Quantum metric fluctuations and Hawking radiation

Abstract

In this Letter we study the gravitational interactions between outgoing configurations giving rise to Hawking radiation and in-falling configurations. When the latter are in their ground state, the near horizon interactions lead to collective effects which express themselves as metric fluctuations and which induce dissipation, as in Brownian motion. This dissipation prevents the appearance of trans-Planckian frequencies and leads to a description of Hawking radiation which is very similar to that obtained from sound propagation in condensed matter models. PACS number(s): 04.70. Dy, In his original derivation [1], Hawking assumed that gravity can be treated classically, i.e. the metric was determined by the energy of the collapsing star but was unaffected by the quantum processes under examination. In this approximation, the radiation field satisfies a linear equation (in the absence of matter interactions) and the in-falling and outgoing field configurations are completely uncorrelated near the black hole horizon. In fact the pairs of quanta generated by its formation are composed of two outgoing quanta, one of each side of it. The external ones form the asymptotic flux whereas their partners fall towards the singularity at r = 0. Upon tracing over these inner configurations one gets an outgoing incoherent flux described by a thermal density matrix.There is nevertheless a precise relationship between the expectation values of the in-falling and the outgoing energy fluxes. Indeed, the asymptotic thermal flux is accompanied by a negative in-falling flux T vv which has, on the horizon, exactly the opposite value and which drives black hole evaporation according toWe work with in Planck units: c =h = M P lanck = 1. v = t+r * is the advanced null time, r * = r+2M ln(r/2M −1) is the tortoise coordinate and M (v) is the time dependent mass which appears in the Vaidya metricThe important point is that (2) offers a good approximation of the near horizon geometry (i.e. |r − 2M | ≪ 2M ) of an evaporating black hole, see [2,3]. This semi-classical description would be perfectly valid if another feature of black hole physics wasn't present, namely the field configurations giving rise to Hawking quanta possess arbitrary high (trans-Planckian) frequencies near the horizon: When measured by in-falling observers at r, the frequency grows aswhere λ is the asymptotic energy of the quantum. This implies that a wave packet centered along the null outgoing geodesic u = t − r * had a frequency ω = λe κu when it emerged from the in-falling star. (κ = 1/4M is the surface gravity and fixes Hawking temperature T H = κ/2π.) Unlike processes characterized by a typical energy scale, the relation ω = λe κu shows that black hole evaporation rests on arbitrary high frequencies.As emphasized by 't Hooft [4], this implies that the gravitational interactions between these configurations and in-falling quanta cannot be neglected, thereby questioning the validity of the semi-classical description. In questioning this validity, two issues should be d...

Black hole entanglement entropy regularized in a freely falling frame

“…The radiative corrections associated with these interactions in the free fall vacuum outside the black hole were studied in Ref. [17,18]. Beyond the local effects that would also be present in the vacuum of flat spacetime, it was found that there are further contributions, dominated by in-modes with Killing frequency ω in ∼ κ.…”

“…This would provide a different resolution to the species problem mentioned above (unless the number of species were of order M p /κ v ). On the other hand, as argued in [17,18], it is also possible that the cutoff arises from quantum gravitational effects in the black hole background, and that its value equals the Planck mass when measured in the static frame. 3 Such a cutoff would appear as Λ ∼ M 2 p /κ v in the freefall frame, and the regulated entropy would be of order A/L 2 p after all.…”

Thermal Radiation From a Fluctuating Event Horizon

“…The gravitational back reaction also raises questions on the applicability of the semiclassical theory of gravity. Within this perspective, models were formulated [22,23] with the aim of studying the effects of fluctuations of the black-hole horizon on the Hawking radiation spectrum. In the absence of knowledge of the precise nature of the metric fluctuations near the horizon, the assumptions made here are that they can be treated classically and their effects on the propagation of quantum fields can be described via random differential equations.…”

Surrounded Vaidya solution by cosmological fields

“…Regarding the positive energy condition for the quintessence background, represented by the relation (16), it is required to have N q (u) 0. The radiation density is given by…”

“…Afterwards, some models to describe the classical essence of this radiation in a language which is free from the usual quantum field theoretic tools and is more familiar to the astrophysicists and relativists, have been introduced. For instance, the Vaidya solution [8][9][10] has provided a simple classical model for the black hole radiation and has been vastly investigated in this regard [11][12][13][14][15][16][17], see also [18][19][20][21][22][23][24][25][26][27][28][29] for more studies. In fact, the Vaidya solution is one of the non-static solutions of the Einstein field equations and can be regarded as a generalization of the static Schwarzschild black hole solution.…”