2}} \quad (8.9) $$

[...] Then one find that the frequency measured by the static observer at $r = r_A$ is

$$ \omega_A = u_A^\mu k_\mu = (g_{00}(r_A))^{-\frac{1}{2}} k_0 \quad (8.10) $$

" [27]. Then

$$ \frac{\omega_A}{\omega_B} = \frac{(g_{00}(r_A))^{-\frac{1}{2}} k_0}{(g_{00}(r_B))^{-\frac{1}{2}} k_0} \approx \frac{1 - \frac{GM}{c^2 r_B}}{1 - \frac{GM}{c^2 r_A}} \approx 1 - \frac{GM}{c^2 r_B} + \frac{GM}{c^2 r_A} \quad (8.11) $$

If the light ray is emitted at $r_A$ and received at $r_B \to \infty$, we have $\omega_B \approx (1 - \frac{GM}{c^2r_A})\omega_A$ : this is an example of gravitational redshift.

**New insights from generally covariant quantum mechanics :**

When the bending of light was introduce above, light was described by a classical massless particle following a path : the computation was initially made for a massive particle and the limit $m = 0$ was chosen at the end of the computation. When the gravitational redshift was considered, light was described by a classical wave with 4-wave vector $k^\mu$. Clearly, general relativity lacks a unified description of light, which takes into account its quantum properties : this is obvious since no testable theory combining general relativity and quantum mechanics has been provided so far. Generally covariant quantum mechanics suggests to describe the photon by a complex electromagnetic field. Interestingly, a paper entitled 'Direct measurement of the quantum wave function' and published in 2011 indicates : "Here we show that the wavefunction can be measured directly by the sequential measurement of two complementary variables of the system. [...] The result is that the real and imaginary components of the wavefunction appear directly on our measurement apparatus. We give an experimental example by directly measuring the transverse spatial wavefunction of a single photon, a task not previously realized by any method. [...] We then further test our method by inducing known magnitude and phase changes to the photons here to prepare a series of modified wavefunctions" [51]. The paper also provides a picture entitled "The measured single-photon wavefunction, $\psi(x)$, and its modulus squared and phase". In [Section 7.2.1] we generalized the usual electromagnetic Lagrangian density $-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu}$ involving the real vector field $A_\mu$ by using $-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu}$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $A_\mu(x^\alpha) = A_\mu(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}} = \epsilon_\mu(x^\alpha)R(x^\alpha)e^{-\frac{iS(x^\alpha)}{\hbar}}$. $\epsilon^\mu(x^\alpha)$ was one of the polarization
vectors introduced in [Section 5.2] such that $\epsilon_\mu \epsilon^{\mu*} = -1$. To simplify the discussion, we assumed that $\epsilon_\mu$ was real. We wrote $\mathcal{R}^2 = \rho$ and computed (see equation (7.109)):

$$ -\frac{1}{4\mu_0} \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu*} = -\left(\partial_\mu A_\nu - \frac{\mathbf{i}\partial_\mu S}{\hbar} A_\nu - \partial_\nu A_\mu + \frac{\mathbf{i}\partial_\nu S}{\hbar} A_\mu\right) \left(\partial^\mu A^\nu + \frac{\mathbf{i}\partial^\mu S}{\hbar} A^\nu - \partial^\nu A^\mu - \frac{\mathbf{i}\partial^\nu S}{\hbar} A^\mu\right) \quad (8.12) \\ = -\frac{1}{4\mu_0} F^{\mu\nu} F_{\mu\nu} + \frac{\rho}{2\mu_0\hbar^2} \eta^{\mu\nu} \partial_\mu S \partial_\nu S + \frac{\rho}{2\mu_0\hbar^2} (\partial_\nu S \epsilon^\nu)^2 $$

We justified that only the term $\frac{\rho}{2\mu_0\hbar^2} \eta^{\mu\nu} \partial_\mu S \partial_\nu S$ is meaningful in the eikonal approximation. In curved spacetime, this term becomes the vanishing lagrangian of a photon, $\frac{\rho}{2\mu_0\hbar^2} g^{\mu\nu} \partial_\mu S \partial_\nu S$ where $\partial_\mu S = p_\mu = \hbar k_\mu$. We are then interested by the equality $g^{\mu\nu} \partial_\mu S \partial_\nu S = 0$ in Schwarzschild's spacetime. We compute:

$$ \begin{aligned} 0 &= g^{\mu\nu} \partial_{\mu} S \partial_{\nu} S = g^{tt} c^{-2} \partial_{t} S \partial_{t} S - g^{rr} \partial_{r} S \partial_{r} S - g^{\varphi\varphi} \partial_{\varphi} S \partial_{\varphi} S \\ 0 &= \frac{1}{1 - \frac{2GM}{c^2 r}} \frac{E^2}{c^2} - \left(1 - \frac{2GM}{c^2 r}\right) p_r^2 - \frac{1}{r^2} L^2 \\ 0 &= \frac{1}{1 - \frac{2GM}{c^2 r}} \frac{E^2}{c^2} - \left[ \left(1 - \frac{2GM}{c^2 r}\right) \frac{p_r^2}{L^2} + \frac{1}{r^2} \right] L^2 \end{aligned} \quad (8.13) $$

The ratio $p_r/L$ can be computed by considering the motion a massive particle since the result is independent of the mass:

$$ \frac{p_r}{L} = \frac{mg_{rr} \frac{dr}{d\lambda}}{mg_{\varphi\varphi} \frac{d\varphi}{d\lambda}} = \frac{1}{1 - \frac{2GM}{c^2 r}} \frac{dr}{r^2 d\varphi} \quad (8.14) $$

where $\lambda$ is any affine parameter. Injecting this result in equation (8.13) with $u = \frac{1}{r}$ we get:

$$ \begin{aligned} 0 &= \frac{1}{1 - \frac{2GM}{c^2 r}} \frac{E^2}{L^2 c^2} - \left[ \frac{1}{1 - \frac{2GM}{c^2 r}} \left( \frac{dr}{r^2 d\varphi} \right)^2 + \frac{1}{r^2} \right] \\ 0 &= \frac{E^2}{L^2 c^2} - \left[ \left( \frac{du}{d\varphi} \right)^2 + u^2 - \frac{2GMu^3}{c^2} \right] \end{aligned} \quad (8.15) $$

Deriving this last expression with respect to $\varphi$ we recover equation (8.16):

$$ \frac{d^2 u}{d\varphi^2} + u = \frac{3GMu^2}{c^2} \quad (8.16) $$

As explained previously, the bending of light by the sun ($\Delta\varphi = \frac{4GM}{c^2 R_s}$ where $R_s$ is the radius of the sun) can be computed from this equation [27][194][195]. Evidently, our prescription $A_\mu(x^\alpha) = A_\mu(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}}$ describes a light ray with 4-wave vector $k^\mu = \frac{p^\mu}{\hbar} = \frac{\partial^\mu S}{\hbar}$ where a light ray corresponds to the eikonal approximation, by definition: as we have seen from equation (8.8) to (8.11), the gravitational redshift directly follows from the description of light by the 4-wave vector $k^\mu$, in Schwarzschild's spacetime. We have then produced a unified description of the bending of light and the gravitational redshift.

L. Freidel declared in 2020: "How can we put together quantum mechanics and dynamical spacetime? That is one of the most fundamental questions [...] However, I think that an experimental breakthrough will be necessary to guide us in the right direction" [57]. The present analysis might constitute an experimental test of complex electrodynamics in curved spacetime.

## 8.6 Gravitons

A paper entitled 'Detecting single gravitons with quantum sensing' was published in 2024. The abstract indicates: "The quantization of gravity is widely believed to result in gravitons-particles of discrete energy that form gravitational waves. But their detection has so far been considered impossible. Here we show that signatures of single graviton exchange can be observed in laboratory experiments. [...] Our results show that single graviton signatures are within reach of experiments. In analogy to the discovery of the photo-electric effect for photons, such signatures can provide the first experimental clue of the quantization of gravity" [196]. We did not include the second quantization as a required tool in the present form of generally covariant quantum mechanics but the theory nevertheless suggests that the hypothetical graviton exists: it is then interesting to examine if the detection of an hypothetical graviton would imply that the gravitational field should be second quantized. We quote T. Y. Ciao: "Historically, in the physical interpretation of [...] quantum field theory (QFT), particularly
in the discussion of the field-particle relationship, the operator formalism of free fields and their Fock space structure has played a central role. The statement that particles are just quanta of the field makes sense only within this formalism, and many difficult interpretive questions hiding behind this statement become discernible when we take a closer look at the logical structure of the formalism. In order to address these questions, we have to pause and consider the core concept of the structure : [second] quantization. [...] The first attempt to formulate a relativistic quantum framework for the description of particles and free fields without recourse to the idea of active quantization was Wigner's representation theory, which, however, can be shown to be compatible with and equivalent to canonical quantization. This, together with the fact that various formalisms of QFT are constructed upon classical space-time [...], has raised a question : 'What is the meaning of being quantal, in contrast with being classical, which may or may not have a classical counterpart?' " [29]. In [Section 5.2] we suggested to identify a particle with a mode solving the field equations, while the field equations determine the spin and mass of the particle. In the same section, we quoted M. Schwartz : "First quantization refers to the discrete modes, for example, of a particle in a box. Second quantization refers to the integer numbers of excitations of each of these modes" [62]. It was also recalled that the first quantization is the consequence of appropriate boundary conditions, in agreement with the Sturm-Liouville theory. In [Section 5.2], we expressed the electromagnetic potential into modes

$$A^{\mu} = \sum_{n} \sqrt{\frac{\hbar}{V\epsilon_{0}\omega_{n}}} (a_{1n}\epsilon_{1}^{\mu} + a_{2n}\epsilon_{2}^{\mu}) e^{-i(\omega_{n}t - k_{3n}x^{3})} \quad (8.17)$$

and we recalled that the second quantization replaces the coefficients $a_{1n}$ and $a_{2n}$ by operators with specific com-mutation rules. It is evidently possible to obtain integer numbers of excitations of each mode without introducing operators, by imposing the condition that $|a_{1n}|^2$ and $|a_{2n}|^2$ should be integer numbers. The integer number of excitations could then be interpreted as integer numbers of photons.

As we have seen in [Section 7.2.1], A. Einstein introduced the concept of energy quanta in 1905, by studying Planck's spectrum. His energy quanta were called 'photon' in 1926. In 1927, A. H. Compton received the Nobel Prize since he had experimentally observed the photon, as he explained in his Nobel lecture [197]. The second quantization of the electromagnetic field was introduced by P. A. M. Dirac the same year, in 1927 [198], while the photon had already been predicted and observed. Precisely, A. Einstein predicted the photon by considering the limit of large frequencies in Planck's spectrum, in agreement with the eikonal approximation which allows to replace wave optics by geometric optics and classical paths. This historical background is completely eluded by T. Y. Ciao because he only discusses quantum field theory. Since he emphasizes the 'many difficult interpretive questions' are hidden behind the particle concept in quantum field theory, it deserves to be noticed that the second quantization of the electromagnetic field has no obvious links with geometric optics, although geometric optics describes the path followed by the photons. Generally covariant quantum mechanics introduces a geometric definition of the photon, valid in the eikonal approximation of the solutions of massless spin-1 field equations (this description being consistent with E. Wigner's representation theory). All fields being treated on equal footing, the theory suggests an analogous description of the graviton.

In 2012, F. Dyson announced during a talk : "This talk is concerned with a different question, whether it is in principle possible to detect individual gravitons, or in other words, whether it is possible to detect the quantization of the gravitational field" [199]. By analogy with the electromagnetic field and matter fields, generally covariant quantum mechanics suggests that the hypothetical graviton exists : interestingly, this subject is not correlated to the second quantization of the gravitational field (although [Section 6] mentions the possibility to perform the canonical quantization of gravity with the Kaluza-Klein theory).

To conclude this section, we quote F. Müller who declared in 2016 : "For nearly a century, how to understand quantum gravity has been one of the defining questions of modern physics - both in terms of trying to identify what those two terms actually mean in the context of each other, and in terms of creating a theory that supports that intersection. Thus far, these efforts have been challenged by a lack of experimental evidence - the two theories operate on such different scales, that having a single source simultaneously showing measurable quantum and gravitational effects is dishearteningly difficult" [200]. The present context was described by J. Ambjorn, during an interview realised in 2011, with the following words : "In the last 30 years there have been no new important discoveries, nothing in fact, and if you wait 30 more years and there are still no new discoveries related to the real world then I think that our community will shrink in a significant way. Perhaps if that happens it is only fair (laughs) since if we have nothing to offer then our existence is useless from that point of view" [57]. The present section illustrated that generally covariant quantum mechanics is already able to enlight existing experimental data. A more in-depth examination of the theory, or its extension (complex general relativity, inclusion of nuclear forces in a Kaluza-Klein mechanism, generalization of [Section 7.1.4] to Navier-Stockes equations usw.), might then allow new discoveries. The theory could also be used to develop other approaches to quantum gravity.
# 9 Conclusion

[Section 1] was devoted to the principle of least action also called 'principe de la moindre action' for historical reasons, as emphasized by C. G. J. Jacobi. The principle states that the action of a system should be stationary under infinitesimal variations (if S is a minimum "-S" is a maximum, the action is then defined up to a sign). A historical account was provided by D. Cline in 2021: "The Age of Enlightenment is a term used to describe a phase in Western philosophy and cultural life in which reason was advocated as the primary source and legitimacy for authority. It developed simultaneously in Germany, France, Britain, the Netherlands, and Italy around the 1650's and lasted until the French Revolution in 1789. The scientific advances of this age include publication of two monumental books 'Philosophiae Naturalis Principia Mathematica' by Newton in 1687 and 'Mécanique analytique' by Lagrange in 1788. These are the definitive two books upon which classical mechanics is built. [...] René Descartes (1596-1650) attempted to formulate the laws of motion in 1644. He talked about conservation of motion (momentum) in a straight line but did not recognize the vector character of momentum. Pierre de Fermat (1601-1665) and René Descartes were two leading mathematicians in the first half of the 17th century. Independently they discovered the principles of analytic geometry and developed some initial concepts of calculus. [...] Fermat revived the principle of least time, which states that "light travels between two given points along the path of shortest time" and was used to derive Snell's law in 1657. This enunciation of variational principles in physics played a key role in the historical development of the principle of least action that underlies the analytical formulations of classical mechanics. [...] The Euler-Lagrangian approach to classical mechanics stems from a deep philosophical belief that the laws of nature are based on the principle of economy. That is, the physical universe follows paths through space and time that are based on extrema principles" [201]. In 1925, M. Planck stated: "The chief law of physics, the pinnacle of the whole system is, in my opinion, the principle of least action" [124]. In 1949, C. Lanczos recalled that "the variational method has often been the focus of philosophical controversies and misinterpretations. Indeed, the idea of enlarging reality by including 'tentative' possibilites and then selecting one of these by the condition that it minimizes a certain quantity, seems to bring a purpose to the flow of natural events" [7]. In 2018, A. Rojo and A. Bloch explained: "The principle of least action originates in the idea that, if nature has a purpose, it should follow a minimum or critical path. This simple principle, and its variants and generalizations, applies to optics, mechanics, electromagnetism, relativity, and quantum mechanics, and provides an essential guide to understanding the beauty of physics" [202]. In [Section 1], we started from the definition of the action: $S = \int L(x^\alpha, \dot{x}^\alpha, \tau)d\tau$. We showed from equations (1.3) to (1.5) that the Euler-Lagrange equations $\frac{\partial L}{\partial x^\alpha} - \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^\alpha} = 0$ implies $\frac{\partial S}{\partial \dot{x}^\alpha} = 0$ which means $S = S(x^\alpha, \tau)$. Using equation (1.11), we showed that $S = S(x^\alpha, \tau)$ conversely implies the Euler-Lagrange equations (it was not needed to introduce variations anymore). Consequently, in [Section 1] we showed:

$$ \underline{\text{Theorem:}} \quad \text{If } S = \int L(x^\alpha, \dot{x}^\alpha, \tau)d\tau \quad \text{then} \quad \frac{\partial L}{\partial x^\alpha} - \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^\alpha} \Leftrightarrow S = S(x^\alpha, \tau) \qquad (9.1) $$

[Section 2] discussed the possibility to derive quantum physics by modifying general relativity: the proper time was treated as a fifth coordinate but the principle of general covariance was not extended to five dimensions. We replaced the usual (vanishing) matter Lagrangian density by the Lagrangian density of a complex scalar field obeying a 5D wave equation. The consistency of this replacement required the introduction of a universal constant $\hbar$ such that $p^\mu = \hbar k^\mu$. Consequently, the wave equation obeyed by the scalar field became the Klein-Gordon equation which reduces to the Schrodinger equation in the non relativistic limit. Then, it was shown that the Lagrangian density of the complex scalar field reduces to the usual (vanishing) matter Lagrangian density in the eikonal approximation: this approximation identifies the phase of the complex field with the mechanical action. Finally, a possible link between the proper time and decay processes of unstable particles was discussed.

[Section 3] illustrated the opposite path: general relativity was derived from quantum physics by introducing general covariance in quantum field theory. In [Section 2], we assumed that the constants c and G exist to deduce that the constant $\hbar$ should exist. In [Section 3], we assumed that the constants c and $\hbar$ exist to deduce that the constant G should exist. The section contained a first derivation of the geodesic equation from quantum physics. This derivation identified the phase of the matter field with the classical action (eikonal approximation).

[Section 4] reconsidered the old Kaluza-Klein theory. It was assumed that the proper time has a hyperbolic internal space: this internal space was used to extend general relativity to five dimensions, allowing to unify Einstein's gravity and Maxwell's electrodynamics in 5D gravity. We then reproduced the analysis given in [Section 2] with a complex scalar field and 5D gravity while we identified the phase of the field with the classical action of particle in a higher dimensional spacetime: we spontaneously obtained the Lagrangian of scalar quantum electrodynamics, a well known quantum field theory.

[Section 5] contained new results in quantum physics. It began by providing a second derivation of the geodesic equation from quantum physics (the first one was given in [Section 3]). In the literature we reviewed, we found three ways to obtain the classical limit from quantum physics: Ehrenfest's theorem, the eikonal approximation, and imposing the condition $\hbar \to 0$ in the Feynman path integral. A flaw in the Ehrenfest theorem was discussed in [Section 5.1]. A new formulation of the theorem, using the proper time in curved spacetime,
was then introduced. A third derivation of the geodesic equation from quantum physics followed. It should be observed that the three derivations of the geodesic equation from quantum physics use the eikonal approximation to recover the classical laws of motion. To describe the path integral, R. P. Feynman explained: "The probability that a particle starting at point 1 at the time $t_1$ will arrive at point 2 at the time $t_2$ is the square of a probability amplitude. The total amplitude can be written as the sum of the amplitudes for each possible path. [...] The amplitude is proportional to some constant times $e^{-iS/\hbar}$ where S is the action for that path. [...] in the limiting case in which Planck's constant $\hbar$ goes to zero, the correct quantum-mechanical laws can be summarized by simply saying: 'Forget about all these probability amplitudes. The particle does go on a special path, namely, that one for which S does not vary in the first approximation'. That's the relation between the principle of least action and quantum mechanics" [11]. R. P. Feynman's statement "a particle starting at point 1 at the time $t_1$ will arrive at point 2 at the time $t_2$" nevertheless suggests that he interpreted particles like photons as pointlike particles (without shapes) involved in his path integral. His point of view was confirmed in [Section 7.2] with another quote since R. P. Feynman also stated that: "The electron is either here, or there, or somewhere else, but wherever it is, it is a point charge. [...] The quantum behavior of atomic objects (electrons, protons, neutrons, photons, and so on) is the same for all, they are all 'particle waves,' or whatever you want to call them" [75]. Nevertheless, it has been experimentally confirmed that single photons are not pointlike [51][52][53][54][55] as discussed in [Section 2]. It is clear that the present theory favors the eikonal approximation to obtain the classical limit from the quantum field equations. It is a standard result that this approximation allows to recover geometric optic from wave optics. The principle of least action is then the bridge between quantum and classical phenomena: when we consider light and/or matter, the eikonal approximation makes it possible to extract the path of classical particles from the wave behavior of quantum fields. Another important result of [Section 5.1] is that the proper time was explicitly introduced in quantum physics: the link between the proper time and decay processes, previously discussed in [Section 2], became clearer. E. Witten declared in 2014: "It might be that trying to make a theory of quantum gravity will help us understand the standard model better, or will give us a better understanding of cosmology or even of quantum mechanics. But at any rate quantum mechanics and gravity are there and we are bound to be curious about how they can work together" [57]. It can then be noticed that [Section 5.1] developed a formulation of quantum physics in curved spacetime, showing how quantum mechanics and gravity can work together. In [Section 5.2] the polarization of the electromagnetic field was reconsidered to suggest that general relativity should be extended to complex numbers. Finally, new results about the Dirac equation were provided in [Section 5.3].

[Section 6] extended the usual ADM formalism of canonical quantum gravity to the new Kaluza-Klein theory introduced in [Section 4]: it enlightened the geometrical meaning of the electromagnetic interaction in the Kaluza-Klein theory.

[Section 7] introduced generally covariant quantum mechanics, in a form defined by the action $S_T$ of equation (7.1). From this action, we derived in [Section 7.1]: Quantum electrodynamics (applied to spin-0 fields), quantum physics in curved spacetime, Einstein's gravity & Maxwell's electrodynamics, hydrodynamics of perfect fluids, Newtonian mechanics and thermodynamics of perfect gases. Technically, our thermodynamics of perfect gases described the behavior of a collection of Newtonian particles. Obviously, the theory described by the action $S_T$ of equation (7.1) can not be fundamental because two fundamental forces are missing (the weak and the strong interactions) while matter is described by a scalar field (fundamental matter particles are described by spinor fields). Nevertheless, the fact that various physical theories can be unified into a single action suggests that the principle of least action could play a preeminent role in a ultimate theory of nature. Up to this point, it is interesting to quote M. Planck again. In 1925, he wrote: "As long as physical science exists, the highest goal to which it aspires is the solution of the problem of embracing all natural phenomena, observed and still to be observed, in one simple principle which will allow all past and, especially, future occurrences to be calculated. [...] Among the more or less general laws, the discovery of which characterize the development of physical science during the last century, the principle of Least Action is at present certainly one which, by its form and comprehensiveness, may be said to have approached most closely to the ideal aim of theoretical inquiry. [...] Maupertuis's real service consisted in his search for a principle that would be, above all, a minimum principle. [...] The interest in the principle of least action was fundamentally based upon the metaphysical idea that the rule of the Deity reveals itself in Nature" [124]. Equation (7.1) defines the action of generally covariant quantum mechanics, which is a field theory: the principle of stationary action determines the equations obeyed by the fields. Identifying the phase of the fields with the mechanical action (which also obey the principle of stationary action) produces the eikonal approximation from which the classical behavior follows. In [Section 7.2], we then discussed the interpretation of quantum mechanics which emerges from generally covariant quantum mechanics. It was suggested that the wave-particle duality is a direct consequence of the complex nature of physical fields. Two incompatible descriptions of multiparticle states widely used in literature were recalled and examined. Finally, the famous collapse of the wave function was studied.

[Section 8] discussed the links between generally covariant quantum mechanics and experimental data: it was observed that interferometry could soon open the possibility to test quantum physics in curved spacetime. In this context, the matter fields take the form $\psi = \mathcal{R}e^{-iS/\hbar}$ which directly leads to the eikonal approximation and
to the framework of generally covariant quantum mechanics. Two famous experimental tests of general relativity are the gravitational redshift and the bending of light by the sun. In the first case, light is described by photons with a wave-like behavior while, in the second case, light is described by photons with a particle-like behavior : it was shown that generally covariant quantum mechanics allows a unified description by considering a complex Maxwell Lagrangian in curved spacetime.

It is clear that the principle of stationary action is of fundamental importance in generally covariant quantum mechanics : as we recalled previously, this principle has important philosophical ramifications. Discussing the history of physical sciences, M. Planck suggested in 1925 that physical laws could be anthropomorphic : "But in what direction does this advance proceed as a whole ? How far can we say that we are actually approaching the goal we seek, namely, the unified system ? The answers to these questions must be of the greatest importance to every physicist who studies the progress of his science. When we are in a position finally to answer them we shall also be in a position to consider the broader question, much discussed today, as to what is the fundamental meaning of the so-called physical universe to us ? Is it merely a practical, though fundamentally arbitrary, creation of our imagination or are we forced to the opposite conception that it reflects real natural phenomena independent of us ? [...] Thus the physiological element affects all physical definitions if all our observations be associated with our sense-perceptions; in short, the whole range of physics, its definitions as well as its entire structure, bears, in a certain sense, an anthropomorphic character" [124]. Similarly, M. Born made the following remark in 1939 : "The ideal would be to condense all laws into a single law, a universal formula, the existence of which was postulated more than a century ago by the great French astronomer, Laplace. [...] We are still far from knowing Laplace's universal formula, but we may be convinced that it will have the form of an extremal principe, not because Nature has a will or a purpose of economy, but because the mechanism of our thinking has no other way of condensing a complicated structure of laws into a short expression [203]". C. Lancsoz also explained in 1949 : "In our day we have witnessed at least one fundamental discovery of unprecedented magnitude, namely Einstein's Theory of General Relativity, which was obtained by mathematical and philosophical speculation of the highest order. Here was a discovery made by a kind of reasoning that a positivist cannot fail to call 'metaphysical' [...] The Principle of General Relativity is automatically satisfied if the fundamental 'action' of the variational principle is chosen as an invariant under any coordinate transformation. Since the differential geometry of Riemann furnishes us such invariants, we have no difficulty in setting up the required field equations. Apart from this, our present knowledge of mathematics does not give us any clue to the formulation of covariant, and at the same time consistent, system of field equations. Hence, in the light of relativity the application of the calculus of variations to the law of nature assumes more than accidental significance" [7]. Another famous physical principle is the principle of relativity, stated as follow by H. Poincaré in 1904 : "the laws of physical phenomena must be the same, either for a fixed observer, or for an observer involved in a uniform translation movement, so that we have and cannot have any means of discerning whether or not we are carried away in such a movement" [204]. This principle was soon generalized to include general covariance. If one agrees with C. Lancsoz asserting that our mathematical knowledge is too limited to directly discover the fundamental laws of nature, new progress in theoretical physics could arise through what he called "metaphysical reasoning" : the principle of relativity (probably generalized to complex transformations) could be used to define invariant Lagrangians on generalized spacetimes : the goal would be to generalize the Kaluza-Klein theory to include the weak and the strong interactions. Then the principle of stationary action would determine the field equations whose solutions would have to be compared to experimental data.

