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Metric tensor at second perturbation order for spherically symmetric space-times

Sergio Mendoza [email protected] Instituto de Astronomía, Universidad Nacional Autónoma de México, AP 70-264, Ciudad de México 04510, México
(5th April 2025)
Abstract

It is shown in this article that if the Einstein Equivalence Principle is valid on a particular metric theory of gravitation in a spherically symmetric space-time, then the time metric component is not equal to the negative of the inverse radial one unless the underlying potential is inversely proportional to the radial coordinate. At the weak field limit of approximation, a general formula is calculated and applied to some useful cases.

General relativity and gravitation; Alternative gravity theories

I Introduction

Finding solutions to extended metric theories of gravity is a difficult task, even when simple static and spherical approximations are assumed. A simple general technique that is often used in the literature to find solutions in spherical space-times is in which the time component of the metric equals the negative of the inverse radial one. This fact is motivated by the validity of this statement in the Schwarzschild solution of general relativity. However, the procedure to build Schwarzschild’s solution does not start with such assumption. It follows as a result of the general solution.

In this article it is shown that if a metric theory of gravity obeys the Einstein Equivalence Principle, then the metric time component is different from the negative inverse radial one. By adopting a Post Newtonian coordinate system of reference and a Post Newtonian gauge it is obtained the correct value of those metric components at the weak field limit of approximation in Schwarzschild-like coordinates.

The article is organised as follows. Section II discusses in general terms the Parametrised Post Newtonian (PPN) formalism at second order of approximation (the weak field limit) in standard Post Newtonian coordinates -which are equivalent to an isotropic space, and in Schwarzschild-like spherical ones. In Section III, the transformation from the isotropic space to the spherical one is performed and the metric components are calculated at second order of approximation. Useful examples of the general result are presented in Section IV and finally we discuss our results in Section V.

II Isotropic space-time.

The geodesic motion of a massive particle with mass mm is obtained by minimising the action [see e.g. 1]:

S=−m​c​∫abds=∫abℒ​dt,S=-mc\int_{a}^{b}{\mathrm{d}s}=\int_{a}^{b}{\mathcal{L}\mathrm{d}t}, (1)

with respect to the space-time coordinates xαx^{\alpha}. In the previous equation, d​s=gα​β​d​xα​d​xβ\mathrm{d}s=g_{\alpha\beta}\mathrm{d}x^{\alpha}\mathrm{d}x^{\beta} represents the interval of space time for a metric gα​βg_{\alpha\beta} and xαx^{\alpha} are a set of chosen coordinates, with x0=tx^{0}=t representing the time coordinate and xkx^{k} spatial coordinates. In here and in what follows we use Einstein’s summation convention and Greek space-time indices run from 0 to 44 while Latin space ones from 11 to 33. We choose a metric signature (+,−,−,−+,-,-,-). At the weakest order of approximation, i.e. when the velocity of light c→∞c\rightarrow\infty, the Lagrangian ℒ\mathcal{L} is given by [see e.g. 1, 2]:

ℒ=−m​c2+12​m​v2−m​ϕ,\mathcal{L}=-mc^{2}+\frac{1}{2}mv^{2}-m\phi, (2)

where ϕ\phi represents the Newtonian scalar potential, which for the particular case of a point mass source MM at the origin of coordinates is given by:

ϕ=−G​Mr,\phi=-\frac{GM}{r}, (3)

in which GG is Newton’s gravitational constant and rr the radial distance to the origin. Combining equations (2) and (1), the time component g00g_{00} of the metric gα​βg_{\alpha\beta} to 𝒪​(1/c2)\mathcal{O}(1/c^{2}) is given by:

g00=g00(0)+g00(2)=1+2​ϕc2.g_{00}={}^{(0)}g_{00}+{}^{(2)}g_{00}=1+\frac{2\phi}{c^{2}}. (4)

In the previous equation and in what follows we will refer to perturbation orders as 𝒪​(2)\mathcal{O}(2), 𝒪​(4)\mathcal{O}(4) meaning 𝒪​(1/c2)\mathcal{O}(1/c^{2}), 𝒪​(1/c4)\mathcal{O}(1/c^{4}) and so on. In equation (4) and in what follows, the left superindex parenthesis in a particular quantity represents the perturbation order at which that particular quantity is approximated. The fact that g00(2)=2​ϕ/c2{}^{(2)}g_{00}=2\phi/c^{2} is a manifestation of the validity of the Einstein Equivalence Principle [2].

