Remarks about Static Back-Reaction on Black Hole Spacetimes

Recently, it has been claimed that the back reaction of vacuum polarization on a black hole spacetime naturally regularizes infinities in the black hole entropy. We examine the back reaction calculation and find no such short-distance cut-off, in contradiction with these recent claims. Moreover, the intuitive expectation that the perturbative calculation breaks down near the event horizon is confirmed. The new surface gravity diverges and the metric is degenerate at the stretched horizon. 04.70.Dy, 04.62.+v, 11.55.Bq Typeset using REVTEX 1 It has recently been claimed [1] that the back-reaction on a black hole spacetime due to vacuum polarization provides a cut-off needed to regularize the entropy. The 1-loop quantum correction to the entropy from a scalar field near the event horizon diverges, as shown by ’t Hooft [2]. The divergence results since an infinite number of states can contribute at the horzion. One might conjecture that if a particle could never actually reach the event horizon, then only a finite number of states would contribute to the entropy and the infinites would be regularized. In Ref. [1], Lousto claims to have found such a brick wall by computing the back reaction due to vacuum polarization on the spacetime. In the corrected metric, he argues, the acceleration required to keep a particle at rest vanishes at some distance outside the stretched horizon and at that point a particle can remain in stable equilibrium. Therefore a natural barrier, and hence a short-distance cut-off, arises. We critically examine this claim and arrive at contrary conclusions. We find that the back-reaction-corrected metric has a new event horizon, stretched by an amount proportional to the Planck length outside the unperturbed horizon. Unlike in Ref. [1], there is no barrier to reaching the horizon. The acceleration required to keep a particle stationary never vanishes. In fact, the acceleration diverges at the stretched horizon. The new surface gravity diverges as well. Not surprisingly, we also find that our perturbative calculation fails before the horizon can be reached. In short, this perturbative back-reaction calculation is unable to shed light on the physics near the event horizon. Following the method employed by Lousto in Ref. [1], we consider the back reaction due to a conformally invariant scalar field in the Boulware vacuum [4]. The one-loop renormalised energy momentum tensor in the spacetime of a Schwarzschild black hole is [5,6] (G=c=1, h6= 1) < B|T ν μ |B >= αM r6 