To conclude this section, we summarize how generally covariant quantum mechanics combines quantum mechanics and general relativity, by quoting H. Poincaré : "When a physicist finds a contradiction between two theories [...], at least one must then be considered false. This is no longer the case if we only seek what we must seek. It is quite possible that they both express true relationships, and that the contradictions exist only in the images we have formed of reality" [9].

J. Kerneïs
France
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# 11 Annexes : Detailed computations

## 11.1 Annex 1 : On the Klein Gordon equation & Fourier analysis

In this annex we illustrate the effect of crossing the phases and amplitudes of 2 signals and we give a detailed derivation of equations (2.21) and (2.22).

### 11.1.1 Complement on Fourier analysis

In the example below, the Fourier transform of two images are computed, and the two images are reconstructed by crossing their amplitudes and phases :

<figure><img src="image_7.png" /><figcaption>Original image : "Las meninas" by D. Velasquez</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Original image : "Las meninas" by P. Picasso</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Phase from D. Velasquez
Amplitude from P. Picasso</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Phase from P. Picasso
Amplitude from D. Velasquez</figcaption></figure>

<caption>Figure 4 : Illustration of the crossing of the phases and amplitudes of the Fourier transforms of two images</caption>

The authors conclude : "In most images [...] the amplitude are always similar and it is really the phase that is the differential element" [...].

### 11.1.2 Computation of $0 = -\hbar^2 \Box_g \psi - m^2 c^2 \psi$ with $\psi = \mathcal{R}e^{-i\frac{\xi}{\hbar}}$ and $\mathcal{R}^2 = \rho$.

$$ 
\begin{aligned}
-\hbar^2 \Box_g \psi - m^2 c^2 \psi &= -\hbar^2 \Box_g (\mathcal{R}e^{-i\frac{\xi}{\hbar}}) - m^2 c^2 \psi = -\hbar^2 (\partial_\mu \partial^\mu + \Gamma^\nu_{\nu\mu} \partial^\mu) \mathcal{R}e^{-i\frac{\xi}{\hbar}} - m^2 c^2 \psi \\
&= -\hbar^2 \partial_\mu \left( (\partial^\mu \mathcal{R})e^{-i\frac{\xi}{\hbar}} - i \frac{\partial^\mu S}{\hbar} \mathcal{R}e^{-i\frac{\xi}{\hbar}} \right) - \hbar^2 \Gamma^\nu_{\nu\mu} (\partial^\mu \mathcal{R})e^{-i\frac{\xi}{\hbar}} + i \hbar \Gamma^\nu_{\nu\mu} \partial^\mu S \mathcal{R}e^{-i\frac{\xi}{\hbar}} - m^2 c^2 \psi \\
&= -\hbar^2 \left( \partial_\mu \partial^\mu \mathcal{R} - 2i \frac{\partial^\mu S}{\hbar} \partial_\mu \mathcal{R} - \frac{\partial_\mu S \partial^\mu S}{\hbar^2} \mathcal{R} - i \frac{\partial_\mu \partial^\mu S}{\hbar} \mathcal{R} + \Gamma^\nu_{\nu\mu} \partial^\mu \mathcal{R} - i \frac{\Gamma^\nu_{\nu\mu} \partial^\mu S}{\hbar} \mathcal{R} \right) e^{-i\frac{\xi}{\hbar}} - m^2 c^2 \psi \quad (11.1) \\
&= \left( -\hbar^2 \frac{\Box_g \mathcal{R}}{\mathcal{R}} + \partial_\mu S \partial^\mu S + \frac{i\hbar}{\rho} \nabla_\mu (\rho \partial^\mu S) \right) \mathcal{R}e^{-i\frac{\xi}{\hbar}} - m^2 c^2 \psi \\
&= \left[ -\hbar^2 \frac{\Box_g \mathcal{R}}{\mathcal{R}} + g^{\mu\nu} \partial_\mu S \partial_\nu S + m^2 c^2 \right] \psi + \frac{i\hbar}{\rho} \nabla_\mu (\rho \partial^\mu S) \psi
\end{aligned}
$$

### 11.1.3 Computation of $S_m = \int \mathcal{L}_m \sqrt{g} d^4 x ds$ with $\psi = \mathcal{R}e^{-i\frac{\xi}{\hbar}}$ and $\mathcal{R}^2 = \rho$

$$ 
\begin{aligned}
S_m &= \int \mathcal{L}_m \sqrt{g} d^4 x ds = \int \left( \frac{\hbar^2}{2m^2 c^2} g^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi^* \partial_\nu \psi) - |\psi|^2 \right) \sqrt{g} d^4 x ds \\
&= \int \left( \rho(g^{\mu\nu} \frac{\partial_\mu S}{mc} \frac{\partial_\nu S}{mc} - 1) + \frac{\hbar^2}{m^2 c^2} g^{\mu\nu} \partial_\mu \mathcal{R} \partial_\nu \mathcal{R} \right) \sqrt{g} d^4 x ds \quad (11.2) \\
&= \int \left( \rho(g^{\mu\nu} \frac{\partial_\mu S}{mc} \frac{\partial_\nu S}{mc} - 1) - \frac{\hbar^2}{m^2 c^2} \mathcal{R} \Box_g \mathcal{R} \right) \sqrt{g} d^4 x ds + \frac{\hbar^2}{m^2 c^2} \int \partial_\mu (\mathcal{R}g^{\mu\nu} \partial_\nu \mathcal{R} \sqrt{g}) d^4 x ds
\end{aligned}
$$

Comparing with equation (11.1), we see that $\mathcal{L}_m$ vanishes up to a boundary term.
## 11.2 Annex 2: On the energy momentum tensor of the Klein Gordon field

From $T_{\alpha\beta} = \frac{\partial \mathcal{L}_m}{\partial g^{\alpha\beta}} - \frac{1}{2}g_{\alpha\beta} \mathcal{L}_m \Leftrightarrow T^{\alpha\beta} = \frac{1}{2} \frac{\hbar^2}{m^2 c^2} (\partial^\alpha \psi \partial^\beta \psi^* + \partial^\beta \psi \partial^\alpha \psi^*) - \frac{g^{\alpha\beta}}{2} \left( \frac{\hbar^2 g_{\mu\nu}}{2m^2c^2} (\partial^\mu \psi \partial^\nu \psi^* + \partial^\nu \psi \partial^\mu \psi^*) - \psi \psi^* \right)$, we compute $\frac{2m^2c^2}{\hbar^2} \nabla_\alpha T^{\alpha\beta}$. We make use of the equations of the motion $\Box_g \psi + \frac{m^2c^2}{\hbar^2} \psi = 0$ and the commutation of the covariant with the metric according to $\nabla_\alpha g^{\mu\nu} = \partial_\alpha g^{\mu\nu} + \Gamma^\mu_{\alpha\lambda} g^{\lambda\nu} + \Gamma^\nu_{\alpha\lambda} g^{\lambda\mu} = 0$.

$$ \begin{aligned} \frac{2m^2c^2}{\hbar^2} \nabla_\alpha T^{\alpha\beta} &= \nabla_\alpha \left[ (\partial^\alpha \psi \partial^\beta \psi^* + \partial^\beta \psi \partial^\alpha \psi^*) - g^{\alpha\beta} \left( \frac{g_{\mu\nu}}{2} (\partial^\mu \psi \partial^\nu \psi^* + \partial^\nu \psi \partial^\mu \psi^*) - \frac{m^2c^2}{\hbar^2} \psi \psi^* \right) \right] \\ &= (\nabla_\alpha \partial^\alpha \psi) \partial^\beta \psi^* + \partial^\alpha \psi \nabla_\alpha \partial^\beta \psi^* + (\nabla_\alpha \partial^\alpha \psi^*) \partial^\beta \psi + \partial^\alpha \psi^* \nabla_\alpha \partial^\beta \psi - \frac{1}{2} \partial^\beta (\partial_\mu \psi \partial^\mu \psi^* + \partial_\mu \psi^* \partial^\mu \psi) + \frac{m^2c^2}{\hbar^2} \psi^* \partial^\beta \psi + \frac{m^2c^2}{\hbar^2} \psi^* \partial^\beta \psi^* \\ &= \left( \Box_g \psi + \frac{m^2c^2}{\hbar^2} \psi \right) \partial^\beta \psi^* + \partial^\alpha \psi \nabla_\alpha \partial^\beta \psi^* + \left( \Box_g \psi + \frac{m^2c^2}{\hbar^2} \psi \right) \partial^\beta \psi^* + \partial^\alpha \psi^* \nabla_\alpha \partial^\beta \psi - \partial^\beta (\partial_\mu \psi \partial^\mu \psi^*) = \partial^\alpha \psi \nabla_\alpha \partial^\beta \psi^* + \partial^\alpha \psi^* \nabla_\alpha \partial^\beta \psi - (\partial^\mu \psi^* \partial^\beta \partial_\mu \psi + \partial_\mu \psi \partial^\beta \partial^\mu \psi^*) \end{aligned} \quad (11.3) $$

We compute separately $\partial^\mu \psi^* \partial^\beta \partial_\mu \psi = \partial^\mu \psi^* \nabla_\mu \partial^\beta \psi + \partial_\nu \psi^* g^{\mu\nu} \Gamma^\sigma_{\mu\lambda} g^{\lambda\beta} \partial_\sigma \psi = \partial^\mu \psi^* \nabla_\mu \partial^\beta \psi + \partial_\sigma \psi g^{\lambda\beta} \Gamma^\sigma_{\mu\lambda} g^{\mu\nu} \partial_\nu \psi^*$ according to:

$$ \partial^\beta \partial_\mu \psi = g^{\sigma\beta} \partial_\sigma \partial_\mu \psi = \partial_\mu (g^{\sigma\beta} \partial_\sigma \psi) - (\partial_\mu g^{\sigma\beta}) \partial_\sigma \psi = \partial_\mu \partial^\beta \psi + (\Gamma^\beta_{\mu\lambda} g^{\lambda\sigma} + \Gamma^\sigma_{\mu\lambda} g^{\lambda\beta}) \partial_\sigma \psi = \nabla_\mu \partial^\beta \psi + \Gamma^\sigma_{\mu\lambda} g^{\lambda\beta} \partial_\sigma \psi \quad (11.4) $$

Similarly we compute $\partial_\mu \psi \partial^\beta \partial^\mu \psi^* = -\partial_\mu \psi g^{\nu\beta} \Gamma^\mu_{\nu\lambda} g^{\sigma\lambda} \partial_\sigma \psi^* - \partial_\mu \psi g^{\nu\beta} \Gamma^\sigma_{\nu\lambda} g^{\mu\lambda} \partial_\sigma \psi^* + \partial^\sigma \psi \nabla_\sigma \partial^\beta \psi^* + \partial_\mu \psi g^{\mu\sigma} \Gamma^\sigma_{\sigma\lambda} g^{\lambda\beta} \partial_\nu \psi^*$ according to:

$$ \begin{aligned} \partial^\beta \partial^\mu \psi^* &= g^{\nu\beta} \partial_\nu (g^{\mu\sigma} \partial_\sigma \psi^*) = g^{\nu\beta} (\partial_\nu g^{\mu\sigma}) \partial_\sigma \psi^* + g^{\nu\beta} g^{\mu\sigma} \partial_\sigma \partial_\nu \psi^* = g^{\nu\beta} (\partial_\nu g^{\mu\sigma}) \partial_\sigma \psi^* + g^{\mu\sigma} \partial_\sigma (g^{\nu\beta} \partial_\nu \psi^*) - g^{\mu\sigma} (\partial_\sigma g^{\nu\beta}) \partial_\nu \psi^* \\ &= -g^{\nu\beta} \Gamma^\mu_{\nu\lambda} g^{\sigma\lambda} \partial_\sigma \psi^* - g^{\nu\beta} \Gamma^\sigma_{\nu\lambda} g^{\mu\lambda} \partial_\sigma \psi^* + g^{\mu\sigma} \partial_\sigma \partial^\beta \psi^* + g^{\mu\sigma} \Gamma^\sigma_{\sigma\lambda} g^{\lambda\beta} \partial_\nu \psi^* + g^{\mu\sigma} \Gamma^\nu_{\sigma\lambda} g^{\lambda\beta} \partial_\nu \psi^* \\ &= -g^{\nu\beta} \Gamma^\mu_{\nu\lambda} g^{\sigma\lambda} \partial_\sigma \psi^* - g^{\nu\beta} \Gamma^\sigma_{\nu\lambda} g^{\mu\lambda} \partial_\sigma \psi^* + g^{\mu\sigma} \nabla_\sigma \partial^\beta \psi^* + g^{\mu\sigma} \Gamma^\nu_{\sigma\lambda} g^{\lambda\beta} \partial_\nu \psi^* \end{aligned} \quad (11.5) $$

Then $\partial^\alpha \psi \nabla_\alpha \partial^\beta \psi^* + \partial^\alpha \psi^* \nabla_\alpha \partial^\beta \psi - (\partial^\mu \psi^* \partial^\beta \partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi \partial^\beta \partial^\mu \psi^*) = -\partial_\sigma \psi g^{\lambda\beta} \Gamma^\sigma_{\mu\lambda} g^{\mu\nu} \partial_\nu \psi^* + \partial_\mu \psi g^{\nu\beta} \Gamma^\sigma_{\nu\lambda} g^{\mu\lambda} \partial_\sigma \psi^* + \partial_\mu \psi g^{\nu\beta} \Gamma^\sigma_{\nu\lambda} g^{\mu\lambda} \partial_\sigma \psi^* - \partial_\mu \psi g^{\mu\sigma} \Gamma^\sigma_{\sigma\lambda} g^{\lambda\beta} \partial_\nu \psi^* = 0$.
We conclude that $\frac{2m^2c^2}{\hbar^2} \nabla_\alpha T^{\alpha\beta} = 0$ in equation (11.3).

With the anzats $\psi = \mathcal{R}e^{-i\frac{S}{\hbar}}$ with $\mathcal{R}^2\rho$, we evaluate $T_{\alpha\beta}$ in the eikonal approximation. The term canceled vanishes according to the equations of motion, see equation (11.1).

$$ \begin{aligned} T_{\alpha\beta} &= \rho \frac{\partial_\alpha S}{mc} \frac{\partial_\beta S}{mc} + \frac{\hbar^2}{m^2c^2} \partial_\alpha \mathcal{R} \partial_\beta \mathcal{R} - \frac{g_{\alpha\beta}}{2} \left( \rho(g^{\mu\nu} \frac{\partial_\mu S}{mc} \frac{\partial_\nu S}{mc} - 1) + \frac{\hbar^2}{m^2c^2} g^{\mu\nu} \partial_\mu \mathcal{R} \partial_\nu \mathcal{R} \right) \\ T_{\alpha\beta} &= \rho \frac{\partial_\alpha S}{mc} \frac{\partial_\beta S}{mc} + \frac{\hbar^2}{m^2c^2} \partial_\alpha \mathcal{R} \partial_\beta \mathcal{R} - \frac{g_{\alpha\beta}}{2} \left( \rho(g^{\mu\nu} \frac{\partial_\mu S}{mc} \frac{\partial_\nu S}{mc} - 1) + \frac{\hbar^2}{m^2c^2} \left( \frac{\partial_\mu (g^{\mu\nu} \mathcal{R} \partial_\nu \mathcal{R} \sqrt{g})}{\sqrt{g}} - \mathcal{R} \Box_g \mathcal{R} \right) \right) \\ T_{\alpha\beta} &= \rho \frac{\partial_\alpha S}{mc} \frac{\partial_\beta S}{mc} + \frac{\hbar^2}{m^2c^2} \partial_\alpha \mathcal{R} \partial_\beta \mathcal{R} - \frac{g_{\alpha\beta}}{2} \frac{\hbar^2}{m^2c^2} \frac{\partial_\mu (g^{\mu\nu} \mathcal{R} \partial_\nu \mathcal{R} \sqrt{g})}{\sqrt{g}} - \frac{g_{\alpha\beta}}{2} \rho \left( \frac{g^{\mu\nu} \partial_\mu S}{mc} \frac{\partial_\nu S}{mc} - 1 - \frac{\hbar^2}{m^2c^2} \frac{\Box_g \mathcal{R}}{\mathcal{R}} \right) \\ T_{\alpha\beta} &= \rho \frac{\partial_\alpha S}{mc} \frac{\partial_\beta S}{mc} + \frac{\hbar^2}{m^2c^2} \left( \partial_\alpha \mathcal{R} \partial_\beta \mathcal{R} - \frac{g_{\alpha\beta}}{2} \frac{\partial_\mu (g^{\mu\nu} \mathcal{R} \partial_\nu \mathcal{R} \sqrt{g})}{\sqrt{g}} \right) \approx \rho \frac{\partial_\alpha S}{mc} \frac{\partial_\beta S}{mc} + \epsilon(\hbar^2) \end{aligned} \quad (11.6) $$
## 11.3 Annex 3 : Einstein's Tensors in the Kaluza-Klein Theory

### 11.3.1 Conventions

In this annexe, we compute the Einstein's tensors associated to the Lagrangian of equation (4.2) :

$$L_m = \rho \left( (g_{\mu\nu} - \Phi^2 \frac{qA_\mu}{Mc} \frac{qA_\nu}{Mc}) \dot{x}^\mu \dot{x}^\nu + (\Phi^2 \frac{qA_\mu}{Mc} \dot{x}^\mu \dot{x}^5 + \Phi^2 \frac{qA_\nu}{Mc} \dot{x}^\nu \dot{x}^5) - \Phi^2 \dot{x}^5 \dot{x}^5 \right) + \rho \dot{x}^6 \dot{x}^6 \quad (11.7)$$

where $M = \sqrt{\frac{\mu_0 g^2}{16\pi G}}$. To simplify computations, we use the following notations :

$$A_{\mu} = -\frac{qA_{\mu}}{Mc} ; \quad F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} = A_{\nu,\mu} - A_{\mu,\nu} = -\frac{q}{Mc}F_{\mu\nu} ; \quad \Phi = i\phi \quad (11.8)$$

such that the Lagrangian takes the form :

$$L_m = \rho \left( (g_{\mu\nu} + \phi^2 A_{\mu} A_{\nu}) \dot{x}^{\mu} \dot{x}^{\nu} + (\phi^2 A_{\mu} \dot{x}^{\mu} \dot{x}^{5} + \phi^2 A_{\nu} \dot{x}^{\nu} \dot{x}^{5}) + \phi^2 \dot{x}^{5} \dot{x}^{5} \right) + \rho \dot{x}^6 \dot{x}^6 \quad (11.9)$$

Since $g_{66}$ is constant ($g_{66} = 1$), we only consider the 5D part of the metric and its inverse, which are :

$$\underline{g_{MN}} = \begin{bmatrix} g_{\mu\nu} + \phi^2 A_{\mu} A_{\nu} & \phi^2 A_{\mu} \\ \phi^2 A_{\nu} & \phi^2 \end{bmatrix} ; \quad \underline{g^{MN}} = \begin{bmatrix} g^{\mu\nu} & -A^{\nu} \\ -A^{\mu} & \frac{1}{\phi^2} + A_{\mu} A^{\mu} \end{bmatrix} \quad (11.10)$$

Then $g^{\mu\nu} = g^{\nu\mu}$. Indices are upped (or lowered) with $g^{\mu\nu}$ (or $g_{\mu\nu}$). $g_{\mu\nu}$, $A_\mu$ and $\phi$ do not depend on $x^5$.

### 11.3.2 Computation of the Christoffel symbols $\underline{\Gamma}^P_{MN}$

We first compute the 5D Christoffel symbols $\underline{\Gamma}^5_{55}$, $\underline{\Gamma}^5_{5\nu}$, $\underline{\Gamma}^5_{\sigma\nu}$, $\underline{\Gamma}^5_{5\sigma}$, $\underline{\Gamma}^5_{5\nu}$, $\underline{\Gamma}^5_{5\nu}$, and $\underline{\Gamma}^\beta_{\mu\nu}$ from the definition :

$$\underline{\Gamma}^P_{MN} = \frac{g^{QP}}{2} \left( \underline{\Gamma}_{QM,N} + \underline{\Gamma}_{NQ,M} - \underline{\Gamma}_{NM,Q} \right) \quad (11.11)$$

$$\begin{aligned} \underline{\Gamma}^5_{55} &= \frac{1}{2} g^{5\mu} (g_{\mu 5,5} + g_{5\mu,5} - g_{55,\mu}) + \frac{1}{2} g^{55} (g_{55,5} + g_{55,5} - g_{55,5}) = -\frac{1}{2} A^\mu (-\partial_\mu \phi^2) \\ &= A^\mu \phi \partial_\mu \phi \end{aligned} \quad (11.12)$$

$$\begin{aligned} \underline{\Gamma}^5_{5\nu} &= \frac{1}{2} g^{5\mu} (g_{\mu 5,\nu} + g_{5\mu,5} - g_{5\nu,\mu}) + \frac{1}{2} g^{55} (g_{55,\nu} + g_{5\nu,5} - g_{55,5}) \\ &= -\frac{A^\mu}{2} (\partial_\nu (\phi^2 A_\mu) - \partial_\mu (\phi^2 A_\nu)) + \frac{1}{2} (\frac{1}{\phi^2} + A_\mu A^\mu) 2\phi \partial_\nu \phi \\ &= -\frac{1}{2} A^\mu \phi^2 F_{\nu\mu} - A^\mu \phi (A_\mu \partial_\nu \phi - A_\nu \partial_\mu \phi) + \left(\frac{\partial_\nu \phi}{\phi} + A_\mu A^\mu \phi \partial_\nu \phi\right) \\ &= -\frac{\phi^2}{2} A^\mu F_{\nu\mu} + A^\mu A_\nu \phi \partial_\mu \phi + \frac{\partial_\nu \phi}{\phi} \end{aligned} \quad (11.13)$$

$$\begin{aligned} \underline{\Gamma}^5_{\sigma\nu} &= \frac{1}{2} g^{5\mu} (g_{\mu\sigma,\nu} + g_{\nu\mu,\sigma} - g_{\sigma\nu,\mu}) + \frac{1}{2} g^{55} (g_{5\sigma,\nu} + g_{5\nu,\sigma} - g_{\sigma\nu,5}) \\ &= -A^\mu \Gamma_{\sigma\nu\mu} - \frac{A^\mu}{2} (\partial_\nu (\phi^2 A_\mu A_\sigma) + \partial_\sigma (\phi^2 A_\nu A_\mu) - \partial_\mu (\phi^2 A_\sigma A_\nu)) \\ &\quad + \frac{1}{2} (\frac{1}{\phi^2} + A_\mu A^\mu) (\partial_\nu (\phi^2 A_\sigma) + \partial_\sigma (\phi^2 A_\nu)) \\ &= -A^\mu \Gamma_{\sigma\nu\mu} - \frac{A^\mu}{2} A_\mu \partial_\nu (\phi^2 A_\sigma) - \frac{A^\mu}{2} \phi^2 A_\sigma \partial_\nu A_\mu - \frac{A^\mu}{2} A_\mu \partial_\sigma (\phi^2 A_\nu) - \frac{A^\mu}{2} \phi^2 A_\nu \partial_\sigma A_\mu \\ &\quad + \frac{A^\mu}{2} \partial_\mu (\phi^2 A_\sigma A_\nu) + \frac{1}{2\phi^2} (\partial_\nu (\phi^2 A_\sigma) + \partial_\sigma (\phi^2 A_\nu)) + \frac{A_\mu A^\mu}{2} (\partial_\nu (\phi^2 A_\sigma) + \partial_\sigma (\phi^2 A_\nu)) \\ &= -A^\mu \Gamma_{\sigma\nu\mu} - \frac{A^\mu}{2} \phi^2 A_\sigma \partial_\nu A_\mu - \frac{A^\mu}{2} \phi^2 A_\nu \partial_\sigma A_\mu + \frac{A^\mu}{2} \phi^2 A_\sigma \partial_\mu A_\nu + \frac{A^\mu}{2} \phi^2 A_\nu \partial_\mu A_\sigma \\ &\quad + A^\mu \phi A_\sigma A_\nu \partial_\mu \phi + (A_\sigma \frac{\partial_\nu \phi}{\phi} + A_\nu \frac{\partial_\sigma \phi}{\phi}) + \frac{1}{2} (\partial_\nu A_\sigma + \partial_\sigma A_\nu) \\ &= -A^\mu \Gamma_{\sigma\nu\mu} + \frac{A^\mu}{2} \phi^2 (A_\nu F_{\mu\sigma} + A_\sigma F_{\mu\nu}) + A^\mu \phi A_\sigma A_\nu \partial_\mu \phi + (A_\sigma \frac{\partial_\nu \phi}{\phi} + A_\nu \frac{\partial_\sigma \phi}{\phi}) \\ &\quad + \frac{1}{2} (\partial_\nu A_\sigma + \partial_\sigma A_\nu) \end{aligned} \quad (11.14)$$
$$
\begin{align}
\underline{\Gamma}_{5\sigma}^{\nu} &= \frac{1}{2} g^{\nu\mu} (g_{\mu 5,\sigma} + g_{\sigma\mu 5,} - g_{5\sigma,\mu}) + \frac{1}{2} g^{\nu 5} (g_{55,\sigma} + g_{\sigma 5,5} - g_{5\sigma 5,}) \nonumber \\
&= \frac{1}{2} g^{\nu\mu} (\partial_{\sigma}(\phi^2 \mathbf{A}_{\mu}) - \partial_{\mu}(\phi^2 \mathbf{A}_{\sigma})) - \frac{\mathbf{A}^{\nu}}{2} (2\phi \partial_{\sigma} \phi) \nonumber \\
&= \frac{1}{2} g^{\nu\mu} (\phi^2 \mathbf{F}_{\sigma\mu} + 2\phi (\mathbf{A}_{\mu} \partial_{\sigma} \phi - \mathbf{A}_{\sigma} \partial_{\mu} \phi)) - \phi \mathbf{A}^{\nu} \partial_{\sigma} \phi \nonumber \\
&= \phi^2 \frac{1}{2} \mathbf{F}_{\sigma}^{\nu} - \phi \mathbf{A}_{\sigma} \partial^{\nu} \phi \tag{11.15}
\end{align}
$$

$$
\begin{align}
\underline{\Gamma}_{55}^{\nu} &= \frac{1}{2} g^{\nu\mu} (g_{\mu 5,5} + g_{5\mu,5} - g_{55,\mu}) + \frac{1}{2} g^{\nu 5} (g_{55,5} + g_{55,5} - g_{55,5}) = \frac{1}{2} g^{\nu\mu} (-2\phi \partial_{\mu} \phi) \tag{11.16} \\
&= -\phi \partial^{\nu} \phi \notag
\end{align}
$$

$$
\underline{\Gamma}_{5\nu}^{\nu} = -\phi \mathbf{A}_{\nu} \partial^{\nu} \phi \qquad (11.17)
$$