In a Post-Newtonian system of reference with a standard Post-Newtonian gauge at 𝒪​(2)\mathcal{O}(2) of approximation, the metric is isotropic and of the form [2]:

d​s2=g00​c2​d​t2+Λ​(𝒙)​d​l2,\mathrm{d}s^{2}=g_{00}c^{2}\mathrm{d}t^{2}+\Lambda(\boldsymbol{x})\mathrm{d}l^{2}, (5)

where the square of the three dimensional length element d​l2=d​xk​d​xk\mathrm{d}l^{2}=\mathrm{d}x_{k}\mathrm{d}x^{k} and 𝒙\boldsymbol{x} is a three dimensional coordinate vector. In spherical-like coordinates (t,r~,θ,φt,\tilde{r},\theta,\varphi) the previous equation is:

d​s2=g00​c2​d​t2+Λ​(𝒙)​(d​r~+r~2​d​Ω),\mathrm{d}s^{2}=g_{00}c^{2}\mathrm{d}t^{2}+\Lambda(\boldsymbol{x})\left(\mathrm{d}\tilde{r}+\tilde{r}^{2}\mathrm{d}\Omega\right), (6)

where the squared angular displacement d​Ω2:=d​θ2+sin2⁡θ​d​φ2\mathrm{d}\Omega^{2}:=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2} for the polar and azimuthal angles θ\theta and φ\varphi respectively. The radial distance is represented by r~\tilde{r}.

At 𝒪​(0)\mathcal{O}(0) the components of the metric g00=1g_{00}=1 and Λ=−1\Lambda=-1. The next 𝒪​(2)\mathcal{O}(2) correction for the time component of the metric is given by equation (4). To find the 𝒪​(2)\mathcal{O}(2) correction to the function Λ​(𝒙)\Lambda(\boldsymbol{x}) note that so far we have not introduced any other function into the discussion but the function ϕ​(r)\phi(r). As such, it is expected that Λ(2)∝ϕ{}^{(2)}\Lambda\propto\phi and so the line element (6) can be rewritten as:

d​s2=(1+2​ϕ​(r~)c2)​c2​d​t2−(1−2​γ​ϕ​(r~)c2)​(d​r~+r~2​d​Ω),\mathrm{d}s^{2}=\left(1+\frac{2\phi(\tilde{r})}{c^{2}}\right)c^{2}\mathrm{d}t^{2}-\left(1-\frac{2\gamma\phi(\tilde{r})}{c^{2}}\right)\left(\mathrm{d}\tilde{r}+\tilde{r}^{2}\mathrm{d}\Omega\right), (7)

Notice that the function Λ(2){}^{(2)}\Lambda has been written to resemble as much as possible the function g00(2){}^{(2)}g_{00} and that is the reason for introducing the constant parameter γ\gamma which is the first PPN parameter as discussed in Section I.

The PPN parameter γ\gamma measures the amount of curvature of space [2] and its precise value must be obtained experimentally. A viable theory of gravity must converge to that value at its 𝒪​(2)\mathcal{O}(2) order of approximation. The deflection of light observed in Solar System experiments yields a value γ=1\gamma=1. This value is fully predicted by general relativity at 𝒪​(2)\mathcal{O}(2) perturbation order of the theory. For the case of MOdified theories of gravity (MOND), it turns out that the same value is also obtained in observations of the deflection of light in individual, groups and clusters of galaxies [3, 4, 5].

III Spherical symmetry

Let us make a coordinate transformation to spherical coordinates (t,r,θ,φt,r,\theta,\varphi) so that the line element (7) is given by:

d​s2=(1+2​ϕ​(r)c2)​c2​d​t2−g11​d​r2−r2​d​Ω2.\mathrm{d}s^{2}=\left(1+\frac{2\phi(r)}{c^{2}}\right)c^{2}\mathrm{d}t^{2}-g_{11}\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}. (8)

In the previous equation we have left the time component of the metric g00=(1+2​ϕ​(r)/c2)g_{00}=\left(1+2\phi(r)/c^{2}\right) unchanged in order to comply with the lowest perturbation order obtained from the action 1, which is coherent to the Einstein Equivalence Principle. The angular displacement d​Ω\mathrm{d}\Omega has been left unchanged with the transformation since the coordinates θ\theta and φ\varphi are physically the same between both systems of reference.

