Multihorizons black hole solutions, photon sphere, and perihelion shift in weak ghost-free Gauss-Bonnet theory of gravity

Among the modified gravitational theories, the ghost-free Gauss-Bonnet (GFGB) theory of gravity has been considered from the viewpoint of cosmology. The best way to check its applicability could be to elicit observable predicts which give guidelines or limitations on the theory, which could be contrasted with the actual observations. In the present study, we derive consistent field equations for GFGB and by applying the equations to a spherically symmetric space-time, we obtain new spherically symmetric black hole (BH) solutions. We study the physical properties of these BH solutions and show that the obtained space-time possesses multihorizons and the Gauss-Bonnet invariants in the space-time are not trivial. We also investigate the thermodynamical quantities related to these BH solutions and we show that these quantities are consistent with what is known in the previous works. Finally, we study the geodesic equations of these solutions which give the photon spheres and we find the perihelion shift for weak GFGB. In addition, we calculate the first-order GFGB perturbations in the Schwarzschild solution and new BH solutions and show that we improve and extend existing results in the past literature on the spherically symmetric solutions.

Figure 1

Schematic plots of the radial coordinate $r$ (a) vs the function $f$ given by Eq. (29); (b) vs the GB invariant given by Eq. (32); (c) the function $h$ vs $\chi$; (d) the potential $V$ vs $\chi$; and (e) the Lagrange multiplier $\lambda (r)$ given by Eq. (29) vs $r$.

Figure 2

Schematic plot of (a) the horizons, $r_1$ and $r_2$, of the BH solution (28); (b) entropy of the BH solution (28); (c) Hawking temperature of the BH solution (28); (d) heat capacity of the BH solution (28); finally, (e) Gibbs’s free energy of the BH solution (28).

Figure 3

Schematic plots of the radial coordinate $r$ (a) vs the function $f$; (b) vs the metric potentials; (c) vs the GB invariant given by Eq. (48); (d) the function $h$ vs $\chi$; (e) the potential $V$ vs $\chi$; and (f) the Lagrange multiplier $\lambda (r)$ vs $r$.

Figure 4

Schematic plot of (a) horizons $r_1$ and $r_2$ of the BH solution (40); (b) Hawking temperature of the BH solution (40); (c) heat capacity of the BH solution (28); and (d) the Gibbs’s free energy of the BH solution (40).