Perturbative solutions of the f(R)theory of gravity in a central gravitational field and some applications
Abstract
Exact solutions of an f(R) theory (of gravity) in a static central (gravitational) field have been studied in the literature quite well, but, to find and study exact solutions in the case of a nonstatic central field are not easy at all. There are, however, approximation methods of finding a solution in a central field which is not necessarily static. It is shown in this article that an approximate solution of an f(R)theory in a general central field, which is not necessary to be static, can be found perturbatively around a solution of the Einstein equation in the general theory of relativity. In particular, vacuum solutions are found for f(R) of general and some special forms. Further, applications to the investigation of a planetary motion and light’s propagation in a central field are presented. An effect of an f(R)gravity is also estimated for the SgrA*–S2 system. The latter gravitational system is much stronger than the Sun–Mercury system, thus the effect could be much stronger and, thus, much more measurable.
1 Introduction
Exact solutions of the f(R) theory in a static central field are studied in [17, 18, 19, 20, 21, 22] but there are also approximation methods for central fields which are not necessarily static [23, 24, 25]. In this article, approximate solutions of the f(R)theory for a general and some special cases in a general central field are found by perturbation around the Einstein equation. Then, we can use the obtained solutions to calibrate parameters of orbits of planets.
 Metric signature in Minkowski space: (\( +, , ,  \)), that is, the infinitesimal distance is calculated aswith Latin letters used for threedimensional spatial indices, and, Greek letters used for fourdimensional spacetime indices.$$\begin{aligned} ds^2&=\eta _{\mu \nu }dx^{\mu }dx^{\nu }= dx^0dx_0+dx^idx_i,\\&=c^2dt^2dx^2dy^2dz^2, \end{aligned}$$
 Riemann curvature tensor:$$\begin{aligned} R^{\alpha }_{~\mu \beta \nu }=\frac{\partial \varGamma ^\alpha _{\mu \beta }}{\partial x^\nu }  \frac{\partial \varGamma ^\alpha _{\mu \nu }}{\partial x^\beta } + \varGamma ^\alpha _{\sigma \nu }\varGamma ^\sigma _{\mu \beta }  \varGamma ^\alpha _{\sigma \beta }\varGamma ^\sigma _{\mu \nu }. \end{aligned}$$

Rank2 curvature tensor (Ricci tensor): \(R_{\mu \nu }=R^\alpha _{~\mu \alpha \nu }.\)

Scalar curvature: \( R=g^{\mu \nu }R_{\mu \nu } \).
 Energymomentum tensor of a macroscopic object:where \( u^\mu = \frac{dx^\mu }{d\tau }=c\frac{dx^\mu }{ds} \), while \( \varepsilon \) and p are the energy density and the pressure, respectively.$$\begin{aligned} T_{\mu \nu } =\frac{1}{c^2}(\varepsilon + p)u_\mu u_\nu  pg_{\mu \nu }, \end{aligned}$$
2 f(R)theory and perturbative solutions
2.1 Vacuum solutions
2.1.1 The case \( f(R)=R2\lambda \) (model I)
2.1.2 The case \( f(R)=R+\lambda R^b \), \( b>0 \) (model II)
2.1.3 The case \(f(R)= R^{1+\varepsilon }\) (model III)
2.2 General perturbative solution
3 Motion in a central field
3.1 Motion of a planet in a central field of a star
3.2 Light’s propagation in the central field of a star
4 Conclusions
Einstein’s general theory of relativity is a triumphant theory but, as mentioned above, it meets several open problems such as the accelerated expansion of the Universe (or dark energy), the cosmic inflation, an integration with quantum theory (quantum gravity), etc. The f(R)theory of gravity was introduced to solve some of these problems. Then, the Einstein equation (2) is replaced by a, more complicated in general, Eq. (6). Usually, solving the latter is problematic and it must be done via an approximation method by imposing appropriate condition(s). As, physically, the f(R)gravity is assumed to be a perturbative theory around Einstein’s GR describing very well most of today observations, we have followed a perturbation approach to solving Eq. (6). However, even with this assumption, it is not always easy to solve Eq. (6) without imposing any further condition. One of the most often imposed conditions is the spherical symmetry being a good approximation in many cases. Therefore, in this article we try to perturbatively solve Eq. (6) in a central field. The corresponding general solution is given in (71)–(74), while the vacuum solution is given in (32)–(35). At a large distance from the gravitational source the solution (32)–(35) can be written in the form (40)–(43) with some particular cases also considered (see 2.1.1–2.1.3). These results, as discussed in Sect. 3, can be applied to investigating planetary and light’s motions in a central field. In comparison with Einstein’s theory, an orbital precession or a trajectory deflection now gets a correction which is a constant for a static central field and varies with time for a nonstatic central field even from a source of a constant mass, unlike the corresponding Einstein’s value which does not change in the same circumstance. In general, a spherically symmetric vacuum solution of Eq. (6) is not stationary, while a spherically symmetric vacuum solution of the Einstein equation is always stationary. In other words, Birkhoff’s theorem in the GR is not valid any more for a general f(R)theory of gravity. This may have interesting consequences (for example, a spherically symmetric pulsating (or expanding or collapsing) object is not disabled to emit gravitational waves as in the GR) being a subject of our next investigation. Following the present method, we will also investigate cosmological equations and models corresponding to the f(R)theory of gravity.
The results obtained above may give an indication for an experimental test of an f(R)theory of gravity. This theory in the considered circumstance can be treated as an Einstein’s GR with an effective mass (\(M_f\)) which may vary with time even in the case keeping the original total mass (M) constant. Let us make some estimation using a real data.
In general, the deviation between the two theories, the GR and the f(R)gravity, is very small but it is measurable if one can invent a measurement technique sensitive as that of the LIGO which is sensitive to a relative length change of an order of around \(10^{20}\).
Footnotes
Notes
Acknowledgements
This research is funded by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under contract No. 103.012017.76.
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