From equation (8) and (7) it follows that:

r=(1−γ​ϕ​(r~)c2)​r~,\displaystyle r=\left(1-\frac{\gamma\phi(\tilde{r})}{c^{2}}\right)\tilde{r}, (9)
g11​d​r2=−(1−2​γ​ϕ​(r~)c2)​d​r~2,\displaystyle g_{11}\mathrm{d}r^{2}=-\left(1-\frac{2\gamma\phi(\tilde{r})}{c^{2}}\right)\mathrm{d}\tilde{r}^{2}, (10)

at 𝒪​(2)\mathcal{O}{(2)}, and so, relation (10) is:

d​r=(1−γ​ϕ​(r~)c2)​d​r~−γ​r~​d​ϕ​(r~)c2.\mathrm{d}r=\left(1-\frac{\gamma\phi(\tilde{r})}{c^{2}}\right)\mathrm{d}\tilde{r}-\frac{\gamma\tilde{r}\mathrm{d}\phi(\tilde{r})}{c^{2}}. (11)

Substitution of this previous result into equation (10) yields at 𝒪​(2)\mathcal{O}(2):

g11​(r)=−(1+2​γ​r~c2​d​ϕ​(r~)d​r~),g_{11}(r)=-\left(1+\frac{2\gamma\tilde{r}}{c^{2}}\frac{\mathrm{d}\phi(\tilde{r})}{\mathrm{d}\tilde{r}}\right), (12)

where we have used the fact that d​ϕ​(r~)=d​r~​d​ϕ​(r~)/d​r~\mathrm{d}\phi(\tilde{r})=\mathrm{d}\tilde{r}\,\mathrm{d}\phi(\tilde{r})/\mathrm{d}\tilde{r}. The function g11​(r)g_{11}(r) in the previous equation is only a function of the spherical radial coordinate rr according to equation (9). To see this in more detail, note that:

ϕ​(r~)=ϕ​(r+γ​r​ϕ​(r~)c2),\phi(\tilde{r})=\phi\left(r+\frac{\gamma r\phi(\tilde{r})}{c^{2}}\right),

and so, performing a Taylor expansion of the previous function up to 𝒪​(2)\mathcal{O}(2) it follows that:

ϕ​(r~)=ϕ​(r)+γ​r​ϕ​(r~)c2​d​ϕ​(r)d​r.\phi(\tilde{r})=\phi(r)+\frac{\gamma r\phi(\tilde{r})}{c^{2}}\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}. (13)

Direct substitution of this relation into equation (12) and using relations (9) and (11) it follows that:

d​ϕ​(r~)d​r~=d​ϕ​(r)d​r~+dd​r~​[γ​r​ϕ​(r~)c2]=[1−γ​ϕ​(r~)c2−γ​r~c2​d​ϕ​(r~)d​r~]​{d​ϕ​(r)d​r~+dd​r~​[γ​r​ϕ​(r~)c2]},\begin{split}\frac{\mathrm{d}\phi(\tilde{r})}{\mathrm{d}\tilde{r}}&=\frac{\mathrm{d}\phi(r)}{\mathrm{d}\tilde{r}}+\frac{\mathrm{d}}{\mathrm{d}\tilde{r}}\left[\frac{\gamma r\phi(\tilde{r})}{c^{2}}\right]\\ &=\left[1-\frac{\gamma\phi(\tilde{r})}{c^{2}}-\frac{\gamma\tilde{r}}{c^{2}}\frac{\mathrm{d}\phi(\tilde{r})}{\mathrm{d}\tilde{r}}\right]\left\{\frac{\mathrm{d}\phi(r)}{\mathrm{d}\tilde{r}}+\frac{\mathrm{d}}{\mathrm{d}\tilde{r}}\left[\frac{\gamma r\phi(\tilde{r})}{c^{2}}\right]\right\},\end{split} (14)

which at 𝒪​(2)\mathcal{O}(2) perturbation order yields:

d​ϕ​(r~)d​r~​(1+γ​rc2​d​ϕ​(r~)d​r)=d​ϕ​(r)d​r+γ​r​ϕc2​d2​ϕ​(r)d​r2,\frac{\mathrm{d}\phi(\tilde{r})}{\mathrm{d}\tilde{r}}\left(1+\frac{\gamma r}{c^{2}}\frac{\mathrm{d}\phi(\tilde{r})}{\mathrm{d}r}\right)=\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}+\frac{\gamma r\phi}{c^{2}}\frac{\mathrm{d}^{2}\phi(r)}{\mathrm{d}r^{2}},

and so:

d​ϕ​(r~)d​r~=d​ϕ​(r)d​r+γ​rc2​{−(d​ϕ​(r)d​r)2+ϕ​d2​ϕ​(r)d​r2}.\frac{\mathrm{d}\phi(\tilde{r})}{\mathrm{d}\tilde{r}}=\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}+\frac{\gamma r}{c^{2}}\left\{-\left(\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}\right)^{2}+\phi\frac{\mathrm{d}^{2}\phi(r)}{\mathrm{d}r^{2}}\right\}. (15)

