Quantum Oppenheimer-Snyder and Swiss Cheese models Jerzy Lewandowski1Yongge Ma2yJinsong Yang3zand Cong Zhang1 4x 1Faculty of Physics University of Warsaw Pasteura 5 02-093 Warsaw Poland

2025-04-29 0 0 494.98KB 8 页 10玖币

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Quantum Oppenheimer-Snyder and Swiss Cheese models

Jerzy Lewandowski,1, ∗Yongge Ma,2, †Jinsong Yang,3, ‡and Cong Zhang1, 4, §

1Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

2Department of Physics, Beijing Normal University, Beijing 100875, China

3School of Physics, Guizhou University, Guiyang 550025, China

4Department Physik, Institut für Quantengravitation, Theoretische Physik III,

Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7/B2, 91058 Erlangen, Germany

By considering the quantum Oppenheimer-Snyder model in loop quantum cosmology, a new

quantum black hole model whose metric tensor is a suitably deformed Schwarzschild one is derived.

The quantum eﬀects imply a lower bound on the mass of the black hole produced by the collapsing

dust ball. For the case of larger masses where the event horizon does form, the maximal extension of

the spacetime and its properties are investigated. By discussing the opposite scenario to the quantum

Oppenheimer-Snyder, a quantum Swiss Cheese model is obtained with a bubble surrounded by the

quantum universe. This model is analogous to black hole cosmology or fecund universes where

the big bang is related to a white hole. Thus our models open a new window to cosmological

phenomenology.

According to Subrahmanyan Chandrasekhar "The

black holes of nature are the most perfect macroscopic

objects there are in the Universe" [1]. Given develop-

ment of quantum models describing spacetime ﬁlled with

dust, the following two questions are addressed in this

letter: What is a black hole (BH) spacetime containing

a collapsing matter ball like? What is a BH spacetime

surrounded by a universe like?

In classical general relativity (GR), our understand-

ing on these two questions is shaped by the the

Oppenheimer-Snyder model [2], which depicts the col-

lapse of the pressureless homogenous dust coupled to the

Friedmann–Lemaître–Robertson–Walker metric. How-

ever, this metric appears to be problematic due to the

Big-Bang singularity. A proposal to resolve this singu-

larity is to replace the Big Bang by a Big Bounce, which

was largely considered by cosmologists for aesthetic rea-

sons [3]. Thus, it is desirable to answer the above two

questions by considering collapsing and bouncing mat-

ters.

Quantum gravity has always been expected to go be-

yond the singularities of the classical GR. Indeed, the

existence of a Big Bounce resolving the Big Bang singu-

larity has found a diverse support in the Loop Quantum

Cosmology (LQC) models (see, e.g., [4–6]). A concrete

bouncing model is the Ashtekar-Pawlowski-Singh (APS)

model, where the bounce is a rigorous result of the funda-

mental discreteness [4]. In this model, the semiclassical

metric tensor has the form

ds2

APS =−dτ2+a(τ)2(d˜r2+ ˜r2dΩ2),(1)

where (τ, ˜r, θ, φ)denotes a coordinate system, dΩ2=

dθ2+ sin2θdφ2. The function a(τ)satisﬁes a deformed

∗jerzy.lewandowski@fuw.edu.pl

†mayg@bnu.edu.cn

‡jsyang@gzu.edu.cn

§czhang@fuw.edu.pl

Friedmann equation

H2:= ˙a

a2

=8πG

3ρ1−ρ

ρc, ρ =M

3π˜r3

0a3,(2)

where the deformation parameter is the critical den-

sity ρc=√3/(32π2γ3G2~)with the Barbero-Immirzi

parameter γ,Mis the mass of the ball of the dust

with the radius a(τ)˜r0in the APS spacetime. Note

that the second equation in (2) with a constant Mis

the consequence of the conversion law ∇µTµν = 0 with

Tµν =ρ(τ)(∇τ)µ(∇τ)ν.Eq. (2) reverts to the usual

Friedmann equation in the classical regime when ρρc.

However in the quantum regime where ρis comparable

with ρc, the equation prevents ρ(τ)from reaching inﬁn-

ity. This property ensures that the metric tensor ds2

APS

is nowhere and never singular. The function a(τ)can be

extended to the whole interval (−∞,∞).

The particles of the dust in the APS spacetime (1)

are the geodesics satisfying ˜r, θ, φ = const. There-

fore, an APS dust ball can be characterized as a region

0≤˜r≤˜r0of the APS spacetime. Then, our quantum (or

rather semiclassical) Oppenheimer-Snyder (qOS) model

assumes the (pseudo) static 1spherically symmetric met-

ric

ds2

MS =−(1 −F(r))dt2+ (1 −G(r))−1dr2+r2dΩ2,(3)

with some functions F(r)and G(r), where (t, r, θ, φ)are

coordinates. The coordinates θand φare joint for the

ball region and the exterior (meaning they are extensions

of each other), whereas the coordinates τ, ˜rare used in

the ball region only, while the coordinates t, r are used

only in the exterior region. Eq. (3) is a minimal as-

sumption if we are to obtain an exact BH metric by the

1By pseudo static, we take into account the case that the Killing

vector ∂tinside the BH is space-like.

arXiv:2210.02253v3 [gr-qc] 28 Feb 2023

junction condition, without employing equations of mo-

tion. As we are going to show, there are close ties be-

tween the models of quantum BH and quantum cosmol-

ogy, providing a possibility to detect the quantum eﬀects

in early universe from BHs. Actually, the resulting met-

ric is a suitably deformed Schwarzschild one, where the

deforming term leads to a BH mass gap. Moreover, the

deformed Schwarzschild metric induces a non-vanishing

eﬀective energy-momentum tensor. This may relate the

quantum eﬀects of BHs with the dark matter.