The last Christoffel symbols has been computed from equation (11.15) with $\nu = \sigma$. We deduce :

$$
\underline{\Gamma}_{5M}^{M} = \underline{\Gamma}_{5\nu}^{\nu} + \underline{\Gamma}_{55}^{5} = 0 \qquad (11.18)
$$

$$
\begin{align}
\underline{\Gamma}_{\mu\nu}^{\beta} &= \frac{1}{2} g^{\beta\sigma} (g_{\sigma\mu,\nu} + g_{\nu\sigma,\mu} - g_{\mu\nu,\sigma}) + \frac{1}{2} g^{\beta 5} (g_{5\mu,\nu} + g_{\nu 5,\mu} - g_{\mu\nu,5}) \nonumber \\
&= \Gamma_{\mu\nu}^{\beta} + \frac{1}{2} g^{\beta\sigma} (\partial_{\nu}(\phi^2 \mathbf{A}_{\sigma} \mathbf{A}_{\mu}) + \partial_{\mu}(\phi^2 \mathbf{A}_{\nu} \mathbf{A}_{\sigma}) - \partial_{\sigma}(\phi^2 \mathbf{A}_{\mu} \mathbf{A}_{\nu})) - \frac{\mathbf{A}^{\beta}}{2} (\partial_{\nu}(\phi^2 \mathbf{A}_{\mu}) + \partial_{\mu}(\phi^2 \mathbf{A}_{\nu})) \nonumber \\
&= \Gamma_{\mu\nu}^{\beta} + \frac{1}{2} g^{\beta\sigma} \left( \underline{\mathbf{A}}_{\sigma} \partial_{\nu} (\phi^2 \underline{\mathbf{A}}_{\mu}) + (\partial_{\nu} \mathbf{A}_{\sigma}) \phi^2 \mathbf{A}_{\mu} + \underline{\mathbf{A}}_{\sigma} \partial_{\mu} (\phi^2 \underline{\mathbf{A}}_{\nu}) + (\partial_{\mu} \mathbf{A}_{\sigma}) \phi^2 \mathbf{A}_{\nu} - \partial_{\sigma}(\phi^2 \mathbf{A}_{\mu} \mathbf{A}_{\nu}) \right) \nonumber \\
&\quad - \frac{\mathbf{A}^{\beta}}{2} \left( \partial_{\nu} (\phi^2 \underline{\mathbf{A}}_{\mu}) + \partial_{\mu} (\phi^2 \underline{\mathbf{A}}_{\nu}) \right) \tag{11.19} \\
&= \Gamma_{\mu\nu}^{\beta} + \frac{1}{2} g^{\beta\sigma} \left( (\partial_{\nu} \mathbf{A}_{\sigma}) \phi^2 \mathbf{A}_{\mu} + (\partial_{\mu} \mathbf{A}_{\sigma}) \phi^2 \mathbf{A}_{\nu} - \mathbf{A}_{\mu} \mathbf{A}_{\nu} \partial_{\sigma} \phi^2 - \mathbf{A}_{\nu} \phi^2 \partial_{\sigma} \mathbf{A}_{\mu} - \phi^2 \mathbf{A}_{\mu} \partial_{\sigma} \mathbf{A}_{\nu} \right) \nonumber \\
&= \Gamma_{\mu\nu}^{\beta} + \frac{1}{2} g^{\beta\sigma} \left( \phi^2 (\mathbf{A}_{\mu} \mathbf{F}_{\nu\sigma} + \mathbf{A}_{\nu} \mathbf{F}_{\mu\sigma}) - 2\phi \mathbf{A}_{\mu} \mathbf{A}_{\nu} \partial_{\sigma} \phi \right) \nonumber \\
&= \Gamma_{\mu\nu}^{\beta} + \frac{\phi^2}{2} (\mathbf{A}_{\mu} \mathbf{F}_{\nu}^{\beta} + \mathbf{A}_{\nu} \mathbf{F}_{\mu}^{\beta}) - \phi \mathbf{A}_{\mu} \mathbf{A}_{\nu} \partial^{\beta} \phi \nonumber
\end{align}
$$

We deduce :

$$
\underline{\Gamma}_{M\nu}^{M} = \underline{\Gamma}_{\sigma\nu}^{\sigma} + \underline{\Gamma}_{5\nu}^{5} = \Gamma_{\sigma\nu}^{\sigma} + \frac{\partial_{\nu}\phi}{\phi} \qquad (11.20)
$$

We summarize :

$$
\begin{align}
\underline{\Gamma}_{55}^{5} &= \mathbf{A}^{\mu}\phi\partial_{\mu}\phi \notag \\
\underline{\Gamma}_{5\nu}^{5} &= -\frac{\phi^2}{2}\mathbf{A}^{\mu}\mathbf{F}_{\nu\mu} + \mathbf{A}^{\mu}\mathbf{A}_{\nu}\phi\partial_{\mu}\phi + \frac{\partial_{\nu}\phi}{\phi} \notag \\
\underline{\Gamma}_{\sigma\nu}^{5} &= -\mathbf{A}^{\mu}\Gamma_{\sigma\nu\mu} + \frac{\mathbf{A}^{\mu}}{2}\phi^2(\mathbf{A}_{\nu}\mathbf{F}_{\mu\sigma} + \mathbf{A}_{\sigma}\mathbf{F}_{\mu\nu}) + \mathbf{A}^{\mu}\phi\mathbf{A}_{\sigma}\mathbf{A}_{\nu}\partial_{\mu}\phi \notag \\
&\quad + (\mathbf{A}_{\sigma}\frac{\partial_{\nu}\phi}{\phi} + \mathbf{A}_{\nu}\frac{\partial_{\sigma}\phi}{\phi}) + \frac{1}{2}(\partial_{\nu}\mathbf{A}_{\sigma} + \partial_{\sigma}\mathbf{A}_{\nu}) \tag{11.21} \\
\underline{\Gamma}_{5\sigma}^{\nu} &= \phi^2\frac{1}{2}\mathbf{F}_{\sigma}^{\nu} - \phi\mathbf{A}_{\sigma}\partial^{\nu}\phi \notag \\
\underline{\Gamma}_{55}^{\nu} &= -\phi\partial^{\nu}\phi \notag \\
\underline{\Gamma}_{5\nu}^{\nu} &= -\phi\mathbf{A}_{\nu}\partial^{\nu}\phi \notag \\
\underline{\Gamma}_{\mu\nu}^{\beta} &= \Gamma_{\mu\nu}^{\beta} + \frac{\phi^2}{2}(\mathbf{A}_{\mu}\mathbf{F}_{\nu}^{\beta} + \mathbf{A}_{\nu}\mathbf{F}_{\mu}^{\beta}) - \phi\mathbf{A}_{\mu}\mathbf{A}_{\nu}\partial^{\beta}\phi \notag
\end{align}
$$

11.3.3 Computation of the Ricci tensors $\underline{R}_{MN}$ and the Ricci Scalar $\underline{R}$

We can now compute the 5D Ricci tensors $\underline{R}_{44}$, $\underline{R}_{\mu 4}$, $\underline{R}_{\mu \nu}$ and the 5D Ricci scalar $\underline{R}$, with the definition :

$$
\underline{R}_{MN} = \partial_P \underline{\Gamma}_{MN}^P - \partial_M \underline{\Gamma}_{PN}^P + \underline{\Gamma}_{PQ}^P \underline{\Gamma}_{MN}^Q - \underline{\Gamma}_{MQ}^P \underline{\Gamma}_{NP}^Q \quad (11.22)
$$

$$
\begin{equation}
\begin{split}
\underline{R}_{MN} ={}& \left( \partial_{\nu} \underline{\Gamma}_{MN}^{\nu} + \partial_{5} \underline{\Gamma}_{MN}^{5} \right) - \left( \partial_{M} \underline{\Gamma}_{\nu N}^{\nu} + \partial_{M} \underline{\Gamma}_{5N}^{5} \right) + \left( \underline{\Gamma}_{\nu \sigma}^{\nu} \underline{\Gamma}_{MN}^{\sigma} + \underline{\Gamma}_{5 \sigma}^{5} \underline{\Gamma}_{MN}^{\sigma} \right. \\
& \qquad \left. + \underline{\Gamma}_{\nu 5}^{\nu} \underline{\Gamma}_{MN}^{5} + \underline{\Gamma}_{5 5}^{5} \underline{\Gamma}_{MN}^{5} \right) - \left( \underline{\Gamma}_{M \sigma}^{\nu} \underline{\Gamma}_{N \nu}^{\sigma} + \underline{\Gamma}_{M \sigma}^{5} \underline{\Gamma}_{N 5}^{\sigma} + \underline{\Gamma}_{M 5}^{\nu} \underline{\Gamma}_{N \nu}^{5} + \underline{\Gamma}_{M 5}^{5} \underline{\Gamma}_{N 5}^{5} \right)
\end{split}
\tag{11.23}
\end{equation}
$$
We will use letter and numbers to indicate which term cancel with which other term. By example, Term[A1] cancels with -Term[-A1]. When useful, the terms which are kept are indicated with the letter K, with the notation Term[K1], such that the reader can see easily where they appear again.

We start with $R_{55}$:

$$ \underline{R_{55}} = \partial_{\nu} \underline{\Gamma}_{55}^{\nu} - \partial_{5} \underline{\Gamma}_{\nu 5}^{\nu} - \partial_{5} \underline{\Gamma}_{55}^{5} + \underline{\Gamma}_{\nu \sigma}^{\nu} \underline{\Gamma}_{55}^{\sigma} + \underline{\Gamma}_{5 \sigma}^{5} \underline{\Gamma}_{55}^{\sigma} \text{[A1]} + \underline{\Gamma}_{\nu 5}^{\nu} \underline{\Gamma}_{55}^{5} + \underline{\Gamma}_{55}^{5} \underline{\Gamma}_{55}^{5} \text{[A2]} - \underline{\Gamma}_{5 \sigma}^{\nu} \underline{\Gamma}_{5 \nu}^{\sigma} - \underline{\Gamma}_{5 \sigma}^{5} \underline{\Gamma}_{55}^{\sigma} \text{[-A1]} - \underline{\Gamma}_{55}^{\nu} \underline{\Gamma}_{5 \nu}^{5} - \underline{\Gamma}_{55}^{5} \underline{\Gamma}_{55}^{5} \text{[-A2]} $$

$$ \underline{R_{55}} = \partial_{\nu} \underline{\Gamma}_{55}^{\nu} + \underline{\Gamma}_{\nu \sigma}^{\nu} \underline{\Gamma}_{55}^{\sigma} + \underline{\Gamma}_{\nu 5}^{\nu} \underline{\Gamma}_{55}^{5} - \underline{\Gamma}_{5 \sigma}^{\nu} \underline{\Gamma}_{5 \nu}^{\sigma} - \underline{\Gamma}_{55}^{\nu} \underline{\Gamma}_{5 \nu}^{5} $$

$$ \underline{R_{55}} = \partial_{\nu} [-\phi \partial^{\nu} \phi] + \left[ \underline{\Gamma}_{\nu \sigma}^{\nu} + \frac{\phi}{2} (\underline{A}_{\nu} \underline{F}_{\sigma}^{\nu} + \underline{A}_{\sigma} \underline{F}_{\nu}^{\nu}) - \phi \underline{A}_{\nu} \underline{A}_{\sigma} \partial^{\nu} \phi \right] [-\phi \partial^{\sigma} \phi] + [-\phi \underline{A}_{\nu} \partial^{\nu} \phi] [\phi \underline{A}^{\mu} \partial_{\mu} \phi] - \left[ \frac{\phi^2}{2} \underline{F}_{\sigma}^{\nu} - \phi \underline{A}_{\sigma} \partial^{\nu} \phi \right] \left[ \frac{\phi^2}{2} \underline{F}_{\nu}^{\sigma} - \phi \underline{A}_{\nu} \partial^{\sigma} \phi \right] \\ - [-\phi \partial^{\nu} \phi] \left[ -\frac{1}{2} \underline{A}^{\mu} \phi \underline{F}_{\nu \mu} + \phi \underline{A}^{\mu} \underline{A}_{\nu} \partial_{\mu} \phi + \frac{\partial_{\nu} \phi}{\phi} \right] $$

$$ \underline{R_{55}} = \left[ -\partial_{\nu} \phi \partial^{\nu} \phi \text{[A3]} - \phi \partial_{\nu} \partial^{\nu} \phi \right] + \left[ -\phi \underline{\Gamma}_{\nu \sigma}^{\nu} \partial^{\sigma} \phi - \frac{\phi^2}{2} \underline{A}_{\nu} \underline{F}_{\sigma}^{\nu} \partial^{\sigma} \phi \text{[A4]} + \phi^2 (\underline{A}_{\nu} \partial^{\nu} \phi)^2 \text{[A6]} \right] \left[ -\phi^2 (\underline{A}_{\nu} \partial^{\nu} \phi)^2 \text{[A6]} \right] \\ + \left[ \frac{\phi^4}{4} \underline{F}_{\sigma}^{\nu} \underline{F}_{\nu}^{\sigma} + \phi^2 \underline{A}_{\nu} \underline{F}_{\sigma}^{\nu} \partial^{\sigma} \phi \text{[2 \times A4]} - \phi^2 (\underline{A}_{\nu} \partial^{\nu} \phi) \text{[A5]} \right] + \left[ -\frac{\phi^2}{2} \underline{A}^{\mu} \underline{F}_{\nu \mu} \partial^{\nu} \phi \text{[A4]} + \phi^2 (\underline{A}_{\nu} \partial^{\nu} \phi)^2 \text{[A5]} + \partial^{\nu} \phi \partial_{\nu} \phi \text{[A3]} \right] $$

$$ \underline{R_{55}} = -\phi \partial_{\nu} \partial^{\nu} \phi - \underline{\Gamma}_{\nu \sigma}^{\nu} \phi \partial^{\sigma} \phi - \frac{\phi^4}{4} \underline{F}_{\sigma}^{\nu} \underline{F}_{\nu}^{\sigma} = -\phi \square_g \phi + \frac{\phi^4}{4} \underline{F}^{\sigma \nu} \underline{F}_{\sigma \nu} $$

Where the above equation defines the covariant laplacian $\square_g$ acting on scalars.

We continue with $R_{5\beta}$:

$$ \underline{R_{5\beta}} = \partial_{\nu} \underline{\Gamma}_{5\beta}^{\nu} - \partial_{5} \underline{\Gamma}_{\nu\beta}^{\nu} - \partial_{5} \underline{\Gamma}_{5\beta}^{5} + \underline{\Gamma}_{\nu\sigma}^{\nu} \underline{\Gamma}_{5\beta}^{\sigma} + \underline{\Gamma}_{5\sigma}^{5} \underline{\Gamma}_{5\beta}^{\sigma} \text{[B1]} + \underline{\Gamma}_{\nu 5}^{\nu} \underline{\Gamma}_{5\beta}^{5} + \underline{\Gamma}_{55}^{5} \underline{\Gamma}_{5\beta}^{5} \text{[B2]} - \underline{\Gamma}_{5\sigma}^{\nu} \underline{\Gamma}_{\beta\nu}^{\sigma} - \underline{\Gamma}_{5\sigma}^{5} \underline{\Gamma}_{\beta 5}^{\sigma} \text{[-B1]} - \underline{\Gamma}_{55}^{\nu} \underline{\Gamma}_{\beta\nu}^{5} - \underline{\Gamma}_{55}^{5} \underline{\Gamma}_{\beta 5}^{5} \text{[-B2]} $$

$$ \underline{R_{5\beta}} = \partial_{\nu} \underline{\Gamma}_{5\beta}^{\nu} + \underline{\Gamma}_{\nu \sigma}^{\nu} \underline{\Gamma}_{5\beta}^{\sigma} + \underline{\Gamma}_{\nu 5}^{\nu} \underline{\Gamma}_{5\beta}^{5} - \underline{\Gamma}_{5 \sigma}^{\nu} \underline{\Gamma}_{\beta \nu}^{\sigma} - \underline{\Gamma}_{55}^{\nu} \underline{\Gamma}_{\beta \nu}^{5} $$

$$ \underline{R_{5\beta}} = \partial_{\nu} \left[ \frac{\phi^2}{2} \underline{F}_{\beta}^{\nu} - \underline{A}_{\beta} \phi \partial^{\nu} \phi \right] + \left[ \underline{\Gamma}_{\sigma \nu}^{\nu} + \frac{\phi^2}{2} \underline{A}_{\nu} \underline{F}_{\sigma}^{\nu} - \phi \underline{A}_{\sigma} \underline{A}_{\nu} \partial^{\nu} \phi \right] \left[ \frac{\phi^2}{2} \underline{F}_{\beta}^{\sigma} - \underline{A}_{\beta} \phi \partial^{\sigma} \phi \right] + [-\phi \underline{A}_{\nu} \partial^{\nu} \phi] \left[ -\frac{\phi^2}{2} \underline{A}^{\mu} \underline{F}_{\beta \mu} + \phi \underline{A}^{\mu} \underline{A}_{\beta} \partial_{\mu} \phi + \frac{\partial_{\beta} \phi}{\phi} \right] \\ - \left[ \frac{\phi^2}{2} \underline{F}_{\sigma}^{\nu} - \phi \underline{A}_{\sigma} \partial^{\nu} \phi \right] \left[ \underline{\Gamma}_{\nu \beta}^{\sigma} + \frac{\phi^2}{2} (\underline{A}_{\nu} \underline{F}_{\beta}^{\sigma} + \underline{A}_{\beta} \underline{F}_{\nu}^{\sigma}) - \phi \underline{A}_{\nu} \underline{A}_{\beta} \partial^{\sigma} \phi \right] \\ - [-\phi \partial^{\nu} \phi] \left[ -\underline{A}^{\mu} \underline{\Gamma}_{\beta \nu \mu} + \frac{\underline{A}^{\mu}}{2} \phi^2 (\underline{A}_{\nu} \underline{F}_{\mu \beta} + \underline{A}_{\beta} \underline{F}_{\mu \nu}) + \underline{A}^{\mu} \phi \underline{A}_{\beta} \underline{A}_{\nu} \partial_{\mu} \phi + (\underline{A}_{\beta} \frac{\partial_{\nu} \phi}{\phi} + \underline{A}_{\nu} \frac{\partial_{\beta} \phi}{\phi}) + \frac{\partial_{\nu} \underline{A}_{\beta} + \partial_{\beta} \underline{A}_{\nu}}{2} \right] $$
$$
\begin{aligned}
\underline{R}_{5\beta} = & \left[ \phi \partial_{\nu} \phi \mathbf{F}_{\beta}^{\nu \text{[K2]}} + \frac{\phi^2}{2} \partial_{\nu} \mathbf{F}_{\beta}^{\nu \text{[K1]}} - \mathbf{A}_{\beta} \partial_{\nu} \phi \partial^{\nu} \phi^{\text{[B3]}} - \phi \partial^{\nu} \phi \partial_{\nu} \mathbf{A}_{\beta}^{\text{[K2]}} - \phi \mathbf{A}_{\beta} \partial_{\nu} \partial^{\nu} \phi^{\text{[K3]}} \right] \\
& + \left[ \frac{\phi^2}{2} \Gamma_{\sigma\nu}^{\nu} \mathbf{F}_{\beta}^{\sigma \text{[K1]}} - \phi \Gamma_{\sigma\nu}^{\nu} \mathbf{A}_{\beta} \partial^{\sigma} \phi^{\text{[K3]}} + \frac{\phi^4}{4} \mathbf{A}_{\nu} \mathbf{F}_{\sigma}^{\nu} \mathbf{F}_{\beta}^{\sigma} \right. \text{[B4]} \\
& - \frac{\phi^3}{2} \mathbf{A}_{\nu} \mathbf{A}_{\beta} \mathbf{F}_{\sigma}^{\nu} \partial^{\sigma} \phi^{\text{[B5]}} - \frac{\phi^3}{2} \mathbf{A}_{\sigma} \mathbf{A}_{\nu} \mathbf{F}_{\beta}^{\sigma} \partial^{\nu} \phi^{\text{[B6]}} + \phi^2 (\mathbf{A}_{\sigma} \partial^{\sigma} \phi)^2 \mathbf{A}_{\beta}^{\text{[B7]}} \right] \\
& + \left[ \frac{\phi^3}{2} \mathbf{A}_{\nu} \mathbf{A}^{\mu} \partial^{\nu} \phi \mathbf{F}_{\beta\mu}^{\text{[B9]}} - \phi^2 (\mathbf{A}_{\nu} \partial^{\nu} \phi)^2 \mathbf{A}_{\beta}^{\text{[B10]}} - \mathbf{A}_{\nu} \partial^{\nu} \phi \partial_{\beta} \phi^{\text{[B11]}} \right] \\
& + \left[ \frac{-\phi^2}{2} \Gamma_{\nu\beta}^{\sigma} \mathbf{F}_{\sigma}^{\nu \text{[K1]}} - \frac{\phi^4}{4} \mathbf{A}_{\nu} \mathbf{F}_{\sigma}^{\nu} \mathbf{F}_{\beta}^{\sigma \text{[B4]}} - \frac{\phi^4}{4} \mathbf{A}_{\beta} \mathbf{F}_{\sigma}^{\nu} \mathbf{F}_{\nu}^{\sigma \text{[K3]}} + \frac{\phi^3}{2} \mathbf{A}_{\nu} \mathbf{A}_{\beta} \mathbf{F}_{\sigma}^{\nu} \partial^{\sigma} \phi^{\text{[B5]}} + \phi \mathbf{A}_{\sigma} \partial^{\nu} \phi \Gamma_{\nu\beta}^{\sigma \text{[B12]}} + \frac{\phi^3}{2} \mathbf{A}_{\sigma} \mathbf{A}_{\nu} \mathbf{F}_{\beta}^{\sigma} \partial^{\nu} \phi^{\text{[B6]}} + \frac{\phi^3}{2} \mathbf{A}_{\sigma} \mathbf{A}_{\beta} \mathbf{F}_{\nu}^{\sigma} \partial^{\nu} \phi^{\text{[B13]}} - \phi^2 \mathbf{A}_{\beta} (\mathbf{A}_{\sigma} \partial^{\sigma} \phi)^2 \right. \text{[B7]} \\
& \left. + \left[ -\phi \partial^{\nu} \phi \mathbf{A}^{\mu} \Gamma_{\beta\nu\mu}^{\text{[B12]}} + \frac{\phi^3 \partial^{\nu} \phi}{2} \mathbf{A}^{\mu} \mathbf{A}_{\nu} \mathbf{F}_{\mu\beta}^{\text{[B9]}} + \frac{\phi^3 \partial^{\nu} \phi}{2} \mathbf{A}^{\mu} \mathbf{A}_{\beta} \mathbf{F}_{\mu\nu}^{\text{[B13]}} + \phi^2 \mathbf{A}_{\beta} (\mathbf{A}_{\nu} \partial^{\nu} \phi)^2 \right. \text{[B10]} + \mathbf{A}_{\beta} \partial_{\nu} \phi \partial^{\nu} \phi^{\text{[B3]}} + \mathbf{A}_{\nu} \partial^{\nu} \phi \partial_{\beta} \phi^{\text{[B11]}} + \frac{\phi}{2} \partial^{\nu} \phi \partial_{\nu} \mathbf{A}_{\beta}^{\text{[K2]}} \right. \\
& \left. + \frac{\phi}{2} \partial^{\nu} \phi \partial_{\beta} \mathbf{A}_{\nu}^{\text{[K2]}} \right]
\end{aligned}
$$

$$
\underline{R}_{5\beta} = \frac{\phi^2}{2} \left( \partial_{\nu} \mathbf{F}_{\beta}^{\nu} + \Gamma_{\sigma\nu}^{\nu} \mathbf{F}_{\beta}^{\sigma} - \Gamma_{\nu\beta}^{\sigma} \mathbf{F}_{\sigma}^{\nu} \right)^{\text{[K1]}} + \phi \left( \partial_{\nu} \phi \mathbf{F}_{\beta}^{\nu} - \partial^{\nu} \phi \partial_{\nu} \mathbf{A}_{\beta} + \frac{1}{2} \partial^{\nu} \phi \partial_{\nu} \mathbf{A}_{\beta} + \frac{1}{2} \partial^{\nu} \phi \partial_{\beta} \mathbf{A}_{\nu} \right)^{\text{[K2]}} + \mathbf{A}_{\beta} \left( -\phi \partial_{\nu} \partial^{\nu} \phi - \phi \Gamma_{\sigma\nu}^{\nu} \partial^{\sigma} \phi - \frac{1}{4} \phi^4 \mathbf{F}_{\sigma}^{\nu} \mathbf{F}_{\nu}^{\sigma} \right)^{\text{[K3]}}
$$

$$
\underline{R}_{5\beta} = \frac{\phi^2}{2} \left( \partial_{\nu} \mathbf{F}_{\beta}^{\nu} + \Gamma_{\sigma\nu}^{\nu} \mathbf{F}_{\beta}^{\sigma} - \Gamma_{\nu\beta}^{\sigma} \mathbf{F}_{\sigma}^{\nu} \right) + \phi \left( \partial_{\nu} \phi \mathbf{F}_{\beta}^{\nu} + \frac{1}{2} \partial^{\nu} \phi \mathbf{F}_{\beta\nu} \right) + \mathbf{A}_{\beta} \left( -\phi \Box_g \phi - \frac{1}{4} \phi^4 \mathbf{F}_{\sigma}^{\nu} \mathbf{F}_{\nu}^{\sigma} \right) = \frac{\phi^2}{2} \nabla_{\nu} \mathbf{F}_{\beta}^{\nu} + \frac{3\phi}{2} \mathbf{F}_{\beta\nu} \partial^{\nu} \phi + \mathbf{A}_{\beta} \underline{R}_{55}
$$

Where $\nabla_{\nu}$ defines the covariant derivative acting on mixed (1, 1) tensors.

