Optical behaviors of black holes in Starobinsky–Bel–Robinson gravity

Abstract

Inspired by M-theory scenarios, we investigate optical properties of black holes in the Starobinsky–Bel–Robinsion gravity. Precisely, we study the shadows and the deflection angle of light rays by this class of black holes in such a novel gravity. First, we approach the shadows of the Schwarzschild-type solutions. As expected, we find perfect circular shadows where the size decreases with a stringy gravity parameter denoted by $\beta$. We reveal that this parameter is constrained by the shadow existence. Combining the Newman–Janis algorithm and the Hamilton–Jacobi mechanism, we examine the shadow behaviors of the rotating solutions in terms of one-dimensional real curves. Precisely, we obtain various sizes and shapes depending on the rotating parameter and the stringy gravity parameter a and $\beta$, respectively. To examine the shadow geometric deformations, we study the astronomical observables and the energy emission rate. As envisaged, we show that a and $\beta$ have an impact on such shadow behaviors. For specific values of a, we remark that the obtained shadow shapes share certain similarities with the ones of the Kerr black holes in the plasma backgrounds. Using the Event Horizon Telescope observational data, we provide predictions for the stringy gravity parameter $\beta$ which could play a relevant role in the M-theory compactifications. After that, we discuss the behaviors of the light rays near to such four dimensional black holes by calculating the deflection angle in terms of a required moduli space.

Appendices

Appendices

A Construction of the rotating SBR black hole from JNA

To get the rotating version of the SBR black hole, we follow the Newman Janis algorithm via certain steps. The first one is based on a change of the involved variables providing a spherically symmetric null surface. Precisely, we consider the following variable change

$$\begin{aligned} du=dt-\frac{dr}{f(r)} \end{aligned}$$

(A.1)

where f(r) is the metric function given by the equation (2.5). In the Eddington-Finkelstein type coordinates, the black hole metric takes the following form

$$\begin{aligned} ds^{2}=-f(r)du^{2}-2du\ dr+r^{2}\left( d\theta ^{2}+\sin ^{2}\theta \ d\phi ^{2}\right) . \end{aligned}$$

(A.2)

The next step is to find a null tetrad $(\ell ^{\mu },n^{\mu },m^{\mu },{\overline{m}}^{\mu })$ satisfying the constraints

$$\begin{aligned} \ell _{\mu } \ell ^{\mu } = m_{\mu } m^{\mu } = n_{\mu } n^{\mu } = \ell _{\mu } m^{\mu } =n_{\mu } m^{\mu } =0 \, , \quad \ell _{\mu } n^{\mu } = - m_{\mu } {{{\bar{m}}}}^{\mu } =1 \end{aligned}$$

(A.3)

where the contra-variant form of the metric can be rewritten as

$$\begin{aligned} g^{\mu \nu }=-\ell ^{\mu }n^{v}-\ell ^{\nu }n^{\mu }+m^{\mu }{\overline{m}}^{\nu }+m^{\nu }{\overline{m}}^{\mu }. \end{aligned}$$

(A.4)

Using the metric expression, the null tetrad vectors are given by

$$\begin{aligned} \ell ^{\mu }= & {} \delta _{1}^{\mu }, \qquad n^{\mu }=\delta _{0}^{\mu }-\frac{1}{2} \left( 1-\frac{r_{s}}{r}+\frac{108 \beta \left( 4\sqrt{2}\pi G r_{s}\right) ^3 }{5 r^8 } \left( \frac{1}{r}-\frac{97 r_{s}}{108 r^2}\right) \right) \delta _{1}^{\mu }\nonumber \\ \end{aligned}$$

(A.5)

$$\begin{aligned} m^{\mu }= & {} \frac{1}{\sqrt{2}r}\left( \delta _{2}^{\mu }+\frac{i}{\sin ^{{}}\theta }\delta _{3}^{\mu }\right) \qquad {\overline{m}}^{\mu }=\frac{1}{ \sqrt{2}r}\left( \delta _{2}^{\mu }-\frac{i}{\sin ^{{}}\theta }\delta _{3}^{\mu }\right) \end{aligned}$$

(A.6)

where ${\overline{m}}^{\mu }$ is the complex conjugate of $m^{\mu }$. In this scenario, the radial coordinate r could take complex values. The tetrad null vectors become

$$\begin{aligned} \ell ^{\mu }= & {} \delta _{1}^{\mu }, \qquad \nonumber \\ n^{\mu }= & {} \delta _{0}^{\mu }-\frac{1}{2} \left[ 1-\frac{r_{s}}{2}\left( \frac{1}{r}+\frac{1}{r*}\right) +\frac{54 \beta \left( 4\sqrt{2}\pi G r_{s}\right) ^3}{5 (r r*)^4}\left( \frac{1}{r}+\frac{1}{r*}-\frac{97 r_{s}}{54 r r*}\right) \right] \delta _{1}^{\mu } \nonumber \\ m^{\mu }= & {} \frac{1}{\sqrt{2}r^{*}}\left( \delta _{2}^{\mu }+\frac{i}{\sin ^{{}}\theta }\delta _{3}^{\mu }\right) , \qquad {\overline{m}}^{\mu }=\frac{1}{ \sqrt{2}r}\left( \delta _{2}^{\mu }-\frac{i}{\sin ^{{}}\theta }\delta _{3}^{\mu }\right) \end{aligned}$$