The substitution of this last result into equation (12) at the same 𝒪​(2)\mathcal{O}(2) perturbation order yields:

g11​(r)=−(1+2​γ​rc2​d​ϕ​(r)d​r)g_{11}(r)=-\left(1+\frac{2\gamma r}{c^{2}}\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}\right) (16)

The previous results can be stated formally in the following way:

Theorem 1.

The general form of the metric in spherical coordinates for any metric theory of gravity at 𝒪​(2)\mathcal{O}(2) perturbation order can be written as:

d​s2=(1+2​ϕ​(r)c2)​c2​d​t2−(1+2​γ​rc2​d​ϕ​(r)d​r)​d​r2−r2​d​Ω2,\mathrm{d}s^{2}=\left(1+\frac{2\phi(r)}{c^{2}}\right)c^{2}\mathrm{d}t^{2}-\left(1+\frac{2\gamma r}{c^{2}}\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}\right)\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}, (17)

or in Schwarzschild-like space-time form:

d​s2=(1+2​ϕ​(r)c2)​c2​d​t2−d​r2(1−2​γ​rc2​d​ϕ​(r)d​r)−r2​d​Ω2.\\ \mathrm{d}s^{2}=\left(1+\frac{2\phi(r)}{c^{2}}\right)c^{2}\mathrm{d}t^{2}-\frac{\mathrm{d}r^{2}}{\left(1-\frac{2\gamma r}{c^{2}}\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}\right)}-r^{2}\mathrm{d}\Omega^{2}. (18)
Corollary 1.1.

The only potential that satisfies a Schwarzschild-like behaviour for which g00=−1/g11g_{00}=-1/g_{11} is one that satisfies ϕ∝1/r1/γ\phi\propto 1/r^{1/\gamma}.

Proof.

The condition g00=−1/g11g_{00}=-1/g_{11} applied to the interval (18) yields γ​d​ln⁡ϕ=−d​ln⁡r\gamma\mathrm{d}\ln\phi=-\mathrm{d}\ln r, i.e. ϕ∝1/r1/γ\phi\propto 1/r^{1/\gamma}. ∎

As mentioned at the end of Section 6, different Solar System experiments show that γ=1\gamma=1 and so the potential ϕ∝1/r\phi\propto 1/r is the only one that satisfies the conditions of the previous Corollary. This corollary is fully satisfied in general relativity, but as we will see in the next Section, it is not satisfied in general metric theories of gravity.

IV Applications

In this section we discuss some of the applications which are important for different cases in many extended theories of gravity which do not necessarily have a Newtonian potential weak-field limit of approximation.

IV.1 Point particle Newtonian-like potentials.

Let us assume that:

ϕ​(r)=A​rk,withk≠0,\phi(r)=Ar^{k},\qquad\text{with}\qquad k\neq 0, (19)

so that:

d​ϕ​(r)d​r=A​k​rk−1,\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}=Akr^{k-1},

and so, from the results of Theorem 1 it follows that:

d​s2=(1+2​A​rkc2)​c2​d​t2+(1+2​γ​A​k​rkc2)​d​r2+r2​d​Ω2,=(1+2​A​rkc2)​c2​d​t2+d​r2(1−2​γ​A​k​rkc2)+r2​d​Ω2\begin{split}\mathrm{d}s^{2}&=\left(1+\frac{2Ar^{k}}{c^{2}}\right)c^{2}\mathrm{d}t^{2}+\left(1+\frac{2\gamma Akr^{k}}{c^{2}}\right)\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2},\\ &=\left(1+\frac{2Ar^{k}}{c^{2}}\right)c^{2}\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{\left(1-\frac{2\gamma Akr^{k}}{c^{2}}\right)}+r^{2}\mathrm{d}\Omega^{2}\end{split} (20)

at perturbation order 𝒪​(2)\mathcal{O}(2).