We now compute $\underline{R}_{\nu\mu}$

$$
\underline{R}_{\nu\mu} = \partial_{\beta} \Gamma_{\nu\mu}^{\beta} - \partial_{\nu} \Gamma_{\beta\mu}^{\beta} - \partial_{\nu} \Gamma_{5\mu}^{5} + \Gamma_{\beta\sigma}^{\beta} \Gamma_{\nu\mu}^{\sigma} + \Gamma_{5\sigma}^{5} \Gamma_{\nu\mu}^{\sigma} + \Gamma_{\beta5}^{\beta} \Gamma_{\nu\mu}^{5} + \Gamma_{55}^{5} \Gamma_{\nu\mu}^{5} - \Gamma_{\nu\sigma}^{\beta} \Gamma_{\mu\beta}^{\sigma} - \Gamma_{\nu\sigma}^{5} \Gamma_{\mu5}^{\sigma} - \Gamma_{\nu5}^{\beta} \Gamma_{\mu\beta}^{5} - \Gamma_{\nu5}^{5} \Gamma_{\mu5}^{5}
$$

$$
\underline{R}_{\nu\mu} = \partial_{\beta} \Gamma_{\nu\mu}^{\beta} - \partial_{\nu} \left( \Gamma_{\beta\mu}^{\beta} + \Gamma_{5\mu}^{5} \right) + \left( \Gamma_{\beta\sigma}^{\beta} + \Gamma_{5\sigma}^{5} \right) \underline{\Gamma}_{\nu\mu}^{\sigma} + \left( \underline{\Gamma}_{\beta5}^{\beta} + \underline{\Gamma}_{55}^{5} \right) \underline{\Gamma}_{\nu\mu}^{5} - \underline{\Gamma}_{\nu\sigma}^{\beta} \underline{\Gamma}_{\mu\beta}^{\sigma} - \underline{\Gamma}_{\nu\sigma}^{5} \underline{\Gamma}_{\mu5}^{\sigma} - \underline{\Gamma}_{\nu5}^{\beta} \underline{\Gamma}_{\mu\beta}^{5} - \underline{\Gamma}_{\nu5}^{5} \underline{\Gamma}_{\mu5}^{5}
$$

We compute each term separately.

$$
\begin{aligned}
\partial_{\beta} \Gamma_{\mu\nu}^{\beta} = \partial_{\beta} \left[ \Gamma_{\mu\nu}^{\beta} + \frac{\phi^2}{2} (\mathbf{A}_{\mu} \mathbf{F}_{\nu}^{\beta} + \mathbf{A}_{\nu} \mathbf{F}_{\mu}^{\beta}) - \phi \mathbf{A}_{\mu} \mathbf{A}_{\nu} \partial^{\beta} \phi \right] \\
= \partial_{\beta} \Gamma_{\mu\nu}^{\beta \text{[K4]}} + \phi \partial_{\beta} \phi \mathbf{A}_{\mu} \mathbf{F}_{\nu}^{\beta \text{[K6]}} + \frac{\phi^2}{2} \mathbf{F}_{\nu}^{\beta} \partial_{\beta} \mathbf{A}_{\mu}^{\text{[K8]}} + \frac{\phi^2}{2} \mathbf{A}_{\mu} \partial_{\beta} \mathbf{F}_{\nu}^{\beta \text{[K6]}} + \phi \mathbf{A}_{\nu} \mathbf{F}_{\mu}^{\beta} \partial_{\beta} \phi^{\text{[K7]}} + \frac{\phi^2}{2} \mathbf{F}_{\mu}^{\beta} \partial_{\beta} \mathbf{A}_{\nu}^{\text{[K8]}} \\
+ \frac{\phi^2}{2} \mathbf{A}_{\nu} \partial_{\beta} \mathbf{F}_{\mu}^{\beta \text{[K7]}} - \mathbf{A}_{\mu} \mathbf{A}_{\nu} \partial^{\beta} \phi \partial_{\beta} \phi^{\text{[C1]}} - \phi \mathbf{A}_{\mu} \mathbf{A}_{\nu} \partial_{\beta} \partial^{\beta} \phi^{\text{[K5]}} - \phi \mathbf{A}_{\nu} \partial_{\beta} \mathbf{A}_{\mu} \partial^{\beta} \phi^{\text{[K7]}} - \phi \mathbf{A}_{\mu} \partial_{\beta} \mathbf{A}_{\nu} \partial^{\beta} \phi^{\text{[K6]}}
\end{aligned}
$$

$$
- \partial_{\mu} \left( \Gamma_{\beta\nu}^{\beta} + \underline{\Gamma}_{5\nu}^{5} \right) = - \partial_{\mu} \left( \Gamma_{\beta\nu}^{\beta} + \frac{\partial_{\nu} \phi}{\phi} \right) = - \partial_{\mu} \Gamma_{\beta\nu}^{\beta \text{[K4]}} - \left( \frac{\partial_{\mu} \partial_{\nu} \phi}{\phi} \right)^{\text{[K9]}} + \left( \frac{\partial_{\mu} \phi \partial_{\nu} \phi}{\phi^2} \right)^{\text{[C2]}}
$$
$$ (\Gamma_{\beta\sigma}^{\beta} + \Gamma_{5\sigma}^{5}) \underline{\Gamma}_{\mu\nu}^{\sigma} = (\Gamma_{\beta\sigma}^{\beta} + \frac{\partial_{\sigma}\phi}{\phi}) [\Gamma_{\mu\nu}^{\sigma} + \frac{\phi^2}{2}(A_{\mu}F_{\nu}^{\sigma} + A_{\nu}F_{\mu}^{\sigma}) - \phi A_{\mu}A_{\nu}\partial^{\sigma}\phi] \\ = \Gamma_{\beta\sigma}^{\beta} \Gamma_{\mu\nu}^{\sigma[\mathbf{K}4]} + \Gamma_{\beta\sigma}^{\beta} \frac{\phi^2}{2} A_{\mu}F_{\nu}^{\sigma[\mathbf{K}6]} + \Gamma_{\beta\sigma}^{\beta} \frac{\phi^2}{2} A_{\nu}F_{\mu}^{\sigma[\mathbf{K}7]} - \phi A_{\mu}A_{\nu}\Gamma_{\beta\sigma}^{\beta} \partial^{\sigma}\phi^{[\mathbf{K}5]} + \frac{\Gamma_{\mu\nu}^{\sigma}}{\phi} \partial_{\sigma}\phi^{[\mathbf{K}9}] + \frac{\phi}{2} A_{\mu}F_{\nu}^{\sigma} \partial_{\sigma}\phi^{[\mathbf{C}3]} + \frac{\phi}{2} A_{\nu}F_{\mu}^{\sigma} \partial_{\sigma}\phi^{[\mathbf{C}4]} - A_{\mu}A_{\nu}\partial_{\sigma}\phi \partial^{\sigma}\phi^{[\mathbf{C}5]} $$

$$ - \underline{\Gamma}_{\nu\sigma}^{\beta} \Gamma_{\mu\beta}^{\sigma} = - \left[ \Gamma_{\nu\sigma}^{\beta} + \frac{\phi^2}{2} (A_{\nu}F_{\sigma}^{\beta} + A_{\sigma}F_{\nu}^{\beta}) - \phi A_{\nu}A_{\sigma} \partial^{\beta} \phi \right] \left[ \Gamma_{\mu\beta}^{\sigma} + \frac{\phi^2}{2} (A_{\mu}F_{\beta}^{\sigma} + A_{\beta}F_{\mu}^{\sigma}) - \phi A_{\mu}A_{\beta} \partial^{\sigma} \phi \right] \\ = - \Gamma_{\nu\sigma}^{\beta} \Gamma_{\mu\beta}^{\sigma[\mathbf{K}4]} - \Gamma_{\nu\sigma}^{\beta} \frac{\phi^2}{2} A_{\mu}F_{\beta}^{\sigma[\mathbf{K}6]} - \Gamma_{\nu\sigma}^{\beta} \frac{\phi^2}{2} A_{\beta}F_{\mu}^{\sigma[\mathbf{K}6]} + \phi A_{\mu}A_{\beta} \Gamma_{\nu\sigma}^{\beta} \partial^{\sigma} \phi^{[\mathbf{C}7]} - \frac{\phi^2}{2} A_{\nu}F_{\sigma}^{\beta} \Gamma_{\mu\beta}^{\sigma[\mathbf{K}7]} - \frac{\phi^4}{4} A_{\nu}F_{\sigma}^{\beta} A_{\mu}F_{\beta}^{\sigma[\mathbf{K}5]} - \frac{\phi^4}{4} A_{\nu}F_{\sigma}^{\beta} A_{\beta}F_{\mu}^{\sigma[\mathbf{C}8]} \\ + \frac{\phi^3}{2} A_{\nu}F_{\sigma}^{\beta} A_{\mu}A_{\beta} \partial^{\sigma} \phi^{[\mathbf{C}9]} - \frac{\phi^2}{2} A_{\sigma}F_{\nu}^{\beta} \Gamma_{\mu\beta}^{\sigma[\mathbf{C}10]} - \frac{\phi^4}{4} A_{\sigma}F_{\nu}^{\beta} A_{\mu}F_{\beta}^{\sigma[\mathbf{C}11]} - \frac{\phi^4}{4} A_{\sigma}F_{\nu}^{\beta} A_{\beta}F_{\mu}^{\sigma[\mathbf{C}12]} + \frac{\phi^3}{2} A_{\sigma}F_{\nu}^{\beta} A_{\mu}A_{\beta} \partial^{\sigma} \phi^{[\mathbf{C}13]} \\ + \Gamma_{\mu\beta}^{\sigma} \phi A_{\nu}A_{\sigma} \partial^{\beta} \phi^{[\mathbf{C}14]} + A_{\mu}F_{\beta}^{\sigma} \frac{\phi^3}{2} A_{\nu}A_{\sigma} \partial^{\beta} \phi^{[\mathbf{C}15]} + A_{\beta}F_{\mu}^{\sigma} \frac{\phi^3}{2} A_{\nu}A_{\sigma} \partial^{\beta} \phi^{[\mathbf{C}16]} - \phi^2 A_{\nu}A_{\mu} (A_{\beta} \partial^{\beta} \phi)^2^{[\mathbf{C}17]} $$

$$ - \underline{\Gamma}_{\nu\sigma}^{5} \Gamma_{\mu\beta}^{\sigma} = - \left[ -A^{\beta} \Gamma_{\sigma\nu\beta} + \frac{A^{\beta}}{2} \phi^2 (A_{\nu}F_{\beta\sigma} + A_{\sigma}F_{\beta\nu}) + A^{\beta} \phi A_{\sigma}A_{\nu} \partial_{\beta} \phi + (A_{\sigma} \frac{\partial_{\nu}\phi}{\phi} + A_{\nu} \frac{\partial_{\sigma}\phi}{\phi}) + \frac{1}{2}(\partial_{\nu}A_{\sigma} + \partial_{\sigma}A_{\nu}) \right] \left[ \phi^2 \frac{1}{2} F_{\mu}^{\sigma} - \phi A_{\mu} \partial^{\sigma} \phi \right] \\ = \frac{\phi^2}{2} F_{\mu}^{\sigma} A^{\beta} \Gamma_{\sigma\nu\beta}^{[-\mathbf{C}6]} - \frac{\phi^4}{4} F_{\mu}^{\sigma} A^{\beta} A_{\nu}F_{\beta\sigma}^{[-\mathbf{C}8]} - \frac{\phi^4}{4} F_{\mu}^{\sigma} A^{\beta} A_{\sigma}F_{\beta\nu}^{[\mathbf{C}22]} - \frac{\phi^3}{2} F_{\mu}^{\sigma} A^{\beta} A_{\sigma}A_{\nu} \partial_{\beta} \phi^{[\mathbf{C}23]} - \frac{\phi}{2} F_{\mu}^{\sigma} A_{\sigma} \partial_{\nu} \phi^{[\mathbf{C}24]} - \frac{\phi}{2} F_{\mu}^{\sigma} A_{\nu} \partial_{\sigma} \phi^{[-\mathbf{C}4]} - \frac{\phi^2}{4} F_{\mu}^{\sigma} \partial_{\nu} A_{\sigma}^{[\mathbf{K}8]} \\ - \frac{\phi^2}{4} F_{\mu}^{\sigma} \partial_{\sigma} A_{\nu}^{[\mathbf{K}8]} - \phi A_{\mu} \partial^{\sigma} \phi A^{\beta} \Gamma_{\sigma\nu\beta}^{[-\mathbf{C}7]} + \frac{\phi^3}{2} A_{\mu} \partial^{\sigma} \phi A^{\beta} A_{\nu}F_{\beta\sigma}^{[-\mathbf{C}9]} + \frac{\phi^3}{2} A_{\mu} \partial^{\sigma} \phi A^{\beta} A_{\sigma}F_{\beta\nu}^{[-\mathbf{C}13]} + \phi^2 A_{\mu} (\partial^{\sigma} \phi A_{\sigma})^2 A_{\nu}^{[\mathbf{C}17]} + A_{\mu} \partial^{\sigma} \phi A_{\sigma} \partial_{\nu} \phi^{[\mathbf{C}25]} + A_{\mu} \partial^{\sigma} \phi A_{\nu} \partial_{\sigma} \phi^{[-\mathbf{C}5]} \\ + \frac{\phi}{2} A_{\mu} \partial^{\sigma} \phi \partial_{\nu} A_{\sigma}^{[\mathbf{K}6]} + \frac{\phi}{2} A_{\mu} \partial^{\sigma} \phi \partial_{\sigma} A_{\nu}^{[\mathbf{K}6]} $$

$$ - \underline{\Gamma}_{\nu 5}^{\beta} \Gamma_{\mu \beta}^{5} = - \left[ \phi^2 \frac{1}{2} F_{\nu}^{\beta} - \phi A_{\nu} \partial^{\beta} \phi \right] \left[ -A^{\sigma} \Gamma_{\mu \beta \sigma} + \frac{A^{\sigma}}{2} \phi^2 (A_{\beta} F_{\sigma \mu} + A_{\mu} F_{\sigma \beta}) + A^{\sigma} \phi A_{\mu} A_{\beta} \partial_{\sigma} \phi + (A_{\mu} \frac{\partial_{\beta} \phi}{\phi} + A_{\beta} \frac{\partial_{\mu} \phi}{\phi}) + \frac{1}{2} (\partial_{\mu} A_{\beta} + \partial_{\beta} A_{\mu}) \right] \\ = \frac{\phi^2}{2} F_{\nu}^{\beta} A^{\sigma} \Gamma_{\mu \beta \sigma}^{[-\mathbf{C}10]} - \frac{\phi^4}{4} F_{\nu}^{\beta} A^{\sigma} A_{\beta} F_{\sigma \mu}^{[-\mathbf{C}12]} - \frac{\phi^4}{4} F_{\nu}^{\beta} A^{\sigma} A_{\mu} F_{\sigma \beta}^{[-\mathbf{C}11]} - \frac{\phi^3}{2} F_{\nu}^{\beta} A_{\mu} A_{\beta} A^{\sigma} \partial_{\sigma} \phi^{[\mathbf{C}18]} - \frac{1}{2} \phi F_{\nu}^{\beta} A_{\mu} \partial_{\beta} \phi^{[-\mathbf{C}3]} - \frac{1}{2} \phi F_{\nu}^{\beta} A_{\beta} \partial_{\mu} \phi^{[\mathbf{C}19]} - \frac{\phi^2}{4} F_{\nu}^{\beta} \partial_{\mu} A_{\beta}^{[\mathbf{K}8]} \\ - \frac{\phi^2}{4} F_{\nu}^{\beta} \partial_{\beta} A_{\mu}^{[\mathbf{K}8]} - \phi A_{\nu} \partial^{\beta} \phi A^{\sigma} \Gamma_{\mu \beta \sigma}^{[-\mathbf{C}14]} + \frac{\phi^3}{2} A_{\nu} \partial^{\beta} \phi A^{\sigma} A_{\beta} F_{\sigma \mu}^{[-\mathbf{C}16]} + \frac{\phi^3}{2} A_{\nu} \partial^{\beta} \phi A^{\sigma} A_{\mu} F_{\sigma \beta}^{[-\mathbf{C}15]} + \phi^2 A_{\nu} A_{\mu} (A^{\sigma} \partial_{\sigma} \phi)^2^{[\mathbf{C}20]} + A_{\nu} \partial^{\beta} \phi A_{\mu} \partial_{\beta} \phi^{[-\mathbf{C}1]} + A_{\nu} \partial^{\beta} \phi A_{\beta} \partial_{\mu} \phi^{[\mathbf{C}21]} \\ + \frac{\phi}{2} A_{\nu} \partial^{\beta} \phi \partial_{\mu} A_{\beta}^{[\mathbf{K}7]} + \frac{\phi}{2} A_{\nu} \partial^{\beta} \phi \partial_{\beta} A_{\mu}^{[\mathbf{K}7]} $$

$$ - \underline{\Gamma}_{\nu 5}^{5} \Gamma_{\mu 5}^{5} = - \left[ - \frac{1}{2} A^{\beta} \phi^2 F_{\nu \beta} + \phi A^{\beta} A_{\nu} \partial_{\beta} \phi + \frac{\partial_{\nu} \phi}{\phi} \right] \left[ - \frac{1}{2} A^{\sigma} \phi^2 F_{\mu \sigma} + \phi A^{\sigma} A_{\mu} \partial_{\sigma} \phi + \frac{\partial_{\mu} \phi}{\phi} \right] \\ = - \frac{\phi^4}{4} A^{\beta} F_{\nu \beta} A^{\sigma} F_{\mu \sigma}^{[-\mathbf{C}22]} + \frac{\phi^3}{2} A^{\beta} F_{\nu \beta} A^{\sigma} A_{\mu} \partial_{\sigma} \phi^{[-\mathbf{C}18]} + \frac{\phi}{2} A^{\beta} F_{\nu \beta} \partial_{\mu} \phi^{[-\mathbf{C}19]} + \frac{\phi^3}{2} A^{\beta} A_{\nu} \partial_{\beta} \phi A^{\sigma} F_{\mu \sigma}^{[-\mathbf{C}23]} - \phi^2 (A^{\beta} \partial_{\beta} \phi)^2 A_{\mu} A_{\nu}^{[-\mathbf{C}20]} - A^{\beta} \partial_{\beta} \phi A_{\nu} \partial_{\mu} \phi^{[-\mathbf{C}21]} \\ + \frac{\phi}{2} A^{\sigma} F_{\mu \sigma} \partial_{\nu} \phi^{[-\mathbf{C}24]} - A^{\sigma} A_{\mu} \partial_{\sigma} \phi \partial_{\nu} \phi^{[-\mathbf{C}25]} - \left( \frac{\partial_{\mu} \phi \partial_{\nu} \phi}{\phi^2} \right)^{[-\mathbf{C}2]} $$
Summing the remaining terms now gives :

$$
\underline{R_{\mu\nu}} = \left( \partial_{\beta}\Gamma^{\beta}_{\mu\nu} - \partial_{\mu}\Gamma^{\beta}_{\beta\nu} + \Gamma^{\beta}_{\beta\sigma}\Gamma^{\sigma}_{\mu\nu} - \Gamma^{\beta}_{\mu\sigma}\Gamma^{\sigma}_{\beta\mu} \right)^{[K4]} + \left( -\phi A_{\mu}A_{\nu}\partial_{\beta}\partial^{\beta}\phi - \phi A_{\mu}A_{\nu}\Gamma^{\beta}_{\beta\sigma}\partial^{\sigma}\phi - \frac{\phi^4}{4}A_{\nu}F^{\beta}_{\sigma}A_{\mu}F^{\sigma}_{\beta} \right)^{[K5]}
$$

$$
+ \left( \frac{\phi^2}{2} A_{\mu} \partial_{\beta} F^{\beta}_{\nu} + \Gamma^{\beta}_{\beta\sigma} \frac{\phi^2}{2} A_{\mu} F^{\sigma}_{\nu} - \Gamma^{\beta}_{\nu\sigma} \frac{\phi^2}{2} A_{\mu} F^{\sigma}_{\beta} + \phi \partial_{\beta} \phi A_{\mu} F^{\beta}_{\nu} + \frac{\phi}{2} A_{\mu} \partial^{\sigma} \phi \partial_{\nu} A_{\sigma} + \frac{\phi}{2} A_{\mu} \partial^{\sigma} \phi \partial_{\sigma} A_{\nu} - \phi A_{\mu} \partial_{\beta} A_{\nu} \partial^{\beta} \phi \right)^{[K6]}
$$

$$
+ \left( \frac{\phi^2}{2} A_{\nu} \partial_{\beta} F^{\beta}_{\mu} + \Gamma^{\beta}_{\beta\sigma} \frac{\phi^2}{2} A_{\nu} F^{\sigma}_{\mu} - \frac{\phi^2}{2} A_{\nu} F^{\sigma}_{\beta} \Gamma^{\sigma}_{\mu\beta} + \phi A_{\nu} F^{\beta}_{\mu} \partial_{\beta} \phi + \frac{\phi}{2} A_{\nu} \partial^{\beta} \phi \partial_{\mu} A_{\beta} + \frac{\phi}{2} A_{\nu} \partial^{\beta} \phi \partial_{\beta} A_{\mu} - \phi A_{\nu} \partial_{\beta} A_{\mu} \partial^{\beta} \phi \right) \quad [K7]
$$

$$
+ \left( \frac{\phi^2}{2} F^{\beta}_{\nu} \partial_{\beta} A_{\mu} - \frac{\phi^2}{4} F^{\beta}_{\nu} \partial_{\mu} A_{\beta} - \frac{\phi^2}{4} F^{\beta}_{\nu} \partial_{\beta} A_{\mu} + \frac{\phi^2}{2} F^{\beta}_{\mu} \partial_{\beta} A_{\nu} - \frac{\phi^2}{4} F^{\sigma}_{\mu} \partial_{\nu} A_{\sigma} - \frac{\phi^2}{4} F^{\sigma}_{\mu} \partial_{\sigma} A_{\nu} \right) \quad [K8]
$$

$$
+ \left( - \frac{\partial_{\mu} \partial_{\nu} \phi}{\phi} + \Gamma_{\mu\nu}^{\sigma} \frac{\partial_{\sigma} \phi}{\phi} \right) \quad [K9]
$$

$$
\begin{aligned}
\underline{R_{\mu\nu}} = (R_{\mu\nu}) &+ A_{\mu}A_{\nu} \left( -\phi\partial_{\beta}\partial^{\beta}\phi - \phi\Gamma^{\beta}_{\beta\sigma}\partial^{\sigma}\phi - \frac{\phi^4}{4}F^{\beta}_{\sigma}F^{\sigma}_{\beta} \right) + A_{\mu} \left( \frac{\phi^2}{2}\partial_{\beta}F^{\beta}_{\nu} + \Gamma^{\beta}_{\beta\sigma}\frac{\phi^2}{2}F^{\sigma}_{\nu} - \Gamma^{\beta}_{\nu\sigma}\frac{\phi^2}{2}F^{\sigma}_{\beta} + \phi\partial_{\beta}\phi F^{\beta}_{\nu} + \frac{\phi}{2}\partial^{\sigma}\phi\partial_{\nu}A_{\sigma} + \phi\frac{1}{2}\partial^{\sigma}\phi\partial_{\sigma}A_{\nu} - \phi\partial_{\beta}A_{\nu}\partial^{\beta}\phi \right) \\
& + A_{\nu} \left( \frac{\phi^2}{2}\partial_{\beta}F^{\beta}_{\mu} + \Gamma^{\beta}_{\beta\sigma}\frac{\phi^2}{2}F^{\sigma}_{\mu} - \frac{\phi^2}{2}F^{\sigma}_{\beta}\Gamma^{\sigma}_{\mu\beta} + \phi F^{\beta}_{\mu}\partial_{\beta}\phi + \frac{\phi}{2}\partial^{\beta}\phi\partial_{\mu}A_{\beta} + \frac{\phi}{2}\partial^{\beta}\phi\partial_{\beta}A_{\mu}u - \phi\partial_{\beta}A_{\mu}\partial^{\beta}\phi \right) + \frac{\phi^2}{2} \left( F^{\beta}_{\nu}(\partial_{\beta}A_{\mu} - \frac{1}{2}\partial_{\mu}A_{\beta} - \frac{1}{2}\partial_{\beta}A_{\mu}) + F^{\beta}_{\mu}(\partial_{\beta}A_{\nu} - \frac{1}{2}\partial_{\nu}A_{\beta} - \frac{1}{2}\partial_{\beta}A_{\nu}) \right)
\end{aligned}
$$

$$
+ \left( - \frac{\partial_{\mu} \partial_{\nu} \phi}{\phi} + \Gamma_{\mu\nu}^{\sigma} \frac{\partial_{\sigma} \phi}{\phi} \right)
$$

$$
\underline{R_{\mu\nu}} = R_{\mu\nu} + A_{\mu}A_{\nu}R_{55} + A_{\mu}\left(\frac{\phi^2}{2}\nabla_{\beta}F^{\beta}_{\nu} + \frac{3}{2}\phi F^{\sigma}_{\nu}\partial_{\sigma}\phi\right) + A_{\nu}\left(\frac{\phi^2}{2}\nabla_{\beta}F^{\beta}_{\mu} + \frac{3}{2}\phi F^{\sigma}_{\mu}\partial_{\sigma}\phi\right) + \frac{\phi^2}{4}\left(F^{\beta}_{\nu}F^{\beta}_{\mu\mu} + F^{\beta}_{\mu}F^{\beta}_{\nu\mu}\right) + \left(-\frac{\partial_{\mu}\partial_{\nu}\phi}{\phi} + \Gamma^{\sigma}_{\mu\nu}\frac{\partial_{\sigma}\phi}{\phi}\right)
$$

$$
\underline{R_{\mu\nu}} = R_{\mu\nu} + A_{\mu}A_{\nu}R_{55} + A_{\mu}(R_{4\nu} - A_{\nu}R_{55}) + A_{\nu}(R_{4\mu} - A_{\mu}R_{55}) + \frac{\phi^2}{2}g^{\alpha\beta}F_{\nu\alpha}F_{\beta\mu} + \left(-\frac{\partial_{\mu}\partial_{\nu}\phi}{\phi} + \Gamma_{\mu\nu}^{\sigma}\frac{\partial_{\sigma}\phi}{\phi}\right)
$$

$$
\underline{R_{\mu\nu}} = R_{\mu\nu} + \frac{\phi^2}{2} g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \left( - \frac{\partial_{\mu} \partial_{\nu} \phi}{\phi} + \Gamma_{\mu\nu}^{\sigma} \frac{\partial_{\sigma} \phi}{\phi} \right) + A_{\mu} \underline{R_{5\nu}} + A_{\nu} \underline{R_{5\mu}} - A_{\mu} A_{\nu} \underline{R_{55}}
$$

