Semiclassical gravity effects near horizon formation

Abstract

We study the magnitude of semiclassical gravity effects near the formation of a black-hole horizon in spherically-symmetric spacetimes. As a probe for these effects we use a quantised massless scalar field. Specifically, we calculate two quantities derived from it: the renormalised stress-energy tensor (a measure of how the field vacuum state affects the spacetime) and the effective temperature function (a generalisation of Hawking temperature related to the energy flux of the field vacuum). The subject of our study are spacetimes which contain a spherical distribution of matter and an empty exterior Schwarzschild region, separated by a surface which is moving in proximity to the Schwarzschild radius r s = 2M , with M the total mass. In particular, we analyse the consequences of three types of surface movement: an oscillation just above r s , a monotonous approach towards r s in infinite time and a crossing of r s at different velocities. For a collapsing matter distribution which follows the expected dynamical evolution in general relativity, we recover the standard picture of black-hole formation and its tenuous semiclassical effects. In more general dynamical regimes, allowing deviations from the standard classical evolution, we obtain a variety of different effects: from the emission of Hawking-like radiation without the formation of a horizon, to large values of the renormalised stress-energy tensor, related to the Boulware vacuum divergence in static spacetimes.

Asymptotic horizon formation, spacetime stretching, and causality

“…Having discussed these static spacetimes, we will now include a time dependence in the metric functions f and g in order to push the formation of the apparent horizon/ asymptotic region at r h out to the limit v → ∞. In a previous work [2], we used several such spacetimes (modeled after collapsing matter) in order to study the relation between characteristics of the geometry around the horizon and Hawking radiation. To do so, we analyzed what particular types of time dependence are necessary in order to trap outgoing light rays within a finite spatial region.…”

“…If both δðvÞ and d h ðvÞ ¼ R h ðvÞ − r h tend to zero sufficiently fast, then after some point in time some of the light rays which are inside the sphere of radius r h remain trapped inside, the outermost of which defines the event horizon. In [2] we assumed that these functions tend to zero at the same rate (e.g., e −v , 1=v, etc. ), and the condition for light-ray confinement turned out to be a relation between this rate and the order of the first nonzero coefficient (and its value if the order is 1) in the expansion of F for the interior (7).…”

“…Quantum fields develop subtle effects when evolving through geometries which generate trapping horizons or through regimes in which the formation of trapping horizons is close to occurring. For instance, in a previous paper [2] we analyzed and compared the form of the Renormalised Stress-Energy Tensor (RSET) [3,4] in several situations of this sort. Particularly, we confirmed the result discussed previously in [5], namely that when the generation of a horizon happens at a slow pace, e.g., when the collapse of a ball of matter occurs at low velocity, the RSET can acquire large values near the horizon.…”

“…Another rather interesting situation analyzed in [2] is one in which there is light-trapping behavior without the presence of actual trapped surfaces. Since Hawking's original calculation, Hawking radiation appeared to be strongly tied up to the existence of a trapping horizon.…”

“…Since Hawking's original calculation, Hawking radiation appeared to be strongly tied up to the existence of a trapping horizon. However, in a series of papers [2,[6][7][8][9] it has been shown that from a purely geometrical point of view it is possible to have Hawking-like radiation without really generating any trapped surfaces. These works consider geometries which tend toward the formation of trapping horizons but only asymptotically in time or, more generally, producing by whatever means an exponential peeling of geodesics during a sufficiently long period of time [8,9].…”