Study of motion around a static black hole in Einstein and Lovelock gravity

We study motion around a static Einstein and pure Lovelock black hole in higher dimensions. It is known that in higher dimensions bound orbits exist only for a pure Lovelock black hole in all even dimensions, $D=2N+2$, where $N$ is the degree of Lovelock polynomial action. In particular, we compute periastron shift and light bending, and the latter is given by one of the transverse spatial components of the Riemann curvature tensor. We also consider the pseudo-Newtonian potentials and Kruskal coordinates.

Figure 1

Variation of $r_{ph}$ as a function of $n$ for Einstein gravity.

Figure 2

Variations of $r_{ph}$ (dashed line), $r_b$ (solid line), and $r_s$ (dotted line) as functions of $N$ for pure Lovelock gravity.

Figure 3

Variation of $E_s$ as a function of $N$ for pure Lovelock gravity.

Figure 4

Cartesian plot for Schwarzschild case: $u^{\prime \prime }+u=1+0.03u^2$, where $u=1/r$, using $M=0.01$ and $l=0.1$ and boundary conditions $u(0)=0.1$ and $u^{\prime }(0)=0$.

Figure 5

Cartesian plot for Einstein $6D$ case: $u^{\prime \prime }+u=3u^2+0.05u^4$, where $u=1/r$, using $M=0.01$ and $l=0.1$ and boundary conditions $u(0)=0.1$ and $u^{\prime }(0)=0$.

Figure 6

Cartesian plot for Pure Lovelock $6D$ case: $u^{\prime \prime }+u=3.535u^{-1/2}+0.17675u^{3/2}$, where $u=1/r$, using $M=0.01$ and $l=0.1$ and boundary conditions $u(0)=0.1$ and $u^{\prime }(0)=0$.

Figure 7

Variation of energy of marginally stable circular orbit for Einstein (dotted line) and pseudo-Newtonian gravity (solid line) with $n$.

Figure 8

Variation of energy of marginally stable circular orbit for pure Lovelock (dotted line) and pseudo-Newtonian (solid line) gravity with $N$.