White Hole Cosmology and Hawking Radiation from Quantum Cosmological Perturbations Hassan Firouzjahi Alireza Talebian

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White Hole Cosmology and Hawking Radiation from

Quantum Cosmological Perturbations

Hassan Firouzjahi∗, Alireza Talebian†

School of Astronomy, Institute for Research in Fundamental Sciences (IPM)

P. O. Box 19395-5531, Tehran, Iran

Abstract

The spacetime inside the white hole is like an anisotropic cosmological background

with the past singularity playing the role of a big bang singularity. The scale factor

along the extended spatial direction is contracting while the scale factor along the two-

sphere is expanding. We consider an eternal Schwarzschild manifold and study quantum

cosmological perturbations generated near the white hole singularity which propagate

towards the past event horizon and to exterior of the black hole. It is shown that an

observer deep inside the white hole and an observer far outside the black hole both

share the same vacuum. We calculate the Hawking radiation associated to these quan-

tum white hole perturbations as measured by an observer in the exterior of the black

hole. Furthermore, we also consider the Hawking radiation for the general case where

the initial cosmological perturbations deep inside the white hole are in “non-vacuum”

state yielding to a deviation from Planck distribution. This analysis suggests that if the

black hole is not entirely black (due to Hawking radiation) then the white hole is not

entirely white either.

∗ﬁrouz@ipm.ir

†talebian@ipm.ir

arXiv:2210.15186v2 [gr-qc] 23 Nov 2022

1 Introduction

Black hole (BH) has played important roles in the developments of theoretical physics. Fur-

thermore, during the past decades numerous observations have suggested the existence of

massive and supermassive black holes at the center of typical galaxies. The recent detections

of gravitational waves by the LIGO and VIRGO collaborations [1,2,3] from the merging of

binary astrophysical black holes put the reality of black holes in cosmos beyond doubt. On

the theoretical side BH physics play key roles in understanding of quantum gravity which

were studied extensively in the past decades.

The physical processes governing the dynamics of the interior of the BH are not well

understood. The prime reason is that the interior region of BH is causally disconnected from

the exterior region. Any in-falling signal smoothly passes through the event horizon and no

signal can escape from inside the event horizon to an outside observer. This phenomena

suggests that the interior of a BH is like a cosmological background bounded by the event

horizon. This interpretation is supported from the fact that inside the horizon, the roles

of coordinates tand ras the time-like and space-like coordinates are switched. There have

been works in the past to treat the interior of a BH as a cosmological background. For

example the idea that the interior of BH may be replaced by a non-singular dS space-time

was studied in [4,5,6,7,8,9,10,11,12], see also [13,14,15,16,17,18] for similar ideas but

in somewhat diﬀerent contexts. The main motivation to replace the interior of BH by the dS

background was to remove the singularity of BH. On the physical ground one may expect that

the singularity of BH is a shortcoming of the classical general relativity. On very small scales,

say on Planck scale, it is expected that the quantum gravity eﬀects can not be neglected. It is

expected that these eﬀects provide mechanisms to resolve the singularity inside the BH. For

example, the idea of maximum curvature of space-time [19,6,7] is an interesting proposal in

this direction.

Like the interior of the BH, the white hole (WH) background is more akin to a cosmological

spacetime in which the global structure of spacetime suggests that the past singularity r=

0 behaves as the onset of big bang singularity in which the signal generated from r= 0

inside the WH propagates towards the future null inﬁnity I+. The WH spacetime can be

viewed as an anisotropic cosmological background with its spatial part having the topology

R×S2known as the Kantowski-Sachs [20] spacetime. On the other hand, perturbations in

FLRW cosmological backgrounds have been studied extensively. Indeed, it is believed that

all structures in observable Universe are generated from tiny quantum ﬂuctuations generated

during primordial inﬂation. It is therefore a natural question to study perturbations inside

the WH as a particular cosmological background. With these discussions in mind, in this

work ﬁrst we study some basic cosmological properties of the WH geometry as part of an

eternal Schwarzschild manifold. Then we study the quantum perturbations of a test scalar

ﬁeld generated deep inside the WH which will propagate towards the past event horizon and

eventually reaching to an observer far outside the BH. We calculate the spectrum of Hawking

radiation as measured by this observer. For a related study see also [21].

It is believed that the WHs are not stable and have disappeared in early universe [22] so

the current analysis may not be directly relevant to observable Universe. However, we treat

WH as part of an eternal BH manifold which exists along with BH as required by the time

reversal symmetry of general relativity. We ﬁnd interesting properties of WH cosmology as an

anisotropic cosmological background while studying quantum ﬁeld theory in WH background

can provide a non-trivial example of quantum ﬁeld theory in curved backgrounds.

Hawking radiation in the space-time of WH has been also studied in [23,24]. In Ref. [23],

it was shown that there is a Hawking radiation associated to a WH spacetime equal to the BH

Hawking temperature when viewed from the outside region of the WH geometry based on BH-

to-WH tunneling scenario [25] during the gravitational collapse process. The tunnelling rate

inside the horizon of a WH for scalar and massive vector particles were calculated using the

Hamilton-Jacobi method. Our study diﬀers form [23] in two ways: ﬁrst we deal with an eternal

WH while in Ref. [23] the WH is considered as a long-lived remnant of BH during gravitational

collapse. Second, we study the quantum ﬂuctuations generated near the WH singularity which

propagate naturally towards the past event horizon. While in [23], Hawking radiation viewed

as a quantum tunnelling eﬀect to the tunnelling rate of particles. Moreover, the authors of

[24] have presented a generalization of the Hawking eﬀect for dynamical trapping horizons by

calculating the tunnelling rate in the Hamilton-Jacobi formalism. They studied the quantum

eﬀects (quantum tunnelling) across various horizons in general, including the WH horizon

(past outer trapping horizon).