The Ricci scalar is finally :

$$
\underline{R} = \underline{g}^{MN} \underline{R}_{MN} = g^{\mu\nu} \underline{R}_{\mu\nu} + 2 \underline{g}^{\mu 5} \underline{R}_{\mu 5} + \underline{g}^{55} \underline{R}_{55}
$$

$$
\underline{R} = g^{\mu\nu} \left( R_{\mu\nu} + \frac{\phi^2}{2} g^{\alpha\beta} \underline{F}_{\nu\alpha} \underline{F}_{\beta\mu} - \frac{\partial_{\mu} \partial_{\nu} \phi}{\phi} + \Gamma^{\sigma}_{\mu\nu} \frac{\partial_{\sigma} \phi}{\phi} + A_{\mu} \underline{R}_{5\nu} + A_{\nu} \underline{R}_{5\mu} - A_{\mu} A_{\nu} \underline{R}_{55} \right) - 2 A^{\mu} \underline{R}_{5\mu} + \left( \frac{1}{\phi^2} + A_{\mu} A^{\mu} \right) \underline{R}_{55}
$$

$$
\underline{R} = R + \frac{\phi^2}{2} \underline{F}^{\mu\beta} \underline{F}_{\beta\mu} - \frac{\Box_g \phi}{\phi} + 2 \underline{A}^{\nu} \underline{R}_{5\nu} \text{ [E1]} - \underline{A}_{\mu} \underline{A}^{\mu} \underline{R}_{55} \text{ [E2]} - 2 \underline{A}^{\mu} \underline{R}_{5\mu} \text{ [-E1]} + \frac{1}{\phi^2} \underline{R}_{55} + \underline{A}_{\mu} \underline{A}^{\mu} \underline{R}_{55} \text{ [-E2]}
$$

$$
\underline{R} = R - \frac{\phi^2}{2} \underline{F}^{\mu\beta} \underline{F}_{\mu\beta} - \frac{\Box_g \phi}{\phi} - \frac{\Box_g \phi}{\phi} + \frac{\phi^2}{4} \underline{F}^{\sigma\nu} \underline{F}_{\sigma\nu} = R - \frac{\phi^2}{4} \underline{F}^{\mu\alpha} \underline{F}_{\mu\alpha} - 2 \frac{\Box_g \phi}{\phi}
$$
We summarize :

$$
\begin{align}
\underline{R_{55}} &= -\phi \square_g \phi + \frac{\phi^4}{4} \mathbf{F}^{\sigma\nu} \mathbf{F}_{\sigma\nu} \nonumber \\
\underline{R_{5\beta}} &= \frac{\phi^2}{2} \nabla_\nu \mathbf{F}_\beta^\nu + \frac{3\phi}{2} \mathbf{F}_{\beta\nu} \partial^\nu \phi + \mathbf{A}_\beta \underline{R_{55}} \nonumber \\
\underline{R_{\mu\nu}} &= R_{\mu\nu} + \frac{\phi^2}{2} g^{\alpha\beta} \mathbf{F}_{\nu\alpha} \mathbf{F}_{\beta\mu} - \frac{\partial_\mu \partial_\nu \phi}{\phi} + \Gamma^\sigma_{\mu\nu} \frac{\partial_\sigma \phi}{\phi} + \mathbf{A}_\mu \underline{R_{5\nu}} + \mathbf{A}_\nu \underline{R_{5\mu}} - \mathbf{A}_\mu \mathbf{A}_\nu \underline{R_{55}} \nonumber \\
\underline{R} &= R - \frac{\phi^2}{4} \mathbf{F}^{\mu\alpha} \mathbf{F}_{\mu\alpha} - 2 \frac{\square_g \phi}{\phi} \tag{11.24}
\end{align}
$$

11.3.4 Computation of the Einstein tensors

We can finally compute the Einstein tensors $G_{MN} = R_{MN} - \frac{1}{2}g_{MN} R$.

$$
\begin{align}
\underline{G}_{55} &= \underline{R}_{55} - \frac{1}{2}g_{55}R = \left(-\phi\square_g\phi + \frac{\phi^4}{4}\mathbf{F}_{\sigma\nu}\mathbf{F}^{\sigma\nu}\right) - \frac{\phi^2}{2}\left(R - \frac{\phi^2}{4}\mathbf{F}^{\mu\alpha}\mathbf{F}_{\mu\alpha} - 2\frac{\square_g\phi}{\phi}\right) \nonumber \\
\underline{G}_{55} &= -\frac{\phi^2}{2}R + \frac{3\phi^4}{8}\mathbf{F}^{\mu\nu}\mathbf{F}_{\mu\nu} \nonumber \\
\underline{G}_{5\beta} &= \underline{R}_{5\beta} - \frac{1}{2}g_{5\beta}R = \left(\frac{\phi^2}{2}\nabla_\nu\mathbf{F}_\beta^\nu + \frac{3\phi}{2}\mathbf{F}_{\beta\nu}\partial^\nu\phi + \mathbf{A}_\beta\underline{R}_{55}\right) - \left(\mathbf{A}_\beta\frac{\phi^2}{2}\underline{R}\right) \nonumber \\
\underline{G}_{5\beta} &= \left(\frac{\phi^2}{2}\nabla_\nu\mathbf{F}_\beta^\nu + \frac{3\phi}{2}\mathbf{F}_{\beta\nu}\partial^\nu\phi\right) + \mathbf{A}_\beta\underline{G}_{55} \nonumber \\
\underline{G}_{\mu\nu} &= \left(R_{\mu\nu} + \frac{\phi^2}{2}g^{\alpha\beta}\mathbf{F}_{\nu\alpha}\mathbf{F}_{\beta\mu} - \frac{\partial_\mu\partial_\nu\phi}{\phi} + \Gamma^\sigma_{\mu\nu}\frac{\partial_\sigma\phi}{\phi} + \mathbf{A}_\mu\underline{R}_{5\nu} + \mathbf{A}_\nu\underline{R}_{5\mu} - \mathbf{A}_\mu\mathbf{A}_\nu\underline{R}_{55}\right) \nonumber \\
&\quad -\frac{1}{2}(g_{\mu\nu} + \phi^2\mathbf{A}_\mu\mathbf{A}_\nu)\left(R - \frac{\phi^2}{4}\mathbf{F}^{\mu\alpha}\mathbf{F}_{\mu\alpha} - 2\frac{\square_g\phi}{\phi}\right) \tag{11.25} \\
\underline{G}_{\mu\nu} &= R_{\mu\nu} + \frac{\phi^2}{2}g^{\alpha\beta}\mathbf{F}_{\nu\alpha}\mathbf{F}_{\beta\mu} - \frac{\partial_\mu\partial_\nu\phi}{\phi} + \Gamma^\sigma_{\mu\nu}\frac{\partial_\sigma\phi}{\phi} - \frac{1}{2}g_{\mu\nu}\left(R - \frac{\phi^2}{4}\mathbf{F}^{\beta\alpha}\mathbf{F}_{\beta\alpha} - 2\frac{\square_g\phi}{\phi}\right) \nonumber \\
&\quad + \mathbf{A}_\mu\underline{R}_{5\nu} + \mathbf{A}_\nu\underline{R}_{5\mu} - \mathbf{A}_\mu\mathbf{A}_\nu\underline{R}_{55} - \frac{1}{2}\phi^2\mathbf{A}_\mu\mathbf{A}_\nu\underline{R} \nonumber \\
\underline{G}_{\mu\nu} &= R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \frac{\phi^2}{2}\left(g^{\alpha\beta}\mathbf{F}_{\nu\alpha}\mathbf{F}_{\beta\mu} + \frac{g_{\mu\nu}}{4}\mathbf{F}^{\beta\alpha}\mathbf{F}_{\beta\alpha}\right) + \left(-\frac{\partial_\mu\partial_\nu\phi}{\phi} + \Gamma^\sigma_{\mu\nu}\frac{\partial_\sigma\phi}{\phi} + g_{\mu\nu}\frac{\square_g\phi}{\phi}\right) \nonumber \\
&\quad + \mathbf{A}_\mu\left(\underline{G}_{5\mu} + \frac{\phi^2}{2}\mathbf{A}_\nu\underline{R}\right) + \mathbf{A}_\nu\left(\underline{G}_{5\mu} + \frac{\phi^2}{2}\mathbf{A}_\mu\underline{R}\right) - \mathbf{A}_\mu\mathbf{A}_\nu\left(\underline{G}_{55} + \frac{\phi^2}{2}\underline{R}\right) - \frac{1}{2}\phi^2\mathbf{A}_\mu\mathbf{A}_\nu\underline{R} \nonumber \\
\underline{G}_{\mu\nu} &= R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \frac{\phi^2}{2}\left(g^{\alpha\beta}\mathbf{F}_{\nu\alpha}\mathbf{F}_{\beta\mu} + \frac{g_{\mu\nu}}{4}\mathbf{F}^{\beta\alpha}\mathbf{F}_{\beta\alpha}\right) + \left(-\frac{\partial_\mu\partial_\nu\phi - \Gamma^\sigma_{\mu\nu}\partial_\sigma\phi}{\phi} + g_{\mu\nu}\frac{\square_g\phi}{\phi}\right) \nonumber \\
&\quad + \mathbf{A}_\mu\underline{G}_{5\mu} + \mathbf{A}_\nu\underline{G}_{5\mu} - \mathbf{A}_\mu\mathbf{A}_\nu\underline{G}_{55} \nonumber
\end{align}
$$

Using $M = \sqrt{\frac{\mu_0 c^2 q^2}{16\pi G}}$ and equation (11.8) we now express the Ricci scalar and Einstein tensors with the original metric :

$$
\begin{align}
\underline{R} &= R + \frac{\Phi^2}{4} \frac{q^2}{M^2 c^2} F^{\alpha\beta} F_{\alpha\beta} - 2 \frac{\square_g \Phi}{\Phi} \nonumber \\
&= R + \frac{8\pi G}{c^4} \left( \frac{\Phi^2}{2\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) - 2 \frac{\square_g \Phi}{\Phi} \nonumber \\
\underline{G}_{55} &= \frac{\Phi^2}{2} R + \frac{3\Phi^4}{8} \frac{q^2}{M^2 c^2} F^{\alpha\beta} F_{\alpha\beta} \nonumber \\
&= \frac{\Phi^2}{2} R + \frac{8\pi G}{c^4} \left( \frac{3\Phi^4}{4\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) \nonumber \\
\underline{G}_{5\mu} &= \frac{\Phi^2}{2} \nabla_\nu \frac{g}{Mc} F_\mu^\nu + \frac{3\Phi}{2} \frac{g}{Mc} F_{\mu\nu} \partial^\nu \Phi - \frac{g A_\mu}{Mc} \underline{G}_{55} \nonumber \\
&= -\frac{\Phi^2}{2} \frac{g}{Mc} \nabla_\nu F_\mu^\nu - \frac{3\Phi}{2} \frac{g}{Mc} F_{\nu\mu} \partial^\nu \Phi - \frac{g A_\mu}{Mc} \underline{G}_{55} \nonumber \\
\underline{G}_{\mu\nu} &= R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R - \frac{8\pi G}{c^4} \frac{\Phi^2}{\mu_0} \left( g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\nu\alpha} \right) \nonumber \\
&\quad + \left( - \frac{\partial_\mu \partial_\nu \Phi - \Gamma^\sigma_{\mu\nu} \partial_\sigma \Phi}{\Phi} - g_{\mu\nu} \frac{\square_g \Phi}{\Phi} \right) - \frac{g A_\mu}{Mc} \underline{G}_{5\mu} - \frac{g A_\nu}{Mc} \underline{G}_{5\mu} - \frac{g A_\mu}{Mc} - \frac{g A_\nu}{Mc} \underline{G}_{55} \tag{11.26}
\end{align}
$$
11.4 Annexe 4 : Details on geodesic equations in the Kaluza Klein theory

In this annex we compute the term $\nabla_M T^{MN}$ for $T^{MN} = \rho \underline{u}^M \underline{u}^N$ and $N = \mu$ or 5. The values of the Christoffel symbols are given in equation (11.21), they correspond to the metric given in equation (11.8) where $A_\mu = -\frac{qA_\mu}{Mc}$, $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu = -\frac{q}{Mc}F_{\mu\nu}$ and $\Phi = i\phi$.

$$
\begin{align}
\frac{\nabla_M T^{MN}}{\nabla_M T^{MN}} &= \frac{\partial_M T^{MN}}{\nabla_M T^{MN}} + \frac{\Gamma^Q_{QM}}{\nabla_M T^{QM}} + \frac{\Gamma^N_{QM}}{\nabla_M T^{QM}} \nonumber \\
&= \partial_\alpha \frac{T^{\alpha N}}{\nabla_M T^{MN}} + \partial_5 \frac{T^{5N}}{\nabla_M T^{MN}} + \left(\frac{\Gamma^\nu_{\nu\alpha}}{\nabla_M T^{MN}} + \frac{\Gamma^5_{5\alpha}}{\nabla_M T^{MN}}\right) \frac{T^{\alpha N}}{\nabla_M T^{MN}} + \left(\frac{\Gamma^\nu_{\nu5}}{\nabla_M T^{MN}} + \frac{\Gamma^5_{55}}{\nabla_M T^{MN}}\right) \frac{T^{5N}}{\nabla_M T^{MN}} + \frac{\Gamma^N_{\alpha\beta}}{\nabla_M T^{MN}} T^{\alpha\beta} + 2\frac{\Gamma^N_{\alpha5}}{\nabla_M T^{MN}} T^{\alpha5} + \frac{\Gamma^N_{55}}{\nabla_M T^{MN}} T^{55} \tag{11.27}
\end{align}
$$

When $N = \mu$ the computation gives :

$$
\begin{align}
\frac{\nabla_M T^{M\mu}}{\nabla_M T^{M\mu}} &= u^\mu \left[ \partial_\alpha (\rho u^\alpha) + \left(\Gamma^\nu_{\nu\alpha} + \frac{\Gamma^5_{5\alpha}}{\Gamma^5_{5\alpha}}\right) \rho u^\alpha \right] + \rho \left[ u^\alpha \partial_\alpha u^\mu + \frac{\Gamma^\mu_{\alpha\beta}}{\Gamma^5_{5\alpha}} u^\alpha u^\beta + 2\Gamma^\mu_{\alpha5} u^\alpha u^5 + \frac{\Gamma^\mu_{55}}{\Gamma^5_{55}} u^5 u^5 \right] \nonumber \\
&= u^\mu \left[ \partial_\alpha (\rho u^\alpha) + \left(\Gamma^\nu_{\nu\alpha} + \frac{\partial_\alpha \phi}{\phi}\right) \rho u^\alpha \right] + \rho \left[ u^\alpha \partial_\alpha u^\mu + \left(\Gamma^\mu_{\alpha\beta} + \phi^2 A_\alpha F_\beta^\mu - \phi A_\alpha A_\beta \partial^\mu \phi \right) u^\alpha u^\beta + \left(\phi^2 F_\alpha^\mu - 2 A_\alpha \phi \partial^\mu \phi \right) u^\alpha \frac{u^5}{\phi} - \phi \partial^\mu \phi \frac{u^5 u^5}{\phi^3} \right] \nonumber \\
&= u^\mu \left[ \nabla_\alpha (\rho u^\alpha) + \frac{\partial_\alpha \phi}{\phi} \rho u^\alpha \right] + \rho \left[ \dot{u}^\mu + \Gamma^\mu_{\alpha\beta} u^\alpha u^\beta + F_\beta^\mu u^\beta \left[ \phi^2 \left( \frac{u^5}{\phi} + A_\alpha u^\alpha \right) \right] - \frac{\partial^\mu \phi}{\phi^3} \left[ \phi^2 \left( \frac{u^5}{\phi} + A_\alpha u^\alpha \right) \right]^2 \right] \tag{11.28} \\
&= u^\mu \left[ \nabla_\alpha (\rho u^\alpha) + \rho \frac{u^\alpha \partial_\alpha \Phi}{\Phi} \right] + \rho \left[ \dot{u}^\mu + \Gamma^\mu_{\alpha\beta} u^\alpha u^\beta + \frac{q F_\beta^\mu u^\beta}{Mc} \left[ \Phi^2 \left( \frac{u^5}{\phi} - \frac{q A_\alpha u^\alpha}{Mc} \right) \right] + \frac{\partial^\mu \Phi}{\Phi^3} \left[ \Phi^2 \left( \frac{u^5}{\phi} - \frac{q A_\alpha u^\alpha}{Mc} \right) \right]^2 \right] \nonumber
\end{align}
$$

When $N = 5$ the computation gives :

$$
\begin{align}
\frac{\nabla_M T^{M5}}{\nabla_M T^{M5}} &= \underline{u}^5 \left[ \partial_\alpha (\rho u^\alpha) + \left(\Gamma^\nu_{\nu\alpha} + \frac{\Gamma^5_{5\alpha}}{\Gamma^5_{5\alpha}}\right) \rho u^\alpha \right] + \rho \left[ u^\alpha \partial_\alpha \underline{u}^5 + \frac{\Gamma^5_{5\alpha}}{\Gamma^5_{5\alpha}} u^\alpha u^5 + 2\frac{\Gamma^5_{55}}{\Gamma^5_{55}} u^5 \right] \nonumber \\
&= \underline{u}^5 \left[ \partial_\alpha (\rho u^\alpha) + \left(\Gamma^\nu_{\nu\alpha} + \frac{\partial_\alpha \phi}{\phi}\right) \rho u^\alpha \right] + \rho \left[ u^\alpha \partial_\alpha \underline{u}^5 + \left(-A_\mu \Gamma^\mu_{\alpha\beta} + \phi^2 A^\mu A_\beta F_{\mu\alpha} + A_\alpha A_\beta \phi A^\mu \partial_\mu \phi + 2A_\beta \frac{\partial_\alpha \phi}{\phi} + \partial_\beta A_\alpha \right) u^\alpha u^5 \right. \nonumber \\
&\qquad \left. + 2\left(-\frac{\phi^2}{2} A^\mu F_{\alpha\mu} + A_\alpha \phi A^\mu \partial_\mu \phi + \frac{\partial_\alpha \phi}{\phi} \right) \underline{u}^5 u^\alpha + \phi A^\mu \partial_\mu \phi \underline{u}^5 \right] \nonumber \\
&= \underline{u}^5 \left[ \nabla_\alpha (\rho u^\alpha) + \frac{\partial_\alpha \phi}{\phi} \rho u^\alpha \right] + \rho \left[ u^\alpha \partial_\alpha \underline{u}^5 - A_\mu \left( \Gamma^\mu_{\alpha\beta} u^\alpha u^5 + F_\alpha^\mu u^\alpha \left[ \phi^2 \left( \underline{u}^5 + A_\beta u^\beta \right) \right] - \frac{\partial^\mu \phi}{\phi^3} \left[ \phi^2 \left( \underline{u}^5 + A_\beta u^\beta \right) \right]^2 \right) \right. \nonumber \\
&\qquad \left. + u^\alpha \partial_\alpha A_\beta u^5 + 2 \frac{u^\alpha \partial_\alpha \phi}{\phi} \left( A_\beta u^5 + \underline{u}^5 \right) \right] \nonumber \\
&= \underline{u}^5 \left[ \nabla_\alpha (\rho u^\alpha) + \frac{\partial_\alpha \phi}{\phi} \rho u^\alpha \right] + \rho \left[ \frac{du^5}{d\tau} + A_\mu \frac{du^\mu}{d\tau} + u^\mu \frac{dA_\mu}{d\tau} + \frac{2}{\phi} \left( A_\alpha u^\alpha + \underline{u}^5 \right) \frac{d\phi}{d\tau} \right] \nonumber \\
&= \underline{u}^5 \left[ \nabla_\alpha (\rho u^\alpha) + \frac{\partial_\alpha \phi}{\phi} \rho u^\alpha \right] + \frac{\rho}{\phi^2} \frac{d}{d\tau} \left[ \phi^2 \left( \underline{u}^5 + A_\mu u^\mu \right) \right] = \underline{u}^5 \left[ \nabla_\alpha (\rho u^\alpha) + \frac{\partial_\alpha \Phi}{\Phi} \rho u^\alpha \right] + \frac{\rho}{\Phi^2} \frac{d}{d\tau} \left[ \Phi^2 \left( \underline{u}^5 - \frac{q A_\alpha u^\alpha}{Mc} \right) \right] \tag{11.29}
\end{align}
$$
11.5 Annexe 5 : Details on Maxwell equations in Kaluza Klein theory

In this annex we reformulate the equation $\Phi^2 \nabla_{\nu} F^{\nu}_{\beta} + 3\Phi F_{\nu\beta} \partial^{\nu} \Phi = \mu_0 j_{\beta}$ : the result is obvious since the metric commutes with the covariant derivative. Then we
derive the continuity equation $\nabla_{\alpha} j^{\alpha} + \frac{j^{\alpha} \partial_{\alpha} \Phi}{\Phi} = 0$.

$$
\mu_0 j_{\beta} = \Phi^2 \nabla_{\nu} F^{\nu}_{\beta} + 3 \Phi F_{\nu\beta} \partial^{\nu} \Phi
$$

$$
\mu_0 j_{\beta} = \Phi^2 (\partial_{\nu} F^{\nu}_{\beta} + \Gamma^{\nu}_{\nu\sigma} F^{\sigma}_{\beta} - \Gamma^{\sigma}_{\beta\nu} F^{\nu}_{\sigma}) + 3 \Phi F_{\nu\beta} \partial^{\nu} \Phi
$$

$$
g_{\alpha\beta}(\mu_0 j^{\alpha}) = \Phi^2 (\partial_{\nu}(g_{\alpha\beta}F^{\nu\alpha}) + \Gamma^{\nu}_{\nu\sigma}g_{\alpha\beta}F^{\sigma\alpha} - \Gamma^{\sigma}_{\beta\nu}g_{\alpha\sigma}F^{\nu\alpha}) + 3g_{\alpha\beta}\Phi F^{\nu\alpha}\partial_{\nu}\Phi
$$

$$
g_{\alpha\beta}(\mu_0 j^{\alpha}) = \Phi^2 (g_{\alpha\beta} \partial_{\nu} F^{\nu\alpha} + \Gamma^{\nu}_{\nu\sigma} g_{\alpha\beta} F^{\sigma\alpha} + (g_{\alpha\beta,\nu} - \Gamma^{\sigma}_{\beta\nu} g_{\alpha\sigma}) F^{\nu\alpha}) + 3g_{\alpha\beta} \Phi F^{\nu\alpha} \partial_{\nu} \Phi
$$

$$
g_{\alpha\beta}(\mu_0 j^{\alpha}) = \Phi^2 (g_{\alpha\beta} \partial_{\nu} F^{\nu\alpha} + \Gamma^{\nu}_{\nu\sigma} g_{\alpha\beta} F^{\sigma\alpha} + (g_{\alpha\beta,\nu} - \frac{1}{2}g_{\alpha\beta,\nu} - \frac{1}{2}g_{\nu\alpha,\beta} + \frac{1}{2}g_{\beta\nu,\alpha}) F^{\nu\alpha}) + 3g_{\alpha\beta} \Phi F^{\nu\alpha} \partial_{\nu} \Phi \quad (11.30)
$$

$$
g_{\alpha\beta}(\mu_0 j^{\alpha}) = \Phi^2 (g_{\alpha\beta} \partial_{\nu} F^{\nu\alpha} + \Gamma^{\nu}_{\nu\sigma} g_{\alpha\beta} F^{\sigma\alpha} + \Gamma_{\nu\alpha\beta} F^{\nu\alpha}) + 3g_{\alpha\beta} \Phi F^{\nu\alpha} \partial_{\nu} \Phi
$$

$$
g_{\alpha\beta}(\mu_0 j^{\alpha}) = \Phi^2 (g_{\alpha\beta} \partial_{\nu} F^{\nu\alpha} + \Gamma^{\nu}_{\nu\sigma} g_{\alpha\beta} F^{\sigma\alpha} + \Gamma_{\nu\alpha\beta} F^{\nu\alpha}) + 3g_{\alpha\beta} \Phi F^{\nu\alpha} \partial_{\nu} \Phi
$$

$$
g_{\alpha\beta}(\mu_0 j^{\alpha}) = g_{\alpha\beta} \Phi^2 (\partial_{\nu} F^{\nu\alpha} + \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha} + \Gamma^{\alpha}_{\nu\sigma} F^{\nu\sigma}) + 3g_{\alpha\beta} \Phi F^{\nu\alpha} \partial_{\nu} \Phi
$$

$$
\mu_0 j^{\alpha} = \Phi^2 \left( \partial_{\nu} F^{\nu\alpha} + \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha} + \underbrace{\Gamma^{\alpha}_{\nu\sigma} F^{\nu\sigma}}_{\Phi} \right) + 3\Phi F^{\nu\alpha} \partial_{\nu} \Phi
$$

$$
\mu_0 j^{\alpha} = \Phi^2 \nabla_{\nu} F^{\nu\alpha} + 3\Phi F^{\nu\alpha} \partial_{\nu} \Phi
$$