(A.7)

where $r^{*}$ denotes the complex conjugate of r. Roughly, the following complex coordinate transformations should be exploited

$$\begin{aligned} {\widetilde{u}}= & {} u-ia\cos \theta , \qquad {\widetilde{r}}=r+ia\cos \theta \end{aligned}$$

(A.8)

$$\begin{aligned} {\widetilde{\theta }}= & {} \theta , \qquad {\widetilde{\varphi }}=\varphi \end{aligned}$$

(A.9)

leading to

$$\begin{aligned} {\widetilde{\ell }}^{\mu }= & {} \delta _{1}^{\mu } \end{aligned}$$

(A.10)

$$\begin{aligned} {\widetilde{n}}^{\mu }= & {} \delta _{0}^{\mu }-\frac{1}{2}\left[ 1-\frac{r_{s} {\widetilde{r}} }{{\widetilde{r}} ^{2}+a^{2}\cos ^{2}{\widetilde{\theta }}}+\frac{54\left( 4\sqrt{2}\pi G r_{s}\right) ^3}{5\left( {\widetilde{r}} ^{2}+a^{2}\cos ^{2}{\widetilde{\theta }}\right) ^5} \left( 2{\widetilde{r}} -\frac{97 r_{s}}{54}\right) \right] \delta _{1}^{\mu }\nonumber \\ \end{aligned}$$

(A.11)

$$\begin{aligned} {\widetilde{m}}^{\mu }= & {} \frac{1}{\sqrt{2}({\widetilde{r}}+ia\cos \widetilde{ \theta })}\left( ia\sin {\widetilde{\theta }}\left( \delta _{0}^{\mu }-\delta _{1}^{\mu }\right) +\delta _{2}^{\mu }+\frac{i}{\sin ^{{}}{\widetilde{\theta }} }\delta _{3}^{\mu }\right) \end{aligned}$$

(A.12)

$$\begin{aligned} \widetilde{{\overline{m}}}^{\mu }= & {} \frac{1}{\sqrt{2}({\widetilde{r}}-ia\cos {\widetilde{\theta }})}\left( -ia\sin {\widetilde{\theta }}\left( \delta _{0}^{\mu }-\delta _{1}^{\mu }\right) +\delta _{2}^{\mu }-\frac{i}{\sin ^{{}} {\widetilde{\theta }}}\delta _{3}^{\mu }\right) . \end{aligned}$$

(A.13)

To obtain the rotating metric solution, certain additional transformations are needed. Indeed, they are given by

$$\begin{aligned} d{\widetilde{u}}= & {} dt-\left( \frac{r^{2}+a^{2}}{\Delta }\right) dr \nonumber \\ d{\widetilde{\varphi }}= & {} d\varphi -\frac{a}{\Delta }\ dr \end{aligned}$$

(A.14)

where the $\Delta$ function is expressed as

$$\begin{aligned} \Delta = \Delta (r) = a^2+r^2 \left( 1-\frac{2 G M}{r}+\frac{1024 \pi ^3 \beta G^6 M^3\left( 108r -194 G M\right) }{5 r^{10}}\right) . \end{aligned}$$

(A.15)

Using such transformations, the covariant components of the metric (A.2) are

$$\begin{aligned} g_{\mu \nu }=\left[ \begin{array}{cccc} \frac{a^{2}\sin ^{2}\theta -\Delta }{\Sigma } &{} 0 &{} 0 &{} \frac{a\sin ^{2}\theta \left[ \Delta -\left( r^{2}+a^{2}\right) \right] }{\Sigma } \\ 0 &{} \frac{\Sigma }{\Delta } &{} 0 &{} 0 \\ 0 &{} 0 &{} \Sigma &{} 0 \\ \frac{a\sin ^{2}\theta \left[ \Delta -\left( r^{2}+a^{2}\right) \right] }{ \Sigma } &{} 0 &{} 0 &{} \frac{\sin ^{2}\theta \left[ \left( r^{2}+a^{2}\right) ^{2}-\Delta a^{2}\sin ^{2}\theta \right] }{\Sigma } \end{array} \right] \end{aligned}$$

(A.16)

where one has used

$$\begin{aligned} \Sigma= & {} r^2+a^2\cos ^2\theta . \end{aligned}$$

(A.17)

The metric of the rotating SBR black hole could be written as follows

$$\begin{aligned} ds^2= & {} -\left( \frac{\Delta (r)-a^2\sin ^2\theta }{\Sigma }\right) dt^2 +\frac{\Sigma }{\Delta (r)}dr^2\nonumber \\{} & {} -2a\sin ^2\theta \left( 1-\frac{\Delta (r)-a^2\sin ^2\theta }{\Sigma }\right) dtd\phi +\Sigma d\theta ^2\nonumber \\{} & {} +\sin ^2\theta \left[ \Sigma +a^2\sin ^2\theta \left( 2-\frac{\Delta (r)-a^2\sin ^2\theta }{\Sigma }\right) \right] d\phi ^2. \end{aligned}$$

(A.18)