Observable strong field effects of extra spacetime dimension in the braneworld black hole

Abstract

Inspired by the string theory, the braneworld picture introduces extra dimensions beyond the four that may have observable non-trivial effects in short distance (strong field) gravity experiments. A case in point is the Randall-Sundrum braneworld picture that projects the $5d$ bulk Weyl tensor onto the $3d$ brane providing a stress tensor in the effective Einstein field equations on the brane. Dadhich, Maartens, Papadopoulos and Rezania (DMPR) derived an exact braneworld black hole solution of the brane vacuum field equations. The solution formally resembles that of Reissner-Nordstr\"{o}m but is physically different from it since the "tidal charge" $\Upsilon$ in the solution is not the electric charge but an imprint from the fifth dimension allowing both signs in the power law modification $\pm \frac{\Upsilon ^{2}}{r^{2}}$ to the Schwarzschild metric $\left( \Upsilon = 0\right)$. The corresponding black holes are designated as DMPR$\pm$. We study here the effect of $\Upsilon$ on strong field lensing observables and compare in the eikonal limit the ring down quasinormal mode (QNM) frequencies of DMPR$-$ with those of DMPR$+$, the two variants of tidal charge modified Schwarzschild black hole ($\Upsilon = 0$). It turns out that the tidal charge can significantly modify the Schwarzschild lensing observables and QNM frequencies. In particular, we find that the Pretorius-Khurana critical exponent $\gamma$ of circular null orbits in the DMPR$-$ black hole has a lower value than that for the Schwarzschild black hole, which indicates a stronger Lyapunov instability suggesting that the accretion disks of DMPR$-$ black holes would appear brighter. The case of the SgrA* black hole is considered for a possible constraint on $\Upsilon$ from the EHT observation of its shadow size.