Vacuum polarization on the brane

TL;DR

This paper calculates the quantum effects of a scalar field near a higher-dimensional black hole on a brane, revealing how these effects vary with the number of extra dimensions.

Contribution

It provides an exact expression for the scalar field's expectation value on the horizon and numerical results elsewhere, considering the impact of extra dimensions.

Findings

Expectation value increases rapidly with more bulk dimensions

Exact formula derived for the horizon case

Numerical results show dimension-dependent behavior

Abstract

We compute the renormalized expectation value of the square of a massless, conformally coupled, quantum scalar field on the brane of a higher-dimensional black hole. Working in the AADD brane-world scenario, the extra dimensions are flat and we assume that the compactification radius is large compared with the size of the black hole. The four-dimensional on-brane metric corresponds to a slice through a higher-dimensional Schwarzschild-Tangherlini black hole geometry and depends on the number of bulk space-time dimensions. The quantum scalar field is in a thermal state at the Hawking temperature. An exact, closed-form expression is derived for the renormalized expectation value of the square of the quantum scalar field on the event horizon of the black hole. Outside the event horizon, this renormalized expectation value is computed numerically. The answer depends on the number of bulk space-time dimensions, with a magnitude which increases rapidly as the number of bulk space-time dimensions increases.