For the case of a Newtonian potential produced by a point mass particle located at the origin of coordinates k=−1k=-1 and A=−G​MA=-GM, where GG is Newton’s constant of gravitation and MM is the mass of the particle producing the gravitational field. With this and using the results of equation (20) and (8) it follows that:

d​s2=(1−2​G​Mr​c2)​c2​d​t2−(1+2​γ​G​Mr​c2)​d​r2−r2​d​Ω2.=(1−2​G​Mr​c2)​c2​d​t2−d​r2(1−2​γ​G​Mr​c2)−r2​d​Ω2.\begin{split}\mathrm{d}s^{2}&=\left(1-\frac{2GM}{rc^{2}}\right)c^{2}\mathrm{d}t^{2}-\left(1+\frac{2\gamma GM}{rc^{2}}\right)\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2}.\\ &=\left(1-\frac{2GM}{rc^{2}}\right)c^{2}\mathrm{d}t^{2}-\frac{\mathrm{d}r^{2}}{\left(1-\frac{2\gamma GM}{rc^{2}}\right)}-r^{2}\mathrm{d}\Omega^{2}.\end{split} (21)

This reproduces Schwarzschild’s metric at 𝒪​(2)\mathcal{O}(2) of approximation if γ=1\gamma=1, which as mentioned before is fulfilled by a wide number of experiments.

IV.2 Logarithmic potential

Logarithmic potentials appear on MOdified Newtonian Dynamics (MOND) gravitation [see e.g. 5, and references therein]. The deep MOND regime is obtained when the acceleration exerted on a test particle is given by:

a=d​ϕ​(r)d​r=−G​M​a0r,a=\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}=-\frac{\sqrt{GMa_{0}}}{r}, (22)

and so, the potential is given by:

ϕ​(r)=−G​M​a0​ln⁡(r/r⋆),\phi(r)=-\sqrt{GMa_{0}}\ln(r/r_{\star}), (23)

where r⋆r_{\star} is an arbitrary distance and a0≈1.2×10−10​m/s2a_{0}\approx 1.2\times 10^{-10}\textrm{m}/\textrm{s}^{2} is Milgrom’s acceleration constant. Direct substitution of the previous two equations into equation (12) yields at perturbation order 𝒪​(2)\mathcal{O}(2):

d​s2=(1−2​G​M​a0c2​ln⁡(r/r⋆))​c2​d​t2−(1−2​γc2​G​M​a0)​d​r2−r2​d​Ω2,=(1−2​G​M​a0c2​ln⁡(r/r⋆))​c2​d​t2−d​r2(1+2​γc2​G​M​a0)−r2​d​Ω2.\begin{split}\mathrm{d}s^{2}&=\left(1-\frac{2\sqrt{GMa_{0}}}{c^{2}}\,\ln(r/r_{\star})\right)c^{2}\mathrm{d}t^{2}-\\ &\left(1-\frac{2\gamma}{c^{2}}\sqrt{GMa_{0}}\right)\mathrm{d}r^{2}-r^{2}\mathrm{d}\Omega^{2},\\ &=\left(1-\frac{2\sqrt{GMa_{0}}}{c^{2}}\,\ln(r/r_{\star})\right)c^{2}\mathrm{d}t^{2}-\\ &\frac{\mathrm{d}r^{2}}{\left(1+\frac{2\gamma}{c^{2}}\sqrt{GMa_{0}}\right)}-r^{2}\mathrm{d}\Omega^{2}.\end{split} (24)

The previous metric was obtained by Mendoza and Olmo [4] and it was verified by Mendoza et al. [3] that the parameter γ=1\gamma=1 from observations of the bending of light of individual, groups and clusters of galaxies.

IV.3 Yukawa-like potential

For the case of analytic f​(R)f(R) theories of gravity it has been shown by Capozziello and De Laurentis [6] that:

ϕ​(r)=−G​Mr​1+δ​er/λδ+1,\phi(r)=-\frac{GM}{r}\frac{1+\delta e^{r/\lambda}}{\delta+1}, (25)

and so:

d​ϕ​(r)d​r=G​Mr​(δ+1)​{1r+δ​er/λ​[1r+1λ]}.\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}=\frac{GM}{r\left(\delta+1\right)}\left\{\frac{1}{r}+\delta e^{r/\lambda}\left[\frac{1}{r}+\frac{1}{\lambda}\right]\right\}. (26)

