Nonthermal nature of incipient extremal black holes

Abstract

We examine particle production from spherical bodies collapsing into extremal Reissner-Nordström black holes. Kruskal coordinates become ill-defined in the extremal case, but we are able to find a simple generalization of them that is good in this limit. The extension allows us to calculate the late-time worldline of the center of the collapsing star, thus establishing a correspondence with a uniformly accelerated mirror in Minkowski spacetime. The spectrum of created particles associated with such uniform acceleration is nonthermal, indicating that a temperature is not defined. Moreover, the spectrum contains a constant that depends on the history of the collapsing object. At first sight this points to a violation of the no-hair theorems; however, the expectation value of the stress-energy-momentum tensor is zero and its variance vanishes as a power law at late times. Hence, both the no-hair theorems and the cosmic censorship conjecture are preserved. The power-law decay of the variance is in distinction to the exponential fall-off of a nonextremal black hole. Therefore, although the vanishing of the stress tensor's expectation value is consistent with a thermal state at zero temperature, the incipient black hole does not behave as a thermal object at any time and cannot be regarded as the thermodynamic limit of a nonextremal black hole, regardless of the fact that the final product of collapse is quiescent.

One-loop corrected thermodynamics of the extremal and nonextremal spinning Banados-Teitelboim-Zanelli black hole

“…The Euclidean topology of an extremal black hole is relatively trivial, and this effectively eliminates the usual horizon contribution (which accounts for the non-extremal entropy [9]) to the Euclidean action. Note that the findings of various other works have since supported the Hawking conjecture [10][11][12][13][14][15].…”

“…To help clarify this apparent conflict, we take note of recent findings by Liberati et al [14]. They have considered an extremal Reissner-Nordstrom black hole undergoing collapse and demonstrated that (in spite of asymptotic particle production) the temperature remains undefinable on account of a non-Planckian distribution.…”

“…5 The Euclidean action can be written in a more convenient form by way of Eq. (14). Let us first define G µν as the left-hand side of this field equation and then express both G tt and T tt (which can be obtained from Eq.…”

Late-time particle creation from gravitational collapse to an extremal Reissner-Nordström black hole

“…However, Liberati, et al [4] pointed out that the generalization to extremal black holes from nonextremal black holes is not trivial. One important difference is that the Kruskal transformation for non-extremal black holes, which plays a crucial role in computing the particle creation, breaks down for extremal black holes.…”

“…The arguments in [4] are reviewed briefly here as follows. Start with the usual form of the Reissner-Nordström (RN) geometry with parameters Q and M…”