2 Background Geometry

In this section we brieﬂy review the necessary preliminaries from BH background and set the

stage for the WH cosmology.

2.1 Black hole preliminaries

We consider the Schwarzschild metric with the following line element

ds2=−1−2GM

rdt2+dr2

1−2GM

r+r2dΩ2,(1)

in which Gis the Newton constant, Mis the mass of the BH as measured by an observer at

inﬁnity while dΩ2represents the metric on a unit two-sphere. The coordinate system (r, t) only

covers the exterior of the whole manifold while becoming singular on the BH event horizon at

r=rS≡2GM. Naively speaking, for the interior of the BH and the WH the roles of tand

rcoordinates are reversed in which tbecomes spacelike while rbecomes timelike. This also

suggests that the interiors of BH and WH are actually dynamical, mimicking cosmological

backgrounds.

To cover the entire manifold, we can use the Kruskal coordinate in which

ds2=32G3M3

re−r/2GM −dT2+ dR2+r2dΩ2,(2)

where (T, R) coordinate is related to the original coordinate (t, r) by

T2−R2=er/2GM 1−r

2GM ,(3)

and

R= tanht

4GM .(4)

It is evident that Tis timelike while Ris spacelike throughout.

It is also very convenient to use the Kruskal coordinate in its lightcone base (U, V ) deﬁned

via

U≡GM(T−R), V ≡GM(T+R),(5)

in which the line element takes the following form

ds2=−32GM

re−r/2GM dUdV+r2dΩ2.(6)

Note that in our convention, the coordinates (U, V ) carry the dimension of length (or time).

A conformal diagram of the entire Schwarzschild manifold is presented in Fig. 1. The

exterior of the BH is the region V > 0, U < 0 while the interior of the BH is the region

U, V > 0 bounded by the future singularity r= 0. The WH region is given by U, V < 0

bounded by the past singularity r= 0. Both the interior of the BH and the WH share

the common property that their backgrounds are dynamical corresponding to anisotropic

cosmological setups. However, the crucial diﬀerence is that the singularity of the BH is in

future, i.e. it represents a big crunch singularity while the singularity of the WH is in the past,

i.e. it represents a big bang singularity. As mentioned before, the WH is unstable and may

not exist in current observable Universe. However, in our treatment we consider an eternal

BH in which a WH is an integral part of the full manifold. In this eternal background, the

WH exists along with BH as required by the time reversal symmetry of the classical general

relativity.

Deﬁning the tortoise coordinate dr∗= (1 −2GM

r)−1drfor the interior of BH and the WH

regions (r < 2GM) we have

r∗=r+ 2GM ln 1−r

2GM ,(r < 2GM).(7)

Restricting our attention to WH region, we note that −∞ < r∗≤0 in which r∗=−∞

corresponds to the past event horizon r= 2GM, V = 0 with U < 0 while r∗= 0 corresponds

to the past singularity at r= 0.

2.2 White hole cosmology

As mentioned above, the WH background represents an anisotropic cosmological setup known

as the Kantowski-Sachs background with the structure of R×S2. To see this more clearly,

Figure 1: The Kruskal diagram. We are interested in quantum perturbations generated

from the past singularity r= 0 inside the WH, crossing the past horizon at V= 0, U < 0

and propagating towards I+. To have a complete Cauchy surface of initial conditions, we

also consider the perturbations generated from left I−and right I−located respectively to

the left and to the right of WH. The wavy green line represents the propagation of the right

mover modes e±iωu while the left movers e±iωv are not shown for brevity.

let us deﬁne the future directed time coordinate dτ=−dr∗so

τ=−r−2GM ln 1−r

2GM .(8)

Correspondingly, the WH singularity is at τ= 0 while the past event horizon U < 0, V = 0

is mapped to τ= +∞. Now deﬁning the spacelike coordinate dx≡dtwith −∞ <x<+∞,

the metric in the WH region takes the following cosmological form

ds2=a(τ)2(−dτ2+ dx2) + r(τ)2dΩ2,(9)

in which the scale factor a(τ) is deﬁned via

a(τ)≡2GM

r(τ)−11

.(10)

It is important to note that r=r(τ).

The metric (9) represents an anisotropic background with two scale factors a(τ) and r(τ).

At the “big bang” singularity τ=r= 0, we have a(τ)→ ∞ while r(τ)→0 so the space

along the xdirection starts oﬀ very large and contracts as time pass by while the scale factor

along the two-sphere is originally zero and stars to expand.

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WhiteHoleCosmologyandHawkingRadiationfromQuantumCosmologicalPerturbationsHassanFirouzjahi*,AlirezaTalebianSchoolofAstronomy,InstituteforResearchinFundamentalSciences(IPM)P.O.Box19395-5531,Tehran,IranAbstractThespacetimeinsidethewhiteholeislikeananisotropiccosmologicalbackgroundwiththepastsingularit...

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White Hole Cosmology and Hawking Radiation from Quantum Cosmological Perturbations Hassan Firouzjahi Alireza Talebian

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