The methodology of LQC could also be applied to

BH models [8–24]. However, the answers do not form a

unique picture. Particularly, according to certain conjec-

ture, the bouncing interior of the BH destroys the Killing

horizon in the future, and the process takes the form of

BH evaporation [9]. According to another proposal, the

reﬂecting interior turns a BH into a white hole (WH), and

the transitional region of spacetime is strictly quantum,

giving the process the character of quantum tunneling

[12, 13]. In both of these cases, the global structure of

the null inﬁnity is similar, there is one scri. Subsequent

models describe spacetime containing a quantum BH dif-

ferently (see, e.g.,[15]). According to them, the bouncing

interior does not aﬀect the global structure of the exte-

rior. Hence, the spacetime looks similar to the Kruskal

diagram of Schwarzschild spacetime, with the only diﬀer-

ence that the singularity becomes an edge of spacetime

on which the metric is still regular. Now the extension

consists in gluing the diagrams, even of an unbounded

number, with the edges.

In the case of the second question, we consider the

opposite scenario: a spherically symmetric, static empty

region of spacetime (a bubble) surrounded by the quan-

tum universe according to the APS model. Precisely, this

scenario consists in removing the ball 0≤˜r≤˜r0from

the APS spacetime, that is considering the APS met-

ric tensor ds2

APS for ˜r≥˜r0. The hole left by the ball

is ﬁlled with a piece of the spacetime (3). Hence this

is a quantum Swiss Cheese (qSC) model whose physical

meaning is quite diﬀerent from the qOS model. Before

the quantum universe bounces, the spherically symmet-

ric bubble is being squised. The question is whether its

radius shrinks below the radius of the horizon, and if

so, what happens next. Indeed, according to the result

shown below, unless the size of the region is of the order

of the Planck length, the horizon does form. The radius

depends on the amount of mass of the dust if it were ﬁll-

ing the bubble. Brieﬂy speaking, the bouncing universe

turns the bubbles into BHs.

In both cases the key role is played by the the dust

space surface (either outer or inner) ˜r= ˜r0in the APS

spacetime, that in the spacetime (3) will be described in

a partially parametric form (t(τ), r(τ), θ, φ)where −∞ <

τ < ∞is the proper time, and the ranges of coordinates

read 0≤θ≤π, 0≤φ < 2π. We glue the spacetimes

by the identiﬁcation (τ, ˜r0, θ, φ)∼(t(τ), r(τ), θ, φ)such

that the induced metric and the extrinsic curvature are

equal on the gluing surfaces that become a single surface

of the dusty part of the spacetime. That will allow us to

unambiguously determine the functions Fand Gas well

as a location of the dust surface in the dust-free spacetime

— asymptotically for r→ ∞ it is tangent to ∂t. Then,

the metric (3) can be obtained as

ds2

MS =−1−2GM

r+αG2M2

r4dt2

+1−2GM

r+αG2M2

r4−1

dr2+r2dΩ2,

(4)

where we introduced the parameter α= 16√3πγ3`2

pwith

`p=√G~denoting the Planck length. Indeed, the cal-

culation to determine the functions Fand Gis quite

straightforward (see Appendix A for details). It is worth

noting that the form (4) of the metric is determined for

r≥rb=αGM

2

,(5)

which results from the fact that the dust surface radius

a(τ)˜r0runs over [rb,∞). Hence the functions F(r)and

G(r)may be deﬁned arbitrarily for r < rb. The parame-

ter Mcoincides with the ADM mass of the metric tensor

(4). As a quantum deformation of the Schwarzschild met-

ric, the spacetime metric tensor (4) coincides with that

derived in [25–27].

The global structure of the spacetime determined by

(4) depends on the number of roots of 1−F(r). It is

convenient to introduce the parameter 0< β < 1by

G2M2=4β4

(1 −β2)3α. (6)

For 0< β < 1/2, that is when

M < Mmin := 4

3√3G√α, (7)

1−F(r)has no real root, implying that the metric (4)

does not admit any horizon. The global causal structure

of the maximally extended spacetime is the same as that

of the Minkowski spacetime. Hence the value

Mmin =16γ√πγ

√3

G(8)

is a lower bound for BHs produced by our models (see

[23, 24, 28] for compatible results). The minimal mass is

of the order of the Planck mass. Its actual value depends

on the value of the Barbero-Immirzi parameter γof LQG,

that is argued to be of order of 0.2[29, 30].

Consider the case of M > Mmin, i.e., 1/2< β < 1.

The function 1−F(r)has exactly two roots

r±=β1±√2β−1

p(1 + β)(1 −β)3√α,

that makes the coordinate tsingular. We extend the met-

ric tensor ds2

MS by following the steps similar to those for

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QuantumOppenheimer-SnyderandSwissCheesemodelsJerzyLewandowski,1,YonggeMa,2,yJinsongYang,3,zandCongZhang1,4,x1FacultyofPhysics,UniversityofWarsaw,Pasteura5,02-093Warsaw,Poland2DepartmentofPhysics,BeijingNormalUniversity,Beijing100875,China3SchoolofPhysics,GuizhouUniversity,Guiyang550025,China4Depart...

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Quantum Oppenheimer-Snyder and Swiss Cheese models Jerzy Lewandowski1Yongge Ma2yJinsong Yang3zand Cong Zhang1 4x 1Faculty of Physics University of Warsaw Pasteura 5 02-093 Warsaw Poland

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