We recognize equation (4.30). The term $\Gamma^\alpha_{\nu\sigma} F^{\nu\sigma}$ cancels because it is the contraction of a symmtric tensor with an antisymmetric tensor. We now apply the 4D covariant
derivative on each side of this equation :

$$
\mu_0 \nabla_{\alpha} j^{\alpha} = \nabla_{\alpha} (\Phi^2 \partial_{\nu} F^{\nu\alpha}) + \nabla_{\alpha} (\Phi^2 \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha}) + 3 \nabla_{\alpha} (\Phi F^{\nu\alpha} \partial_{\nu} \Phi)
$$

$$
\begin{equation}
\begin{split}
\mu_0 \nabla_{\alpha} j^{\alpha} = {}& (2\Phi \partial_{\alpha} \Phi \partial_{\nu} F^{\nu\alpha} + \Phi^2 \underbrace{\partial_{\alpha} \partial_{\nu} F^{\nu\alpha}}_{\Phi} + \Phi^2 \Gamma^{\lambda}_{\lambda\alpha} \partial_{\nu} F^{\nu\alpha}) \\
& + (2\Phi \partial_{\alpha} \Phi \Gamma^{\nu}_{\nu\sigma} F^{\nu\alpha} + \Phi^2 (\partial_{\alpha} \Gamma^{\nu}_{\nu\sigma}) F^{\sigma\alpha} + \Phi^2 \Gamma^{\nu}_{\nu\sigma} \partial_{\alpha} F^{\sigma\alpha} + \Phi^2 \underbrace{\Gamma^{\lambda}_{\lambda\alpha} \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha}}_{\Phi}) \\
& + (3 \underbrace{F^{\nu\alpha} \partial_{\nu} \Phi}_{\Phi} \partial_{\alpha} \Phi + 3 \Phi \underbrace{F^{\nu\alpha} \partial_{\alpha} \partial_{\nu} \Phi}_{\Phi} + 3 \Phi \partial_{\alpha} F^{\nu\alpha} \partial_{\nu} \Phi + 3 \Phi \Gamma^{\sigma}_{\sigma\alpha} F^{\nu\alpha} \partial_{\nu} \Phi)
\end{split}
\tag{11.31}
\end{equation}
$$

Calling $g$ the determinant of $g_{\mu\nu}$, the term $(\partial_\alpha \Gamma^\nu_{\nu\sigma}) F^{\nu\alpha}$ can be written $\frac{1}{2}(\partial_\nu \partial_\alpha g) F^{\nu\alpha} = 0$, while $\Phi^2 \Gamma^\lambda_{\lambda\alpha} \partial_\nu F^{\nu\alpha} + \Phi^2 \Gamma^\nu_{\nu\sigma} \partial_\alpha F^{\sigma\alpha} = 0$. The remaining terms are :

$$
\mu_0 \nabla_{\alpha} j^{\alpha} = 2\Phi \partial_{\alpha} \Phi \partial_{\nu} F^{\nu\alpha} + 2\Phi \partial_{\alpha} \Phi \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha} + 3\Phi \partial_{\alpha} F^{\nu\alpha} \partial_{\nu} \Phi + 3\Phi \Gamma^{\sigma}_{\sigma\alpha} F^{\nu\alpha} \partial_{\nu} \Phi
$$

$$
\mu_0 \nabla_{\alpha} j^{\alpha} = -\Phi \partial_{\alpha} \Phi \partial_{\nu} F^{\nu\alpha} - \Phi \partial_{\alpha} \Phi \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha}
$$

$$
\mu_0 \nabla_{\alpha} j^{\alpha} = -(\Phi^2 \partial_{\nu} F^{\nu\alpha} + \Phi^2 \Gamma^{\nu}_{\nu\sigma} F^{\sigma\alpha}) \frac{\partial_{\alpha} \Phi}{\Phi} \qquad (11.32)
$$

$$
\mu_0 \nabla_{\alpha} j^{\alpha} = -(\mu_0 j^{\alpha} \frac{\partial_{\alpha} \Phi}{\Phi} - \frac{3 \Phi F^{\nu\alpha} \partial_{\nu} \Phi \partial_{\alpha} \Phi}{\Phi})
$$

$$
0 = \mu_0 (\nabla_{\alpha} j^{\alpha} + \frac{j^{\alpha} \partial_{\alpha} \Phi}{\Phi})
$$
## 11.6 Annexe 6 : Divergence of the 4D Einstein tensor in Kaluza Klein theory

In this Annex, we compute $\nabla_{\mu} G^{\mu\nu} = \nabla_{\mu} (g^{\alpha\mu} g^{\beta\nu} G_{\alpha\beta})$ where :

$$ G_{\alpha\beta} = 8\pi G \left( \Phi^2 \left[ \frac{g^{\lambda\sigma} F_{\alpha\lambda} F_{\sigma\beta} + \frac{1}{4} g_{\alpha\beta} F^{\lambda\sigma} F_{\lambda\sigma}}{\mu_0} \right] + \left[ \frac{\partial_{\alpha} \partial_{\beta} \Phi - \Gamma^{\sigma}_{\alpha\beta} \partial_{\sigma} \Phi - g_{\alpha\beta} \Box g \Phi}{8\pi G \Phi} \right] + [\rho u_{\alpha} u_{\beta}] \right) \quad (11.33) $$

We compute each term separately. We start with $g^{\alpha\mu}g^{\beta\nu}(g^{\lambda\sigma}F_{\alpha\lambda}F_{\sigma\beta} + \frac{1}{4}g_{\alpha\beta}F^{\lambda\sigma}F_{\lambda\sigma}) = F^{\mu\lambda}F_{\lambda}^{\nu} + \frac{1}{4}g^{\mu\nu}F^{\lambda\sigma}F_{\lambda\sigma}$ and we use the property $\nabla_{\mu}g_{\alpha\beta} = \nabla_{\mu}g^{\lambda\sigma} = 0$ :

$$
\begin{align*}
& \nabla_{\mu} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \\
&= (\partial_{\mu} F^{\mu\lambda}) F_{\lambda}^{\nu} + F^{\mu\lambda} \partial_{\mu} F_{\lambda}^{\nu} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda} F_{\lambda}^{\nu} + \Gamma_{\mu\sigma}^{\nu} F^{\mu\lambda} F_{\lambda}^{\sigma} + \frac{1}{4} g^{\mu\nu} (\partial_{\mu} F^{\lambda\sigma} F_{\lambda\sigma} + F^{\lambda\sigma} \partial_{\mu} F_{\lambda\sigma}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + F^{\mu\lambda} (\partial_{\mu} F_{\lambda}^{\nu} + \Gamma_{\mu\sigma}^{\nu} F_{\lambda}^{\sigma}) + \frac{1}{4} g^{\mu\nu} (\partial_{\mu} (g^{\lambda\alpha} g^{\beta\sigma} F_{\alpha\beta}) F_{\lambda\sigma} + F^{\lambda\sigma} \partial_{\mu} F_{\lambda\sigma}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + F^{\mu\lambda} g^{\nu\sigma} \partial_{\mu} F_{\lambda\sigma} + F^{\mu\lambda} (F_{\lambda\sigma} \partial_{\mu} g^{\nu\sigma} + \Gamma_{\mu\sigma}^{\nu} F_{\lambda\beta} g^{\beta\sigma}) + \frac{g^{\mu\nu}}{4} (\partial_{\mu} g^{\lambda\alpha} g^{\beta\sigma} F_{\alpha\beta} F_{\lambda\sigma} + \partial_{\mu} g^{\beta\sigma} F_{\alpha\beta} F_{\lambda\sigma} g^{\lambda\alpha} + \partial_{\mu} F_{\alpha\beta} F_{\lambda\sigma} g^{\alpha\lambda} g^{\beta\sigma} + F^{\lambda\sigma} \partial_{\mu} F_{\lambda\sigma}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + F^{\lambda\sigma} g^{\nu\mu} \partial_{\lambda} F_{\sigma\mu} + F^{\mu\lambda} F_{\lambda\sigma} (-\Gamma_{\mu\sigma}^{\sigma} g^{\nu\beta}) + \frac{1}{4} g^{\mu\nu} (\partial_{\mu} g^{\sigma\beta} F_{\beta}^{\lambda} F_{\sigma\lambda} + \partial_{\mu} g^{\beta\sigma} F_{\beta}^{\lambda} F_{\lambda\sigma}) + \frac{1}{2} g^{\mu\nu} F^{\lambda\sigma} \partial_{\mu} F_{\lambda\sigma} \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + \frac{1}{2} (F^{\lambda\sigma} g^{\nu\mu} \partial_{\lambda} F_{\sigma\mu} + F^{\sigma\lambda} g^{\nu\mu} \partial_{\sigma} F_{\lambda\mu}) + F^{\mu\lambda} F_{\lambda\sigma} (-\Gamma_{\mu\sigma}^{\sigma} g^{\nu\beta}) + \frac{1}{4} g^{\mu\nu} (\partial_{\mu} g^{\sigma\beta} F_{\beta}^{\lambda} F_{\sigma\lambda} + \partial_{\mu} g^{\beta\sigma} F_{\beta}^{\lambda} F_{\lambda\sigma}) + \frac{1}{2} g^{\mu\nu} F^{\lambda\sigma} \partial_{\mu} F_{\lambda\sigma} \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + \frac{1}{2} g^{\mu\nu} F^{\lambda\sigma} (\partial_{\lambda} F_{\sigma\mu} + \partial_{\sigma} F_{\mu\lambda}) + F^{\mu\lambda} F_{\lambda\sigma} (-\Gamma_{\mu\sigma}^{\sigma} g^{\nu\beta}) + \frac{1}{2} g^{\mu\nu} \partial_{\mu} g^{\sigma\beta} F_{\beta}^{\lambda} F_{\sigma\lambda} + \frac{1}{2} g^{\mu\nu} F^{\lambda\sigma} \partial_{\mu} F_{\lambda\sigma} \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + \frac{1}{2} g^{\mu\nu} F^{\lambda\sigma} \underbrace{(\partial_{\lambda} F_{\sigma\mu} + \partial_{\sigma} F_{\lambda\mu} + \partial_{\mu} F_{\lambda\sigma})}_{F_{\lambda\sigma}} + F_{\lambda\sigma} (-F^{\mu\lambda} \Gamma_{\mu\sigma}^{\sigma} g^{\nu\beta} - \frac{1}{2} g^{\mu\nu} \partial_{\mu} g^{\sigma\beta} F_{\beta}^{\lambda}) \quad (11.34) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + F_{\lambda\sigma} (-F^{\mu\lambda} \Gamma_{\mu\sigma}^{\sigma} g^{\nu\beta} + \frac{1}{2} \Gamma_{\mu\alpha\beta} g^{\sigma\alpha} g^{\mu\nu} F_{\beta}^{\lambda} + \frac{1}{2} \Gamma_{\mu\alpha\gamma} g^{\sigma\gamma} g^{\mu\nu} F_{\beta}^{\lambda}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + F_{\lambda\sigma} (-F^{\mu\lambda} \Gamma_{\mu\sigma\gamma} g^{\nu\gamma} g^{\sigma\alpha} + \frac{1}{2} \Gamma_{\mu\alpha\mu} g^{\sigma\alpha} g^{\gamma\nu} F^{\mu\lambda} + \frac{1}{2} \Gamma_{\mu\alpha\gamma} g^{\sigma\gamma} g^{\alpha\nu} F^{\mu\lambda}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + F_{\lambda\sigma} F^{\mu\lambda} (-\Gamma_{\mu\alpha\gamma} g^{\sigma\gamma} g^{\nu\alpha} + \frac{1}{2} \Gamma_{\alpha\mu\gamma} g^{\sigma\gamma} g^{\alpha\nu} + \frac{1}{2} \Gamma_{\alpha\mu\gamma} g^{\sigma\gamma} g^{\alpha\nu}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + g^{\sigma\gamma} g^{\alpha\nu} F_{\lambda\sigma} F^{\mu\lambda} (\frac{1}{2} \Gamma_{\alpha\gamma\mu} - \frac{1}{2} \Gamma_{\alpha\mu\gamma}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + g^{\alpha\nu} F_{\lambda}^{\gamma} F^{\mu\lambda} \frac{1}{4} (g_{\mu\alpha,\gamma} + g_{\gamma\mu,\alpha} - g_{\alpha\gamma,\mu} - g_{\gamma\alpha,\mu} - g_{\mu\gamma,\alpha} + g_{\alpha\mu,\gamma}) \\
&= (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} - g^{\alpha\nu} \underbrace{\frac{1}{2} F_{\lambda}^{\gamma} F^{\lambda\mu}}_{(g_{\mu\alpha,\gamma} - g_{\alpha\gamma,\mu})} (g_{\mu\alpha,\gamma} - g_{\alpha\gamma,\mu})
\end{align*}
$$

The last term vanishes because it is the contraction of the symmetric tensor $F_{\lambda}^{\gamma}F^{\lambda\mu}$ with the antisymmetric tensor $g_{\mu\alpha,\gamma} - g_{\alpha\gamma,\mu}$.
Using equations (4.29), (4.30), (11.30), we deduce :

$$
\begin{align*}
\nabla_{\mu} \left[ \frac{\Phi^2}{\mu_0} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \right] &= \frac{\Phi^2}{\mu_0} (\partial_{\mu} F^{\mu\lambda} + \Gamma_{\mu\sigma}^{\mu} F^{\sigma\lambda}) F_{\lambda}^{\nu} + 2 \frac{\partial_{\mu} \Phi}{\Phi} \left[ \frac{\Phi^2}{\mu_0} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \right] \\
&= F_{\lambda}^{\nu} (j^{\lambda} - 3 \frac{\partial_{\sigma} \Phi}{\mu_0 \Phi} \Phi^2 F^{\sigma\lambda}) + 2 \frac{\partial_{\mu} \Phi}{\Phi} \left[ \frac{\Phi^2}{\mu_0} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \right] \\
&= \frac{q}{Mc} F_{\lambda}^{\nu} \rho u^{\lambda} \left[ \Phi^2 \left( u^5 - \frac{qA_{\sigma}}{Mc} u^{\sigma} \right) \right] - 3 \frac{\partial_{\sigma} \Phi}{\Phi} \Phi^2 F^{\sigma\lambda} F_{\lambda}^{\nu} + 2 \frac{\partial_{\mu} \Phi}{\mu_0 \Phi} \left[ \frac{\Phi^2}{\mu_0} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \right] \tag{11.35} \\
&= \frac{q}{Mc} F_{\lambda}^{\nu} \rho u^{\lambda} \left[ \Phi^2 \left( u^5 - \frac{qA_{\sigma}}{Mc} u^{\sigma} \right) \right] - \frac{\partial_{\mu} \Phi}{\Phi} \left[ \frac{\Phi^2}{\mu_0} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \right] + \frac{\partial_{\mu} \Phi}{\Phi} \frac{3g^{\mu\nu}}{4\mu_0} \Phi^2 F^{\lambda\sigma} F_{\lambda\sigma} \\
&= \frac{q}{Mc} F_{\lambda}^{\nu} \rho u^{\lambda} \left[ \Phi^2 \left( u^5 - \frac{qA_{\sigma}}{Mc} u^{\sigma} \right) \right] - \frac{\partial_{\mu} \Phi}{\Phi} \left[ \frac{\Phi^2}{\mu_0} (F^{\mu\lambda} F_{\lambda}^{\nu} + \frac{1}{4} g^{\mu\nu} F^{\lambda\sigma} F_{\lambda\sigma}) \right] + \frac{\partial_{\mu} \Phi}{\Phi} \frac{1}{8\pi G} \left( -\frac{1}{2} g^{\mu\nu} R + \rho [\Phi^2 (u^5 - \frac{qA_{\alpha}}{Mc})^2] \right)
\end{align*}
$$

We used equation (4.27) in the last line. To compute the divergence of the energy momentum tensor of the scalar field, we use the following property, which hold for any tensor $T_{\alpha\beta}$

$$
\begin{align}
\nabla_{\mu} T^{\mu\nu} &= \nabla_{\mu} (g^{\mu\alpha} g^{\nu\beta} T_{\alpha\beta}) = \partial_{\mu} (g^{\mu\alpha} g^{\nu\beta} T_{\alpha\beta}) + \Gamma^{\mu}_{\mu\sigma} g^{\sigma\alpha} g^{\nu\beta} T_{\alpha\beta} + \Gamma^{\nu}_{\mu\sigma} g^{\mu\alpha} g^{\sigma\beta} T_{\alpha\beta} = g^{\mu\alpha} g^{\nu\beta} \partial_{\mu} T_{\alpha\beta} + (\partial_{\mu} g^{\mu\alpha} + \Gamma^{\mu}_{\mu\sigma} g^{\sigma\alpha}) g^{\nu\beta} T_{\alpha\beta} + (\partial_{\mu} g^{\nu\beta} + \Gamma^{\nu}_{\mu\sigma} g^{\sigma\beta}) g^{\mu\alpha} T_{\alpha\beta} \nonumber \\
&= g^{\mu\alpha} g^{\nu\beta} \partial_{\mu} T_{\alpha\beta} - \Gamma^{\alpha}_{\mu\sigma} g^{\sigma\mu} g^{\nu\beta} T_{\alpha\beta} - \Gamma^{\beta}_{\mu\sigma} g^{\alpha\mu} g^{\nu\sigma} T_{\alpha\beta} = g^{\mu\alpha} g^{\nu\beta} \partial_{\mu} T_{\alpha\beta} - \Gamma^{\sigma}_{\mu\alpha} g^{\alpha\mu} g^{\nu\beta} T_{\sigma\beta} - \Gamma^{\sigma}_{\mu\beta} g^{\alpha\mu} g^{\nu\beta} T_{\alpha\sigma} = g^{\alpha\mu} g^{\nu\beta} (\partial_{\mu} T_{\alpha\beta} - \Gamma^{\sigma}_{\mu\alpha} T_{\sigma\beta} - \Gamma^{\sigma}_{\mu\beta} T_{\alpha\sigma}) \tag{11.36} \\
&= g^{\mu\alpha} g^{\nu\beta} \nabla_{\mu} T_{\alpha\beta} \nonumber
\end{align}
$$

We first consider $T_{\alpha\beta} = \partial_{\alpha}\partial_{\beta}\Phi - \Gamma^{\lambda}_{\alpha\beta}\partial_{\lambda}\Phi - g_{\alpha\beta}\Box g_{\beta}\Phi$

$$
\begin{align}
g^{\mu\alpha} g^{\nu\beta} \nabla_{\mu} T_{\alpha\beta} &= g^{\mu\alpha} g^{\nu\beta} \left[ \nabla_{\mu} (\partial_{\alpha} \partial_{\beta} \Phi - \Gamma^{\sigma}_{\alpha\beta} \partial_{\sigma} \Phi - g_{\alpha\beta} \Box g_{\beta} \Phi) \right] \nonumber \\
&= g^{\mu\alpha} g^{\nu\beta} \left[ \partial_{\mu} T_{\alpha\beta} - \Gamma^{\sigma}_{\mu\alpha} T_{\beta\sigma} - \Gamma^{\sigma}_{\mu\beta} T_{\alpha\sigma} \right] \nonumber \\
&= g^{\mu\alpha} g^{\nu\beta} \left[ \partial_{\mu} (\partial_{\alpha} \partial_{\beta} \Phi - \Gamma^{\lambda}_{\alpha\beta} \partial_{\lambda} \Phi) - g_{\alpha\beta,\mu} \Box g_{\beta} \Phi - g_{\alpha\beta} \partial_{\mu} \Box g_{\beta} \Phi - \Gamma^{\sigma}_{\mu\alpha} (\partial_{\sigma} \partial_{\beta} \Phi - \Gamma^{\lambda}_{\sigma\beta} \partial_{\lambda} \Phi) + \Gamma^{\sigma}_{\mu\alpha} g_{\sigma\beta} \Box g_{\beta} \Phi - \Gamma^{\sigma}_{\mu\beta} (\partial_{\sigma} \partial_{\alpha} \Phi - \Gamma^{\lambda}_{\sigma\alpha} \partial_{\lambda} \Phi) + \Gamma^{\sigma}_{\mu\beta} g_{\sigma\alpha} \Box g_{\beta} \Phi \right] \nonumber \\
&= g^{\mu\alpha} g^{\nu\beta} \left[ \partial_{\mu} (\partial_{\alpha} \partial_{\beta} \Phi - \Gamma^{\lambda}_{\alpha\beta} \partial_{\lambda} \Phi) - g_{\alpha\beta} \partial_{\mu} \Box g_{\beta} \Phi - \Gamma^{\sigma}_{\mu\alpha} (\partial_{\sigma} \partial_{\beta} \Phi - \Gamma^{\lambda}_{\sigma\beta} \partial_{\lambda} \Phi) - \Gamma^{\sigma}_{\mu\beta} (\partial_{\sigma} \partial_{\alpha} \Phi - \Gamma^{\lambda}_{\sigma\alpha} \partial_{\lambda} \Phi) + (-g_{\alpha\beta,\mu} + \Gamma^{\sigma}_{\mu\alpha} g_{\sigma\beta} + \Gamma^{\sigma}_{\mu\beta} g_{\sigma\alpha}) \Box g_{\beta} \Phi \right] \nonumber \\
&= g^{\mu\alpha} g^{\nu\beta} \partial_{\beta} \partial_{\mu} \partial_{\alpha} \Phi - g^{\mu\alpha} g^{\nu\beta} \partial_{\mu} (\Gamma^{\lambda}_{\alpha\beta} \partial_{\lambda} \Phi) - g^{\mu\alpha} g^{\nu\beta} g_{\alpha\beta} \partial_{\mu} (\partial_{\lambda} \partial^{\lambda} \Phi) - g^{\mu\alpha} g^{\nu\beta} g_{\alpha\beta} \partial_{\mu} (\Gamma^{\lambda}_{\lambda\sigma} \partial^{\sigma} \Phi) \nonumber \\
&\quad + g^{\mu\alpha} g^{\nu\beta} (-\Gamma^{\sigma}_{\mu\alpha} \partial_{\sigma} \partial_{\beta} \Phi + \Gamma^{\sigma}_{\mu\alpha} \Gamma^{\lambda}_{\sigma\beta} \partial_{\lambda} \Phi - \Gamma^{\sigma}_{\mu\beta} \partial_{\sigma} \partial_{\alpha} \Phi + \Gamma^{\sigma}_{\mu\beta} \Gamma^{\lambda}_{\sigma\alpha} \partial_{\lambda} \Phi) \tag{11.37} \\
&= g^{\mu\alpha} g^{\nu\beta} \partial_{\beta} \partial_{\mu} \partial_{\alpha} \Phi - g^{\mu\alpha} g^{\nu\beta} (\partial_{\mu} \Gamma^{\lambda}_{\alpha\beta}) \partial_{\lambda} \Phi - g^{\mu\alpha} g^{\nu\beta} \Gamma^{\lambda}_{\alpha\beta} \partial_{\mu} \partial_{\lambda} \Phi - g^{\nu\beta} \partial_{\beta} \partial_{\lambda} \partial^{\lambda} \Phi - g^{\nu\beta} \partial_{\beta} (\Gamma^{\lambda}_{\lambda\sigma} \partial^{\sigma} \Phi) \nonumber \\
&\quad + g^{\mu\alpha} g^{\nu\beta} (-\Gamma^{\sigma}_{\mu\alpha} \partial_{\sigma} \partial_{\beta} \Phi + \Gamma^{\sigma}_{\mu\alpha} \Gamma^{\lambda}_{\sigma\beta} \partial_{\lambda} \Phi - \Gamma^{\sigma}_{\mu\beta} \partial_{\sigma} \partial_{\alpha} \Phi + \Gamma^{\sigma}_{\mu\beta} \Gamma^{\lambda}_{\sigma\alpha} \partial_{\lambda} \Phi) \nonumber \\
&= g^{\nu\beta} \partial_{\beta} \partial_{\mu} (g^{\mu\alpha} \partial_{\alpha} \Phi) - (\partial_{\mu} g^{\alpha\mu}) g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi - (g^{\nu\beta} \partial_{\beta} \partial_{\mu} g^{\alpha\mu}) \partial_{\alpha} \Phi - (g^{\nu\beta} \partial_{\beta} g^{\alpha\mu}) \partial_{\mu} \partial_{\alpha} \Phi - g^{\nu\beta} \partial_{\beta} \partial_{\lambda} \partial^{\lambda} \Phi - g^{\nu\beta} \partial_{\beta} (\Gamma^{\lambda}_{\lambda\sigma} \partial^{\sigma} \Phi) \nonumber \\
&\quad + g^{\mu\alpha} g^{\nu\beta} (-\partial_{\mu} \Gamma^{\lambda}_{\alpha\beta}) \partial_{\lambda} \Phi - \Gamma^{\lambda}_{\alpha\beta} \partial_{\mu} \partial_{\lambda} \Phi - \Gamma^{\sigma}_{\mu\alpha} \partial_{\sigma} \partial_{\beta} \Phi + \Gamma^{\sigma}_{\mu\alpha} \Gamma^{\lambda}_{\sigma\beta} \partial_{\lambda} \Phi - \Gamma^{\sigma}_{\mu\beta} \partial_{\sigma} \partial_{\alpha} \Phi + \Gamma^{\sigma}_{\mu\beta} \Gamma^{\lambda}_{\sigma\alpha} \partial_{\lambda} \Phi) \nonumber
\end{align}
$$

