Conformal invariance and apparent universality of semiclassical gravity

Abstract

In a recent work, it has been pointed out that certain observables of the massless scalar field theory in a static spherically symmetric background exhibit a universal behavior at large distances. More precisely, it was shown that, unlike what happens in the case where the coupling to the curvature $\ensuremath{\xi}$ is generic, for the special cases $\ensuremath{\xi}=0$ and $\ensuremath{\xi}=1/6$ the large distance behavior of the expectation value $⟨T^{\ensuremath{\mu}}{}_{\ensuremath{\nu}}⟩$ turns out to be independent of the internal structure of the gravitational source. Here, we address a higher dimensional generalization of this result: We first compute the difference between a black hole and a static spherically symmetric star for the observables $⟨{\ensuremath{\phi}}^{2}⟩$ and $⟨T^{\ensuremath{\mu}}{}_{\ensuremath{\nu}}⟩$ in the far field limit. Thus, we show that the conformally invariant massless scalar field theory in a static spherically symmetric background exhibits such a universality phenomenon in $D\ensuremath{\ge}4$ dimensions. Also, using the one-loop effective action, we compute $⟨T^{\ensuremath{\mu}}{}_{\ensuremath{\nu}}⟩$ for a weakly gravitating object. These results lead to the explicit expression of the expectation value $⟨T^{\ensuremath{\mu}}{}_{\ensuremath{\nu}}⟩$ for a Schwarzschild-Tangherlini black hole in the far field limit. As an application, we obtain quantum corrections to the gravitational potential in $D$ dimensions, which for $D=4$ are shown to agree with the one-loop correction to the graviton propagator previously found in the literature.