# Effects of geometric optics in conformal Weyl gravity

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## Abstract

We have investigated effect of geometric optics as the rotation of polarization vector of light in spacetime of gravitational compact object in the fourth-order theory of conformal Weyl gravity. The Pineault–Roeder method is applied to the rotating Weyl metric, and analytical results are obtained in the limit of weak field and or slow rotation. For the photon traveling parallel to the symmetry axes from the equatorial plane to infinity, the rotation of the polarization plane depends on the Weyl parameter \(\gamma \) on the contrary to the Kerr spacetime where there is no rotation of polarization plane for this case.

## Mathematics Subject Classification

83C10 83C25 83C57## 1 Introduction

*k*are integration constants and defined as follows: \(\beta =\frac{GM}{c^2}\ (\hbox {cm})\) is geometrized mass, where

*M*is the mass of the (spherically symmetric) source and

*G*is the universal gravitational constant; and others \(\gamma \ (\hbox {cm}^{-1})\) and \(k\ (\hbox {cm}^{-2})\) are required by conformal gravity. The theory comes to the standard Schwarzschild solution (\(\gamma =0=k\)), and to the Schwarzschild–de Sitter (\(\gamma =0\)), as well as the term \(-kr^{2}\), means a background De Sitter spacetime, which is important only over cosmological distances, while

*k*has a very small value, the term \(\gamma r\) becomes significant over galactic distance scales. In the review letter, [5] of Philip Mannheim has been discussed how gravitational theory can enable to be constructed candidate alternatives to the standard theory in which the dark matter and dark energy problems could then be resolved.

Moreover, in this paper, we study the effect of geometric optics as the rotation of polarization vector of light in the spacetime of rotating compact object in conformal Weyl gravity model.

This paper is organized as follows. In Sect. 2, it has been investigated the rotation of polarization vector using general Pienault–Roeder approach [12]. In Sect. 3, there have been considered two physical scenarios: in the first case, the source is in the equatorial plane, and the observer above the plane with \(\theta < \pi /2\), which photons propagating parallel to and a distance *b* from the symmetry axis (\(L=0\)). In the second case, the source and the observer are in the symmetry axis (\(\theta =0\)), \(r_{o} > r_{s}\). Section 5 summarizes the main results obtained.

Throughout, we use a space-like signature \((-,+,+,+)\). Greek indices run from 0 to 3 and Latin ones from 1 to 3.

## 2 Rotation of polarization vector of electromagnetic waves propagating in spacetime of compact object in conformal Weyl gravity

*s*are constants satisfying the constraint \(uv-\bar{r}s=0\). The angular momentum of the source is represented by

*j*. The fourth-order Kerr solution includes the Kerr solution (\(u=-2MG/c^2,\ p=1,\ v=\bar{r}=s=k=0\)) and the Kerr–de Sitter solution (\(u=2MG/c^2,\ p=1-kj^2,\ v=\bar{r}=s=0,\ k=\Lambda /3;\ \Lambda \) being the cosmological constant) as special cases.

*J*of the gravitating object per unit mass. In the paper, [12] have been considered the applications of geometrical optics to the Kerr metric in the weak-field approximation (\(j \ll M\)) using Newman–Penrose formalism. Here, we applied this formalism to Weyl gravity.

*D*denotes covariant derivative in the direction of the wave vector \(k^{\mu }\).

*B*is complex, and \(\chi \) is real. When vector \(k^{\mu }\) is tangent to the null congruence, the spin coefficient \(\kappa \equiv -Dk_{\mu }m^{\mu }\) is zero [11]. Furthermore, assumption that the null tetrad has to propagate parallelly along the null congruence implies disappearance of other two spin coefficients: \(\varrho =\pi =0\). Consequently, the plane spanned by vectors \(k^{\mu }\) and \(a^{\mu }\) can be identified with the polarization plane which is parallelly propagated in the direction of vector \(k^{\mu }\), and, in turn, the polarization vector can be identified with the \(a^{\mu }\) vector of the Newman–Penrose formalism.

*a*as follows:

*b*from the symmetry axis (\(L=0\)). In the second case, the source and the observer are in the symmetry axis (\(\theta =0\)), \(r_{o} > r_{s}\). The source and the observer are modeled by timelike curves \(r=r_{s}\) and \(r=r_{o}\), respectively.

## 3 In particular case

### 3.1 Source in the black hole equatorial plane

*b*from symmetry axis:

### 3.2 Source and observer in the symmetry axis

## 4 Astrophysical implications

## 5 Conclusion

In this paper, we have studied the rotation of polarization vector in the null congruence direction in the gravitational field of rotating massive object in conformal Weyl gravity using the Pineault–Roeder approach [12]. For the photon traveling parallel to the symmetry axes from the equatorial plane to infinity, the rotation of the polarization plane (33) depends on the Weyl parameter \(\gamma \) on the contrary to the Kerr spacetime [12] where there is no rotation of polarization plane for this case. Here, if we admit that Weyl parameter \(\gamma >0\) is positive, then, in this case, the polarization vector is counter-rotating with respect to the spin of the black hole as the angle in Eq. (33) would be negative.

To get the estimation for the value of Weyl parameter \(\gamma \), one should compare the observational results with the theoretical results. In Ref. [6] has been gotten expression for integration constant in the theory as \(\gamma \le 10^{-28} \mathrm{cm}^{-1}\) which was in good agreement with the observed galactic rotation curves without the need for dark matter. Regarding solar system experiments, in Ref. [13], there has been used CASSINI Doppler Datas and got an upper bound for Weyl parameter as \(\gamma \sim 10^{-21}\mathrm{cm}^{-1}\). Authors of the Ref. [15] from the correction to geodetic effect obtained \(\gamma \le 1.5\times 10^{-20} \mathrm{cm}^{-1}\). In the recent paper [3], there has been estimated lower limit for parameter \(\gamma \) as \(\gamma \le 2\times 10^{-20}\mathrm {cm}^{-1} \ \) using the experimental results of Ref. [9] which was set on the Earth as central body on the precise measurement of the gravitational redshift by the interference of matter waves. Here, we estimated Weyl parameter as \(\gamma \le 2\times 10^{-21} \mathrm{cm}^{-1}\) using some observational data of blazar in our quasiclassical approximation.

## Notes

### Acknowledgements

This research is supported in part by Projects no. VA-FA-F-2-008 and no. EFA-Ftex-2018-8 of the Uzbekistan Ministry of Innovation Development, and by the Abdus Salam International Centre for Theoretical Physics through Grant no. OEA-NT-01. This research is partially supported by Erasmus+ exchange grant between SU and NUUz. A.A. acknowledges the TWAS associateship programm for the support.

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