We now isolate the quantity $-(\partial_{\mu} g^{\alpha\mu}) g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi - (g^{\nu\beta} \partial_{\beta} \partial_{\mu} g^{\alpha\mu}) \partial_{\alpha} \Phi - g^{\nu\beta} \partial_{\beta} (\Gamma^{\lambda}_{\lambda\sigma} \partial^{\sigma} \Phi)$:
$$
\begin{align}
& - (\partial_{\mu} g^{\alpha\mu}) g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi - (g^{\nu\beta} \partial_{\beta} \partial_{\mu} g^{\alpha\mu}) \partial_{\alpha} \Phi - g^{\nu\beta} \partial_{\beta} (\Gamma_{\lambda\sigma}^{\lambda} \partial^{\sigma} \Phi) \nonumber \\
&= (\Gamma_{\sigma\mu}^{\mu} g^{\alpha\sigma} + \Gamma_{\sigma\mu}^{\alpha} g^{\mu\sigma}) g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi - (g^{\nu\beta} \partial_{\beta} \partial_{\mu} g^{\alpha\mu}) \partial_{\alpha} \Phi - g^{\nu\beta} \partial_{\beta} (\Gamma_{\lambda\mu}^{\mu} g^{\alpha\lambda} \partial_{\alpha} \Phi) \nonumber \\
&= (\Gamma_{\sigma\mu}^{\mu} g^{\alpha\sigma} + \Gamma_{\sigma\mu}^{\alpha} g^{\mu\sigma}) g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi - (g^{\nu\beta} \partial_{\beta} \partial_{\mu} g^{\alpha\mu}) \partial_{\alpha} \Phi - g^{\nu\beta} \partial_{\beta} (\Gamma_{\lambda\mu}^{\mu} g^{\alpha\lambda} \partial_{\alpha} \Phi) - g^{\nu\beta} \Gamma_{\lambda\mu}^{\mu} g^{\alpha\lambda} \partial_{\beta} \partial_{\alpha} \Phi \nonumber \\
&= (\Gamma_{\sigma\mu}^{\mu} g^{\alpha\sigma} + \Gamma_{\sigma\mu}^{\alpha} g^{\mu\sigma} - \Gamma_{\lambda\mu}^{\mu} g^{\alpha\lambda}) g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi - g^{\nu\beta} \partial_{\beta} (\partial_{\mu} g^{\alpha\mu} + \Gamma_{\lambda\mu}^{\mu} g^{\alpha\lambda}) \partial_{\alpha} \Phi \tag{11.38} \\
&= \Gamma_{\sigma\mu}^{\alpha} g^{\mu\sigma} g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi + g^{\nu\beta} \partial_{\beta} (\Gamma_{\lambda\mu}^{\alpha} g^{\mu\lambda}) \partial_{\alpha} \Phi \nonumber \\
&= \Gamma_{\sigma\mu}^{\alpha} g^{\mu\sigma} g^{\nu\beta} \partial_{\beta} \partial_{\alpha} \Phi + g^{\nu\beta} (\partial_{\beta} \Gamma_{\lambda\mu}^{\alpha} g^{\mu\lambda} \partial_{\alpha} \Phi + g^{\nu\beta} \Gamma_{\lambda\mu}^{\alpha} (\partial_{\beta} g^{\mu\lambda}) \partial_{\alpha} \Phi) \nonumber \\
&= \Gamma_{\alpha\mu}^{\sigma} g^{\mu\alpha} g^{\nu\beta} \partial_{\beta} \partial_{\sigma} \Phi + g^{\nu\beta} (\partial_{\beta} \Gamma_{\alpha\mu}^{\sigma} g^{\mu\alpha} \partial_{\lambda} \Phi - \Gamma_{\lambda\mu}^{\sigma} \Gamma_{\beta\sigma}^{\mu} g^{\nu\beta} g^{\sigma\lambda} \partial_{\alpha} \Phi - \Gamma_{\lambda\mu}^{\sigma} \Gamma_{\beta\sigma}^{\lambda} g^{\nu\beta} g^{\sigma\mu} \partial_{\alpha} \Phi) \nonumber \\
&= g^{\nu\beta} g^{\alpha\mu} (\partial_{\beta} \Gamma_{\alpha\mu}^{\lambda} \partial_{\lambda} \Phi + \Gamma_{\sigma\mu}^{\lambda} \partial_{\beta} \partial_{\lambda} \Phi - 2 \Gamma_{\sigma\mu}^{\lambda} \Gamma_{\beta\alpha}^{\sigma} \partial_{\lambda} \Phi) \nonumber
\end{align}
$$

We inject this result in equation (11.37) :

$$
\begin{align}
g^{\mu\alpha} g^{\nu\beta} \nabla_{\mu} T_{\alpha\beta} &= -g^{\nu\beta} (\partial_{\beta} g^{\lambda\mu}) \partial_{\mu} \partial_{\lambda} \Phi + g^{\nu\beta} g^{\alpha\mu} (\partial_{\beta} \Gamma_{\alpha\mu}^{\lambda} \partial_{\lambda} \Phi + \underbrace{\Gamma_{\sigma\mu}^{\lambda} \partial_{\beta} \partial_{\lambda} \Phi}_{\sim} - 2 \Gamma_{\sigma\mu}^{\lambda} \Gamma_{\beta\alpha}^{\sigma} \partial_{\lambda} \Phi) \nonumber \\
&\quad + g^{\mu\alpha} g^{\nu\beta} (-\underbrace{(\partial_{\mu} \Gamma_{\alpha\beta}^{\lambda})}_{\sim} \partial_{\lambda} \Phi - \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} \partial_{\lambda} \Phi - \underbrace{\Gamma_{\mu\alpha}^{\sigma} \partial_{\sigma} \partial_{\beta} \Phi}_{\sim} \nonumber \\
&\quad + \Gamma_{\mu\alpha}^{\sigma} \Gamma_{\alpha\beta}^{\lambda} \partial_{\lambda} \Phi - \Gamma_{\mu\beta}^{\lambda} \partial_{\lambda} \partial_{\alpha} \Phi + \Gamma_{\mu\beta}^{\sigma} \Gamma_{\alpha\sigma}^{\lambda} \partial_{\lambda} \Phi) \nonumber \\
&= -g^{\nu\beta} (\partial_{\beta} g^{\lambda\mu} + \Gamma_{\alpha\beta}^{\lambda} g^{\mu\alpha}) \partial_{\mu} \partial_{\lambda} \Phi - g^{\nu\beta} \Gamma_{\mu\beta}^{\lambda} g^{\mu\alpha} \partial_{\lambda} \partial_{\alpha} \Phi \nonumber \\
&\quad + g^{\mu\alpha} g^{\nu\beta} (\partial_{\beta} \Gamma_{\alpha\mu}^{\lambda} - \partial_{\mu} \Gamma_{\alpha\beta}^{\lambda} - 2 \Gamma_{\sigma\mu}^{\lambda} \Gamma_{\beta\sigma}^{\sigma} + \Gamma_{\mu\alpha}^{\sigma} \Gamma_{\sigma\beta}^{\lambda} + \Gamma_{\mu\beta}^{\sigma} \Gamma_{\sigma\alpha}^{\lambda}) \partial_{\lambda} \Phi \tag{11.39} \\
&= -g^{\nu\beta} (\partial_{\beta} g^{\lambda\mu} + \Gamma_{\alpha\beta}^{\lambda} g^{\mu\alpha}) \partial_{\mu} \partial_{\lambda} \Phi - g^{\nu\beta} \Gamma_{\mu\beta}^{\lambda} g^{\mu\alpha} \partial_{\lambda} \partial_{\alpha} \Phi \nonumber \\
&\quad + g^{\mu\alpha} g^{\nu\beta} (\partial_{\beta} \Gamma_{\alpha\mu}^{\lambda} - \partial_{\mu} \Gamma_{\alpha\beta}^{\lambda} - \Gamma_{\sigma\mu}^{\lambda} \Gamma_{\beta\sigma}^{\alpha} + \Gamma_{\mu\alpha}^{\sigma} \Gamma_{\sigma\beta}^{\lambda}) \partial_{\lambda} \Phi \nonumber \\
&= -g^{\nu\beta} (\partial_{\beta} g^{\lambda\mu} + \underbrace{\Gamma_{\alpha\beta}^{\lambda} g^{\mu\alpha}}_{\sim} + \Gamma_{\mu\beta}^{\mu} g^{\alpha\lambda}) \partial_{\mu} \partial_{\lambda} \Phi + g^{\mu\alpha} g^{\nu\beta} \underbrace{R_{\alpha\mu\beta}^{\lambda} \partial_{\lambda} \Phi}_{\sim} \nonumber
\end{align}
$$

where $R_{\alpha\mu\beta}^{\lambda}$ is the Riemann tensor. We now compute $g^{\mu\alpha} R_{\alpha\mu\beta}^{\lambda} = g^{\mu\alpha}(\partial_{\beta}\Gamma_{\alpha\mu}^{\lambda} - \partial_{\mu}\Gamma_{\alpha\beta}^{\lambda} + \Gamma_{\mu\alpha}^{\sigma}\Gamma_{\sigma\beta}^{\lambda} - \Gamma_{\sigma\mu}^{\lambda}\Gamma_{\beta\alpha}^{\sigma})$. Each term is computed separately.

$$
\begin{align}
g^{\mu\alpha} \partial_{\beta} \Gamma_{\alpha\mu}^{\lambda} &= \partial_{\beta} (g^{\mu\alpha} \Gamma_{\alpha\mu}^{\lambda}) - \Gamma_{\alpha\mu}^{\lambda} \partial_{\beta} g^{\mu\alpha} \nonumber \\
&= \partial_{\beta} \left( g^{\mu\alpha} \frac{g^{\lambda\gamma}}{2} (g_{\gamma\alpha,\mu} + g_{\mu\gamma,\alpha} - g_{\alpha\mu,\gamma}) \right) - \Gamma_{\alpha\mu}^{\lambda} \partial_{\beta} g^{\mu\alpha} \nonumber \\
&= \partial_{\beta} \left( g^{\mu\alpha} g^{\lambda\gamma} (-\Gamma_{\alpha\gamma\mu} + g_{\mu\gamma,\alpha}) \right) - \Gamma_{\alpha\mu}^{\lambda} \partial_{\beta} g^{\mu\alpha} \tag{11.40} \\
&= \partial_{\beta} \left( -g^{\lambda\gamma} \Gamma_{\alpha\gamma}^{\alpha} + g^{\lambda\gamma} \partial_{\alpha} (g_{\mu\gamma} g^{\mu\alpha}) \right) - g^{\lambda\gamma} g_{\mu\gamma} \partial_{\alpha} g^{\alpha\mu} - \Gamma_{\alpha\mu}^{\lambda} \partial_{\beta} g^{\mu\alpha} \nonumber \\
&= \partial_{\beta} \left( -g^{\lambda\gamma} \Gamma_{\alpha\gamma}^{\alpha} - g^{\lambda}_{\mu} \partial_{\alpha} g^{\alpha\mu} \right) - \Gamma_{\alpha\mu}^{\lambda} \partial_{\beta} g^{\mu\alpha} \nonumber \\
&= -g^{\lambda\gamma} \partial_{\beta} \Gamma_{\alpha\gamma}^{\alpha} - \Gamma_{\alpha\gamma}^{\alpha} \partial_{\beta} g^{\lambda\gamma} \text{ [F1]} - \partial_{\beta} \partial_{\alpha} g^{\alpha\lambda} \text{ [F2]} - \Gamma_{\alpha\mu}^{\lambda} \partial_{\beta} g^{\mu\alpha} \nonumber
\end{align}
$$

$$
\begin{align}
-g^{\mu\alpha} \partial_{\mu} \Gamma_{\alpha\beta}^{\lambda} &= -\partial_{\mu} (g^{\mu\alpha} \Gamma_{\alpha\beta}^{\lambda}) + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \nonumber \\
&= -\partial_{\mu} \left( g^{\mu\alpha} \frac{g^{\lambda\gamma}}{2} (g_{\gamma\alpha,\beta} + g_{\beta\gamma,\alpha} - g_{\alpha\beta,\gamma}) \right) + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \nonumber \\
&= -\partial_{\mu} (g^{\mu\alpha} g^{\lambda\gamma} (-\Gamma_{\beta\gamma\alpha} + g_{\alpha\gamma,\beta})) + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \nonumber \\
&= -\partial_{\mu} (g^{\mu\alpha} g^{\lambda\gamma} (-\Gamma_{\beta\gamma\alpha} + g_{\alpha\gamma,\beta})) + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \tag{11.41} \\
&= -\partial_{\mu} (-g^{\lambda\gamma} \Gamma_{\beta\gamma}^{\mu} + g^{\lambda\gamma} (\partial_{\beta}(g^{\mu\alpha} \overline{g_{\alpha\gamma}}) - g_{\alpha\gamma} \partial_{\beta} g^{\mu\alpha})) + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \nonumber \\
&= -\partial_{\mu} (-g^{\lambda\gamma} \Gamma_{\beta\gamma}^{\mu} - g^{\lambda}_{\alpha} \partial_{\beta} g^{\alpha\mu}) + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \nonumber \\
&= g^{\lambda\gamma} \partial_{\mu} \Gamma_{\beta\gamma}^{\mu} + \Gamma_{\beta\gamma}^{\mu} \partial_{\mu} g^{\lambda\gamma} + \partial_{\mu} \partial_{\beta} g^{\lambda\mu} \text{ [-F2]} + \Gamma_{\alpha\beta}^{\lambda} \partial_{\mu} g^{\mu\alpha} \text{ [F3]} \nonumber
\end{align}
$$

$$
\begin{align}
g^{\mu\alpha} \Gamma_{\alpha\mu}^{\sigma} \Gamma_{\beta\sigma}^{\lambda} &= g^{\mu\alpha} \frac{g^{\sigma\delta}}{2} (g_{\delta\alpha,\mu} + g_{\mu\delta,\alpha} - g_{\alpha\mu,\delta}) \frac{g^{\lambda\gamma}}{2} (g_{\gamma\beta,\sigma} + g_{\sigma\gamma,\beta} - g_{\beta\sigma,\gamma}) \nonumber \\
&= g^{\alpha\mu} \frac{g^{\sigma\delta}}{2} (-\Gamma_{\delta\alpha\mu} + g_{\mu\delta,\alpha}) \frac{g^{\lambda\gamma}}{2} (-\Gamma_{\gamma\beta\sigma} + g_{\sigma\gamma,\beta}) \nonumber \\
&= g^{\lambda\gamma} (-\Gamma_{\delta\alpha}^{\alpha} - g_{\mu\delta} \partial_{\alpha} g^{\alpha\mu}) (-\Gamma_{\gamma\beta}^{\delta} - g_{\sigma\gamma} \partial_{\beta} g^{\sigma\delta}) \tag{11.42} \\
&= g^{\lambda\gamma} \Gamma_{\delta\alpha}^{\sigma} \Gamma_{\gamma\beta}^{\delta} + \Gamma_{\delta\alpha}^{\sigma} \partial_{\beta} g^{\lambda\delta} + g^{\lambda\gamma} \Gamma_{\gamma\beta\mu} \partial_{\alpha} g^{\alpha\mu} + g_{\mu\delta} \partial_{\alpha} g^{\alpha\mu} \partial_{\beta} g^{\lambda\delta} \nonumber \\
&= g^{\lambda\gamma} \Gamma_{\delta\alpha}^{\sigma} \Gamma_{\gamma\beta}^{\delta} + \Gamma_{\delta\alpha}^{\sigma} \partial_{\beta} g^{\lambda\delta} + g^{\lambda\gamma} (-\Gamma_{\mu\beta\gamma} + g_{\mu\gamma,\beta}) \partial_{\alpha} g^{\alpha\mu} - \underbrace{\partial_{\alpha} g^{\mu\alpha} g^{\lambda\delta}}_{\sim} \partial_{\beta} g_{\mu\delta} \nonumber \\
&= g^{\lambda\gamma} \Gamma_{\delta\alpha}^{\sigma} \Gamma_{\gamma\beta}^{\delta} + \Gamma_{\delta\alpha}^{\sigma} \partial_{\beta} g^{\lambda\delta} \text{ [-F1]} - \Gamma_{\mu\beta}^{\lambda} \partial_{\alpha} g^{\alpha\mu} \text{ [-F3]} \nonumber
\end{align}
$$
$$g^{\mu\alpha}\Gamma_{\sigma\mu}^{\lambda}\Gamma_{\beta\alpha}^{\sigma} = g^{\mu\gamma}\Gamma_{\sigma\mu}^{\lambda}\Gamma_{\beta\gamma}^{\sigma} \qquad (11.43)$$

We now add the remaining terms to get

$$
\begin{align}
g^{\mu\alpha} R_{\alpha\mu\beta}^{\lambda} &= g^{\lambda\gamma}(-\partial_{\beta}\Gamma_{\alpha\gamma}^{\alpha} + \partial_{\mu}\Gamma_{\beta\gamma}^{\mu} + \Gamma_{\delta\alpha}^{\alpha}\Gamma_{\gamma\beta}^{\delta}) - g^{\mu\gamma}\Gamma_{\sigma\mu}^{\lambda}\Gamma_{\beta\gamma}^{\sigma} + \Gamma_{\beta\gamma}^{\mu}\partial_{\mu}g^{\lambda\gamma} - \Gamma_{\alpha\mu}^{\lambda}\partial_{\beta}g^{\mu\alpha} \nonumber \\
&= g^{\lambda\gamma}(-\partial_{\beta}\Gamma_{\alpha\gamma}^{\alpha} + \partial_{\mu}\Gamma_{\beta\gamma}^{\mu} + \Gamma_{\delta\alpha}^{\alpha}\Gamma_{\gamma\beta}^{\delta}) \nonumber \\
&\quad - g^{\mu\gamma}\Gamma_{\sigma\mu}^{\lambda}\Gamma_{\beta\gamma}^{\sigma} - \Gamma_{\beta\gamma}^{\mu}\Gamma_{\mu\sigma}^{\lambda}g^{\gamma\sigma} - \Gamma_{\beta\gamma}^{\mu}\Gamma_{\mu\sigma}^{\gamma}g^{\lambda\sigma} + \Gamma_{\alpha\mu}^{\lambda}\Gamma_{\beta\sigma}^{\mu}g^{\sigma\alpha} + \Gamma_{\alpha\mu}^{\lambda}\Gamma_{\beta\sigma}^{\sigma}g^{\mu\alpha} \nonumber \\
&= g^{\lambda\gamma}(-\partial_{\beta}\Gamma_{\alpha\gamma}^{\alpha} + \partial_{\mu}\Gamma_{\beta\gamma}^{\mu} + \Gamma_{\sigma\alpha}^{\alpha}\Gamma_{\gamma\beta}^{\sigma}) \nonumber \\
&\quad - \frac{g^{\mu\gamma}\Gamma_{\sigma\mu}^{\lambda}\Gamma_{\beta\gamma}^{\sigma}}{\Gamma_{\beta\gamma}^{\mu}\Gamma_{\mu\sigma}^{\lambda}g^{\gamma\sigma}} - \Gamma_{\beta\gamma}^{\mu}\Gamma_{\mu\sigma}^{\lambda}g^{\gamma\sigma} - \Gamma_{\beta\sigma}^{\mu}\Gamma_{\mu\gamma}^{\sigma}g^{\lambda\gamma} + \frac{\Gamma_{\alpha\mu}^{\lambda}\Gamma_{\beta\gamma}^{\mu}g^{\gamma\sigma}}{\Gamma_{\beta\gamma}^{\mu}\Gamma_{\mu\sigma}^{\lambda}g^{\sigma\mu}} + \frac{\Gamma_{\alpha\mu}^{\lambda}\Gamma_{\beta\sigma}^{\sigma}g^{\mu\gamma}}{\Gamma_{\beta\sigma}^{\mu}\Gamma_{\mu\gamma}^{\sigma}g^{\gamma\mu}} \nonumber \\
&= g^{\lambda\gamma}(-\partial_{\beta}\Gamma_{\alpha\gamma}^{\alpha} + \partial_{\mu}\Gamma_{\beta\gamma}^{\mu} + \Gamma_{\sigma\alpha}^{\alpha}\Gamma_{\gamma\beta}^{\sigma} - \Gamma_{\beta\sigma}^{\mu}\Gamma_{\mu\gamma}^{\sigma}) \nonumber \\
&= g^{\lambda\gamma}R_{\beta\gamma}
\end{align}
$$

We can use this result to rewrite equation (11.39)

$$g^{\mu\alpha}g^{\nu\beta}\nabla_{\mu}T_{\alpha\beta} = g^{\mu\alpha}g^{\nu\beta}R_{\alpha\mu\beta}^{\lambda}\partial_{\lambda}\Phi = g^{\nu\beta}g^{\lambda\gamma}R_{\beta\gamma}\partial_{\lambda}\Phi = R^{\nu\lambda}\partial_{\lambda}\Phi \qquad (11.45)$$

We immediately deduce

$$\nabla_{\mu}(g^{\mu\alpha}g^{\nu\beta}\left[\frac{\partial_{\alpha}\partial_{\beta}\Phi - \Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\Phi - g_{\alpha\beta}\Box_{g}\Phi}{8\pi G\Phi}\right]) = \frac{R^{\nu\lambda}\partial_{\lambda}\Phi}{8\pi G\Phi} - \frac{\partial_{\mu}\Phi}{\Phi}g^{\mu\alpha}g^{\nu\beta}\left[\frac{\partial_{\alpha}\partial_{\beta}\Phi - \Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\Phi - g_{\alpha\beta}\Box_{g}\Phi}{8\pi G\Phi}\right] \quad (11.46)$$

With the above equation and equation (11.35), we can now compute $\nabla_{\mu}G^{\mu\nu}$ where $G^{\mu\nu}$ is given by equation (11.33)

$$
\begin{align}
0 &= \nabla_{\mu}G^{\mu\nu} \nonumber \\
&= \frac{q}{Mc}F_{\lambda}^{\nu}\rho u^{\lambda}[\Phi^{2}(u^{5} - \frac{qA_{\sigma}}{Mc}u^{\sigma})] - \frac{\partial_{\mu}\Phi}{\Phi}[\frac{\Phi^{2}}{\mu_{0}}(F^{\mu\lambda}F_{\lambda}^{\nu} + \frac{1}{4}g^{\mu\nu}F^{\lambda\sigma}F_{\lambda\sigma})] - \frac{\partial_{\mu}\Phi}{\Phi}\frac{1}{8\pi G}\frac{1}{2}g^{\mu\nu}R \nonumber \\
&\quad + \frac{\partial_{\mu}\Phi}{\Phi}\rho[\Phi^{2}(u^{5} - \frac{qA_{\alpha}}{Mc})^{2}] + \frac{R^{\nu\mu}\partial_{\mu}\Phi}{8\pi G\Phi} - \frac{\partial_{\mu}\Phi}{\Phi}g^{\mu\alpha}g^{\nu\beta}\left[\frac{\partial_{\alpha}\partial_{\beta}\Phi - \Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\Phi - g_{\alpha\beta}\Box_{g}\Phi}{8\pi G\Phi}\right] \nonumber \\
&\quad + \nabla_{\mu}(\rho u^{\mu}u^{\nu}) \nonumber \\
&= \left[\frac{q}{Mc}F_{\lambda}^{\nu}\rho u^{\lambda}[\Phi^{2}(u^{5} - \frac{qA_{\sigma}}{Mc}u^{\sigma})] + \rho(\dot{u}^{\nu} + \Gamma_{\sigma\mu}^{\nu}u^{\sigma}u^{\mu}) + \frac{\partial_{\mu}\Phi}{\Phi^{3}}\rho[\Phi^{2}(u^{5} - \frac{qA_{\alpha}}{Mc})^{2}]\right] \nonumber \\
&\quad + \frac{\partial_{\mu}\Phi}{\Phi}\left[\frac{R^{\nu\mu} - \frac{1}{2}g^{\mu\nu}R}{8\pi G} - \frac{\Phi^{2}}{\mu_{0}}(F^{\mu\lambda}F_{\lambda}^{\nu} + \frac{1}{4}g^{\mu\nu}F^{\lambda\sigma}F_{\lambda\sigma}) \right. \nonumber \\
&\qquad \left. - g^{\mu\alpha}g^{\nu\beta}\frac{\partial_{\alpha}\partial_{\beta}\Phi - \Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\Phi - g_{\alpha\beta}\Box_{g}\Phi}{8\pi G\Phi}\right] + u^{\nu}\left[\partial_{\mu}(\rho u^{\mu}) + \Gamma_{\alpha\mu}^{\mu}\rho u^{\alpha}\right] \nonumber \\
&= \left[\rho(\dot{u}^{\nu} + \Gamma_{\sigma\mu}^{\nu}u^{\sigma}u^{\mu}) + \frac{q}{Mc}F_{\lambda}^{\nu}\rho u^{\lambda}[\Phi^{2}(u^{5} - \frac{qA_{\sigma}}{Mc}u^{\sigma})] + \frac{\partial_{\mu}\Phi}{\Phi^{3}}\rho[\Phi^{2}(u^{5} - \frac{qA_{\alpha}}{Mc})^{2}]\right] \nonumber \\
&\quad + u^{\nu}\left[\partial_{\mu}(\rho u^{\mu}) + \Gamma_{\alpha\mu}^{\mu}\rho u^{\alpha} + \rho\frac{u^{\mu}\partial_{\mu}\Phi}{\Phi}\right] \tag{11.47}
\end{align}
$$

We recognize equation (4.21).
11.7 Annex 7 : Details on covariant quantum mechanics

In this annex we justify the results given in equation (5.23) and (5.24).