Substitution of these last two expressions on the results of Theorem 1 yields:

d​s2={1−2​G​Mr​c2​(δ+1)​(1+δ​er/λ)}​c2​d​t2−{1+2​γ​G​Mr​c2​(δ+1)​(1+δ​e−r/λ)+2​γ​G​Mλ​(1+δ)​δ​e−r/λ}​d​r2−r2​d​Ω2,={1−2​G​Mr​c2​(δ+1)​(1+δ​er/λ)}​c2​d​t2−d​r2{1−2​γ​G​Mr​c2​(δ+1)​(1+δ​e−r/λ)−2​γ​G​Mλ​(1+δ)​δ​e−r/λ}−r2​d​Ω2.\begin{split}\mathrm{d}s^{2}&=\left\{1-\frac{2GM}{rc^{2}\left(\delta+1\right)}\left(1+\delta e^{r/\lambda}\right)\right\}c^{2}\mathrm{d}t^{2}-\\ &\left\{1+\frac{2\gamma GM}{rc^{2}\left(\delta+1\right)}\left(1+\delta e^{-r/\lambda}\right)+\frac{2\gamma GM}{\lambda\left(1+\delta\right)}\delta e^{-r/\lambda}\right\}\mathrm{d}r^{2}-\\ &r^{2}\mathrm{d}\Omega^{2},\\ &=\left\{1-\frac{2GM}{rc^{2}\left(\delta+1\right)}\left(1+\delta e^{r/\lambda}\right)\right\}c^{2}\mathrm{d}t^{2}-\\ &\frac{\mathrm{d}r^{2}}{\left\{1-\frac{2\gamma GM}{rc^{2}\left(\delta+1\right)}\left(1+\delta e^{-r/\lambda}\right)-\frac{2\gamma GM}{\lambda\left(1+\delta\right)}\delta e^{-r/\lambda}\right\}}-\\ &r^{2}\mathrm{d}\Omega^{2}.\end{split} (27)

or equivalently:

ds2={1−2​G​Mr​c2​(δ+1)​(1+δ​er/λ)}​c2​d​t2−d​r2{1−2​γ​G​Mc2​(δ+1)​[1r+δ​er/λ​(1r+1λ)]}+r2​d​Ω2.\begin{split}\mathrm{d}&s^{2}=\left\{1-\frac{2GM}{rc^{2}\left(\delta+1\right)}\left(1+\delta e^{r/\lambda}\right)\right\}c^{2}\mathrm{d}t^{2}-\\ &\frac{\mathrm{d}r^{2}}{\left\{1-\frac{2\gamma GM}{c^{2}\left(\delta+1\right)}\left[\frac{1}{r}+\delta e^{r/\lambda}\left(\frac{1}{r}+\frac{1}{\lambda}\right)\right]\right\}}+r^{2}\mathrm{d}\Omega^{2}.\end{split} (28)

The radial component of the metric in the previous equation does not converge to the results presented by Capozziello et al. [7], Capozziello and de Laurentis [8], Capozziello and De Laurentis [6], De Martino et al. [9, 10], De Laurentis et al. [11], Cruz-Osorio et al. [12]. This fact occurs because in these works authors assumed that the time and radial component of the metric satisfy g00=−1/g11g_{00}=-1/g_{11} which according to the results of Corollary 1.1 is only valid for potentials ϕ​(r)∝1/r\phi(r)\propto 1/r at 𝒪​(2)\mathcal{O}(2) of approximation for γ=1\gamma=1.

V Discussion

The intention of this article has been to show that if the Einstein Equivalence Principle is to be valid in a spherically symmetric space-time, then the interval:

d​s2≠(1+2​ϕ​(r)c2)​c2​d​t2−d​r2(1+2​ϕ​(r)c2)−r2​d​Ω2,\mathrm{d}s^{2}\neq\left(1+\frac{2\phi(r)}{c^{2}}\right)c^{2}\mathrm{d}t^{2}-\frac{\mathrm{d}r^{2}}{\left(1+\frac{2\phi(r)}{c^{2}}\right)}-r^{2}\mathrm{d}\Omega^{2}, (29)

for any metric theory of gravity in spherical Schwarzschild-like coordinates (t,r,θ,φt,r,\theta,\varphi). This fact follows from the remarks of Theorem 1 and Corollary 1.1 that were calculated at 𝒪​(2)\mathcal{O}(2) of approximation, and hence imply the general result of equation (29) at any other order of approximation.

Acknowledgements

This work was supported by PAPIIT DGAPA-UNAM (IN110522) and CONACyT (26344).

References