Computation of equation (5.23) where $\mathcal{O} = \mathcal{O}(x^\alpha)$

$$
\begin{align}
\psi^*[\Box_g(\mathcal{O}\psi)]\sqrt{g} &= \psi^*[\partial_\mu(g^{\mu\nu}\partial_\nu(\mathcal{O}\psi)) + \Gamma^\lambda_{\lambda\mu}g^{\mu\nu}\partial_\nu(\mathcal{O}\psi)]\sqrt{g} \nonumber \\
&= \psi^*[\partial_\mu(g^{\mu\nu}(\partial_\nu\mathcal{O})\psi + g^{\mu\nu}\mathcal{O}\partial_\nu\psi) + \Gamma^\lambda_{\lambda\mu}g^{\mu\nu}((\partial_\nu\mathcal{O})\psi + (\mathcal{O}\partial_\nu\psi))]\sqrt{g} \nonumber \\
&= \psi^*\partial_\mu g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g} + \psi^*g^{\mu\nu}\partial_\mu(\partial_\nu\mathcal{O})\psi\sqrt{g} + \psi^*g^{\mu\nu}(\partial_\mu\psi)(\partial_\nu\mathcal{O})\sqrt{g} \nonumber \\
&\quad + \psi^*\mathcal{O}\partial_\mu(g^{\mu\nu}\partial_\nu\psi)\sqrt{g} + \psi^*g^{\mu\nu}(\partial_\mu\mathcal{O})(\partial_\nu\psi)\sqrt{g} + \psi^*\Gamma^\lambda_{\lambda\mu}g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g} \nonumber \\
&\quad + \psi^*\Gamma^\lambda_{\lambda\mu}g^{\mu\nu}\mathcal{O}(\partial_\nu\psi)\sqrt{g} \nonumber \\
&= \psi^*\partial_\mu g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g} + \partial_\mu(\psi^*g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g}) - \underbrace{\psi^*\partial_\mu g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g}}_{\psi^*g^{\mu\nu}(\partial_\nu\mathcal{O})(\partial_\mu\psi)\sqrt{g}} - (\partial_\mu\psi^*)g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g} \nonumber \\
&\quad - \psi^*g^{\mu\nu}(\partial_\nu\mathcal{O})(\partial_\mu\psi)\sqrt{g} - (\partial_\mu\psi^*)g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\sqrt{g} - \psi^*g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\Gamma^\lambda_{\lambda\mu\lambda}\sqrt{g} \nonumber \\
&\quad + \psi^*g^{\mu\nu}(\partial_\nu\mathcal{O})(\partial_\mu\psi)\sqrt{g} + \psi^*\mathcal{O}\partial_\mu(g^{\mu\nu}\partial_\nu\psi)\sqrt{g} + \psi^*g^{\mu\nu}(\partial_\mu\mathcal{O})(\partial_\nu\psi)\sqrt{g} \nonumber \\
&\quad + \psi^*g^{\mu\nu}(\partial_\nu\mathcal{O})\psi\Gamma^\lambda_{\lambda\mu\lambda}\sqrt{g} + \psi^*\Gamma^\lambda_{\lambda\mu}g^{\mu\nu}\mathcal{O}(\partial_\nu\psi)\sqrt{g} \nonumber \\
&= \partial_\mu(g^{\mu\nu}(\partial_\nu\mathcal{O})\psi^*\psi\sqrt{g}) + \psi^*\mathcal{O}[\partial_\mu(g^{\mu\nu}\partial_\nu\psi) + \Gamma^\lambda_{\lambda\mu}g^{\mu\nu}\mathcal{O}(\partial_\nu\psi)]\sqrt{g} \nonumber \\
&\quad + (\psi^*\partial^\nu\psi - \psi\partial^\nu\psi^*)\partial_\nu\mathcal{O}\sqrt{g} \nonumber \\
&= \partial_\mu(|\psi|^2\partial^\mu\mathcal{O})\sqrt{g} + \psi^*\mathcal{O}[\Box_g\psi]\sqrt{g} + (\psi^*\partial^\nu\psi - \psi\partial^\nu\psi^*)\partial_\nu\mathcal{O}\sqrt{g} \tag{11.48}
\end{align}
$$

Then we get

$$
\psi^*[\Box_g(\mathcal{O}\psi)]\sqrt{g} - \psi^*\mathcal{O}[\Box_g\psi]\sqrt{g} = \partial_\mu(|\psi|^2\partial^\mu\mathcal{O})\sqrt{g} + (\psi^*\partial^\nu\psi - \psi\partial^\nu\psi^*)\partial_\nu\mathcal{O}\sqrt{g} \quad (11.49)
$$

$$
\psi^*[\mathcal{O}\Box_g\psi - \Box_g(\mathcal{O}\psi)]\sqrt{g} = (\psi\partial^\nu\psi^* - \psi^*\partial^\nu\psi)\partial_\nu\mathcal{O}\sqrt{g} - \partial_\mu(|\psi|^2\partial^\mu\mathcal{O})\sqrt{g} \quad (11.50)
$$

Computation of equation (5.24) :

The last term of (11.50) does not contribute to the action under appropriate conditions : neglecting it, we can introduce $\hat{H}_g = \frac{\hbar^2}{2mc} \Box_g + \frac{1}{2}mc$ and write

$$
\begin{align}
\psi^* \left[ \mathcal{O} \frac{\hbar^2}{2mc} \Box_g - \frac{\hbar^2}{2mc} \Box_g (\mathcal{O}\psi) \psi \right] \sqrt{g} &= \frac{\hbar^2}{2mc} (\psi \partial^\nu \psi^* - \psi^* \partial^\nu \psi) \partial_\nu \mathcal{O} \sqrt{g} \nonumber \\
\psi^* \left[ \mathcal{O} \frac{\hbar^2}{2mc} \Box_g \psi + \frac{1}{2}mc\mathcal{O}\psi - \frac{\hbar^2}{2mc} \Box_g (\mathcal{O}\psi) - \frac{1}{2}mc\mathcal{O}\psi \right] \sqrt{g} &= \frac{\hbar^2}{2mc} (\psi \partial^\nu \psi^* - \psi^* \partial^\nu \psi) \partial_\nu \mathcal{O} \sqrt{g} \nonumber \\
\psi^* \left[ \mathcal{O} \left( \frac{\hbar^2}{2mc} \Box_g + \frac{1}{2}mc \right) \psi - \left( \frac{\hbar^2}{2mc} \Box_g + \frac{1}{2}mc \right) (\mathcal{O}\psi) \right] \sqrt{g} &= \frac{\hbar^2}{2mc} (\psi \partial^\nu \psi^* - \psi^* \partial^\nu \psi) \partial_\nu \mathcal{O} \sqrt{g} \tag{11.51} \\
\psi^* \left[ \mathcal{O} \hat{H}_g \psi - \hat{H}_g (\mathcal{O}\psi) \right] \sqrt{g} &= \frac{\hbar^2}{2mc} (\psi \partial^\nu \psi^* - \psi^* \partial^\nu \psi) \partial_\nu \mathcal{O} \sqrt{g} \nonumber \\
\psi^* \left[ [\mathcal{O}; \hat{H}_g]_- \psi \right] \sqrt{g} &= \frac{\hbar^2}{2mc} (\psi \partial^\nu \psi^* - \psi^* \partial^\nu \psi) \partial_\nu \mathcal{O} \sqrt{g} \nonumber \\
\psi^* \left[ \frac{1}{i\hbar} \left[ [\mathcal{O}; \hat{H}_g]_- \psi \right] \right] \sqrt{g} &= \frac{-i\hbar}{2mc} (\psi \partial^\nu \psi^* - \psi^* \partial^\nu \psi) \partial_\nu \mathcal{O} \sqrt{g} \nonumber
\end{align}
$$
## 11.8 Annex 8 : Linearized 5D Gravity

In this annex we examine the gauge degrees of freedom in the Kaluza-Klein theory. We start by writing the linearized theory of 5D gravity, 5D quantities being underlined to distinguish them from 4D quantities : the metric is written $g_{MN} = \eta_{MN} + \bar{h}_{MN}$ and the harmonic gauge becomes :

$$ \underline{g}^{MN} \Gamma_{MN}^{Q} = 0 \rightarrow \eta^{MN} \frac{\eta^{PQ}}{2} (h_{PM,N} + h_{NP,M} - h_{MN,P}) = \partial_{N} \underline{h}^{QN} - \frac{1}{2} \partial^{Q} \underline{h} = \partial_{N} (\underline{h}^{QN} - \frac{1}{2} \eta^{QN} \underline{h}) = 0 \quad (11.52) $$

where $\underline{h} = \eta^{MN}h_{MN}$. We can then write the harmonic gauge in the form $\partial_{N}\underline{h}^{QN}$ with $\bar{h}_{MN} = h_{MN} - \frac{1}{2}\eta_{MN}\underline{h}$. By analogy with 4D gravity, the Einstein tensor takes the form $G_{MN} = -\frac{1}{2}\Box \bar{h}_{MN}$ which produces a 5D wave equation, $\Box$ being the 5D Laplacian. We now choose the Kaluza-Klein metric given in equation (4.9) with the condition $\Phi = 1$ and we only keep the linear terms (the terms $\frac{qA_{\mu}}{Mc} \frac{qA_{\nu}}{Mc}$ are neglected) :

$$ \underline{g}_{MN} = \left[ \begin{matrix} g_{\mu\nu} - \frac{\Phi^2 q A_{\mu} q A_{\nu}}{Mc^2 Mc} & \frac{\Phi^2 q A_{\mu}}{Mc} \\ \frac{\Phi^2 q A_{\nu}}{Mc} & -\Phi^2 \end{matrix} \right] \rightarrow \underline{g}_{MN} = \left[ \begin{matrix} \eta_{\mu\nu} + h_{\mu\nu} & \frac{q A_{\mu}}{Mc} \\ \frac{q A_{\nu}}{Mc} & -1 \end{matrix} \right] \quad (11.53) $$

where all fields depend on $x^{\alpha}$ but not $x^5$. Neglecting non linear terms implies in particular that all terms of the form $E_i E_j$ are neglected. Then $\underline{h}_{\mu\nu} = h_{\mu\nu}$, $\underline{h}_{5\mu} = \frac{qA_{\mu}}{Mc}$, $\underline{h}_{55} = 0$ and $\underline{h} = \eta^{MN}h_{MN} = \eta^{\mu\nu}h_{\mu\nu} = h$. The 5D harmonic gauge condition becomes :

$$ 0 = \partial_N (\underline{h}^{QN} - \frac{1}{2} \underline{\eta}^{QN} \underline{h}) = \partial_5 (\underline{h}^{Q5} - \frac{1}{2} \underline{\eta}^{Q5} \underline{h}) + \partial_\mu (\underline{h}^{Q\mu} - \frac{1}{2} \underline{\eta}^{Q\mu} \underline{h}) \quad (11.54) $$

When $Q = \nu$ we recover the 4D harmonic gauge condition $\partial_{\mu}(\underline{h}^{\nu\mu} - \frac{1}{2}\underline{\eta}^{\nu\mu}\underline{h}) = \partial_{\mu}\bar{h}^{\nu\mu} = 0$ while, when $Q = 5$ we obtain the Lorenz gauge $\partial_{\mu}(\frac{qA_{\mu}}{Mc} - \frac{1}{2}\underline{\eta}^{5\mu}\underline{h}) = \partial_{\mu}\frac{qA_{\mu}}{Mc} = 0$. The 5D gauge change takes the form :

$$ \bar{h}_{MN} \rightarrow \bar{h}_{MN} + \partial_M \xi_N + \partial_N \xi_M - \eta_{MN} \partial_P \xi^P = \bar{h}_{MN} + \partial_M \xi_N + \partial_N \xi_M - \eta_{MN} (\partial_5 \xi^5 + \partial_\alpha \xi^\alpha) \quad (11.55) $$

When $M = \mu$ with $N = \nu$ it reduces to the previously introduced 4D gauge change $\bar{h}_{\mu\nu} \rightarrow \bar{h}_{\mu\nu} + \partial_{\mu}\xi_{\nu} + \partial_{\nu}\xi_{\mu} - \eta_{\mu\nu}\partial_{\alpha}\xi^{\alpha}$ and, when $M = \mu$ with $N = 5$ it produces the electromagnetic gauge change $\frac{qA_{\mu}}{Mc} \rightarrow \frac{qA_{\mu}}{Mc} + \partial_{5}\xi_{\mu} + \partial_{\mu}\xi_{5} + \eta_{5\mu}\partial_{\alpha}\xi^{\alpha} = \frac{qA_{\mu}}{Mc} + \partial_{\mu}\xi_{5}$. Counting simultaneously the 2 possible polarizations of electromagnetic waves and the 2 possible polarizations of gravitational waves can be done in the linearized approximation of the 5D Kaluza-Klein theory. The gravitational part is not affected while the electromagnetic part is automatically produced : the 5D gauge invariance produces the 4D gravitational gauge invariance and the electromagnetic gauge invariance while the 5D harmonic gauge produces the 4D gravitational harmonic gauge and the electromagnetic Lorenz gauge.
## 11.9 Annex 9 : On the product of Householder matrices

In this annex we give a detailed computation of equation (5.78) : We compute $a_{11}, a_{12}, a_{21}, a_{22}$ in the following equation.

$$ -H_{ij}(\vec{x}) = \begin{bmatrix} 1 - 2x_1^2 & -2x_1x_2 & 0 \\ -2x_1x_2 & 1 - 2x_2^2 & 0 \\ 0 & 0 & 1 \end{bmatrix} ; \quad -H_{ij}(\vec{p}) = \begin{bmatrix} 1 - 2p_1^2 & -2p_1p_2 & 0 \\ -2p_1p_2 & 1 - 2p_2^2 & 0 \\ 0 & 0 & 1 \end{bmatrix} ; \quad H_i^j(\vec{p})H_{jk}(\vec{p}) = \begin{bmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad (11.56) $$

where $x_1^2 + x_2^2 = p_1^2 + p_2^2 = 1$. The angle $\theta$ between the vectors $\vec{r} = (x_1, x_2, 0)$ and $\vec{p} = (p_1, p_2, 0)$ can be computed according to $\cos\theta = x_1p_1 + x_2p_2$ and $\sin\theta = x_2p_1 - x_1p_2$. Using the trigonometric identities $\cos2\theta = \cos^2\theta - \sin^2\theta$ and $\sin2\theta = 2\cos\theta\sin\theta$, direct computation gives

$$ \begin{aligned} a_{11} &= 1 - 2x_1^2 - 2p_1^2 + 4x_1^2p_1^2 + 4x_1x_2p_1p_2 \\ &= (x_1^2 + x_2^2)(p_1^2 + p_2^2) - 2x_1^2(p_1^2 + p_2^2) - 2p_1^2(x_1^2 + x_2^2) + 4x_1^2p_1^2 + 4x_1x_2p_1p_2 \\ &= x_1^2p_1^2 + x_1^2p_2^2 + x_2^2p_1^2 + x_2^2p_2^2 - 2x_1^2p_1^2 - 2x_1^2p_2^2 - 2p_1^2x_1^2 - 2p_1^2x_2^2 + 4x_1^2p_1^2 + 4x_1x_2p_1p_2 \\ &= x_1^2p_1^2 + x_2^2p_2^2 - x_1^2p_2^2 - x_2^2p_1^2 + 4x_1x_2p_1p_2 \\ &= (x_1p_1 + x_2p_2)^2 - (x_1p_2 - x_2p_1)^2 \\ &= \cos^2\theta - \sin^2\theta = \cos 2\theta \end{aligned} \qquad (11.57) $$

$$ \begin{aligned} a_{22} &= 1 - 2x_2^2 - 2p_2^2 + 4x_2^2p_2^2 + 4x_1x_2p_1p_2 \\ &= (x_1^2 + x_2^2)(p_1^2 + p_2^2) - 2x_2^2(p_1^2 + p_2^2) - 2p_2^2(x_1^2 + x_2^2) + 4x_2^2p_2^2 + 4x_1x_2p_1p_2 \\ &= x_1^2p_1^2 + x_1^2p_2^2 + x_2^2p_1^2 + x_2^2p_2^2 - 2x_2^2p_1^2 - 2x_2^2p_2^2 - 2p_2^2x_1^2 - 2p_2^2x_2^2 + 4x_2^2p_2^2 + 4x_1x_2p_1p_2 \\ &= x_1^2p_1^2 + x_2^2p_2^2 - x_1^2p_2^2 - x_2^2p_1^2 + 4x_1x_2p_1p_2 \\ &= (x_1p_1 + x_2p_2)^2 - (x_1p_2 - x_2p_1)^2 \\ &= \cos^2\theta - \sin^2\theta = \cos 2\theta \end{aligned} \qquad (11.58) $$

$$ \begin{aligned} a_{12} &= -2p_1p_2 + 4x_1^2p_1p_2 - 2x_1x_2 + 4p_2^2x_1x_2 \\ &= -2(p_1p_2(x_1^2 + x_2^2) - 2x_1^2p_1p_2 + x_1x_2(p_1^2 + p_2^2) - 2p_2^2x_1x_2) \\ &= -2(-p_1p_2x_1^2 + p_1p_2x_2^2 + x_1x_2p_1^2 - x_1x_2p_2^2) \\ &= -2((x_1p_1 + x_2p_2)(x_2p_1 - x_1p_2)) \\ &= -2\cos\theta\sin\theta = -\sin 2\theta \end{aligned} \qquad (11.59) $$

$$ \begin{aligned} a_{21} &= -2p_1p_2 + 4x_2^2p_1p_2 - 2x_1x_2 + 4p_1^2x_1x_2 \\ &= -2(p_1p_2(x_1^2 + x_2^2) - 2x_2^2p_1p_2 + x_1x_2(p_1^2 + p_2^2) - 2p_1^2x_1x_2) \\ &= -2(p_1p_2x_1^2 - p_1p_2x_2^2 - x_1x_2p_1^2 + x_1x_2p_2^2) \\ &= -2((x_1p_1 + x_2p_2)(x_1p_2 - x_2p_1)) \\ &= -2\cos\theta(-\sin\theta) = \sin 2\theta \end{aligned} \qquad (11.60) $$

Then,

$$ H_i^j(\vec{p})H_{jk}(\vec{p}) = \begin{bmatrix} \cos 2\theta & -\sin 2\theta & 0 \\ \sin 2\theta & \cos 2\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad (11.61) $$
11.10 Annex 10 : Algebraic properties of angular momentum operators

In this Annex, we closely follow D. J. Griffith [122]. $\vec{J}$ is used to designate the orbital angular momentum operator $\hat{L} = -i\hbar(\vec{r} \times \vec{\nabla})$ or the spin matrices $\vec{S}$. We only assume the relation :

$$[\mathbf{J}_i, \mathbf{J}_j]_- = \epsilon_{ijk} i \hbar \mathbf{J}_k \qquad (11.62)$$

where $i, j, k = 1, 2, 3$. We define $\mathbf{J}^2 = |\vec{J}|^2 = \mathbf{J}_i^2 + \mathbf{J}_j^2 + \mathbf{J}_k^2$. Then $\mathbf{J}_1$ commutes with $\mathbf{J}^2$ since :

$$ \begin{aligned} [\mathbf{J}^2, \mathbf{J}_1]_- &= [\mathbf{J}_1^2, \mathbf{J}_1]_- + [\mathbf{J}_2, \mathbf{J}_1]_- + [\mathbf{J}_3, \mathbf{J}_1]_- \\ &= \mathbf{J}_2[\mathbf{J}_2, \mathbf{J}_1]_- + [\mathbf{J}_2, \mathbf{J}_1]_-\mathbf{J}_2 + \mathbf{J}_3[\mathbf{J}_3, \mathbf{J}_1]_- + [\mathbf{J}_3, \mathbf{J}_1]_-\mathbf{J}_3 \end{aligned} \qquad (11.63) \\ = \mathbf{J}_2(-i\hbar\mathbf{J}_3) + (-i\hbar\mathbf{J}_3)\mathbf{J}_2 + \mathbf{J}_3(i\hbar\mathbf{J}_2) + (i\hbar\mathbf{J}_2)\mathbf{J}_3 $$

Similarly $[\mathbf{J}^2, \mathbf{J}_2]_- = [\mathbf{J}^2, \mathbf{J}_3]_- = 0$. $\mathbf{J}^2$ is then compatible with all components of $\vec{J}$ so we are interested by simultanous eigenstates $|\Lambda\rangle$ of $\mathbf{J}^2$ and (say) $\mathbf{J}_3$ :

$$ \mathbf{J}^2|\Lambda\rangle = \lambda\hbar^2|\Lambda\rangle \quad ; \quad \mathbf{J}_3|\Lambda\rangle = m_j\hbar|\Lambda\rangle \qquad (11.64) $$

We define the ladder operator $\mathbf{J}_{\pm} = \mathbf{J}_1 \pm i\mathbf{J}_2$. Its commutator with $\mathbf{J}_3$ is :

$$ [\mathbf{J}_3, \mathbf{J}_{\pm}]_- = [\mathbf{J}_3, \mathbf{J}_1]_- \pm i[\mathbf{J}_3, \mathbf{J}_2]_- = i\hbar\mathbf{J}_2 \pm (-i\hbar\mathbf{J}_1) = \pm\hbar\mathbf{J}_{\pm} \qquad (11.65) $$

while of course $[\mathbf{J}^2, \mathbf{J}_{\pm}]_- = 0$. If $|\Lambda\rangle$ is an eigenstate of $\mathbf{J}^2$ and $\mathbf{J}_3$, so also is $\mathbf{J}_{\pm}|\Lambda\rangle$ since

$$ \mathbf{J}^2(\mathbf{J}_{\pm}|\Lambda\rangle) = \mathbf{J}_{\pm}(\mathbf{J}^2|\Lambda\rangle) = \mathbf{J}_{\pm}(\lambda\hbar^2|\Lambda\rangle) = \lambda\hbar^2\mathbf{J}_{\pm}|\Lambda\rangle \qquad (11.66) $$

$|\Lambda\rangle$ and $\mathbf{J}_{\pm}|\Lambda\rangle$ are then eigenstates of $\mathbf{J}^2$ with the same eigenvalue $\lambda$. Using equation (11.65), we have :

$$ \mathbf{J}_3\mathbf{J}_{\pm}|\Lambda\rangle = (\mathbf{J}_3\mathbf{J}_{\pm} - \mathbf{J}_{\pm}\mathbf{J}_3)|\Lambda\rangle + \mathbf{J}_{\pm}\mathbf{J}_3|\Lambda\rangle = \pm\hbar\mathbf{J}_{\pm}|\Lambda\rangle + \mathbf{J}_{\pm}m_j\hbar|\Lambda\rangle = (m_j \pm 1)\hbar\mathbf{J}_{\pm}|\Lambda\rangle \qquad (11.67) $$

So $\mathbf{J}_{\pm}|\Lambda\rangle$ is an eigenfunction of $\mathbf{J}_3$ with the new eigenvalue $m_j \pm 1$. Since $\mathbf{J}_3^2|\Lambda\rangle \le \mathbf{J}^2|\Lambda\rangle = \lambda\hbar^2|\Lambda\rangle$, there is necessarily a maximum value $|\Lambda_{max}\rangle$ and a minimum value $|\Lambda_{min}\rangle$ such that $\mathbf{J}_+|\Lambda_{max}\rangle = \mathbf{J}_-|\Lambda_{min}\rangle = 0$. Let $\hbar j$ be the eigenvalue of $|\Lambda_{max}\rangle$ according to $\mathbf{J}_3|\Lambda_{max}\rangle = \hbar j|\Lambda_{max}\rangle$ and similarly $\mathbf{J}_3|\Lambda_{min}\rangle = \hbar \bar{j}|\Lambda_{min}\rangle$. Now

$$ \begin{aligned} \mathbf{J}_{\pm}\mathbf{J}_{\mp} &= (\mathbf{J}_1 \pm i\mathbf{J}_2)(\mathbf{J}_1 \mp i\mathbf{J}_2) = \mathbf{J}_1^2 + \mathbf{J}_2^2 \mp i(\mathbf{J}_1\mathbf{J}_2 - \mathbf{J}_2\mathbf{J}_1) = \mathbf{J}^2 - \mathbf{J}_3^2 \pm \hbar\mathbf{J}_3 \end{aligned} \qquad (11.68) \\ \Rightarrow \mathbf{J}^2 = \mathbf{J}_{\pm}\mathbf{J}_{\mp} + \mathbf{J}_3^2 \mp \hbar\mathbf{J}_3 $$

It follows that :

$$ \begin{aligned} \mathbf{J}^2|\Lambda_{max}\rangle &= (\mathbf{J}_-\mathbf{J}_+ + \mathbf{J}_3^2 + \hbar\mathbf{J}_3)|\Lambda_{max}\rangle = (0 + \hbar^2j^2 + \hbar^2j)|\Lambda_{max}\rangle = \hbar^2j(j+1) \end{aligned} \qquad (11.69) \\ \mathbf{J}^2|\Lambda_{min}\rangle &= (\mathbf{J}_+\mathbf{J}_- + \mathbf{J}_3^2 - \hbar\mathbf{J}_3)|\Lambda_{min}\rangle = (0 + \hbar^2\bar{j}^2 - \hbar^2\bar{j})|\Lambda_{min}\rangle = \hbar^2\bar{j}(\bar{j}-1) \end{aligned} $$

Such that $\lambda = j(j+1) = \bar{j}(\bar{j}-1)$ which implies $\bar{j} = -j$ since $\bar{j} < -j$ is the minimum eigenvalue. The eigenvalues of $\mathbf{J}_3$ are $m_j\hbar$ where $m_j$ goes from $-j$ to $j$ in $N$ integer steps such that $j = -j + N$ or $2j = N$ : $j$ must be an integer or a half-integer.