Modeling black hole evaporative mass evolution via radiation from moving mirrors Michael R. R. Good1Alessio Lapponi2 3yOrlando Luongo4 5 6 7zand Stefano Mancini4 7x 1Department of Physics Energetic Cosmos Laboratory

2025-05-06 0 0 615.14KB 13 页 10玖币

侵权投诉

Modeling black hole evaporative mass evolution via radiation from moving mirrors

Michael R. R. Good,1, ∗Alessio Lapponi,2, 3, †Orlando Luongo,4, 5, 6, 7, ‡and Stefano Mancini4, 7, §

1Department of Physics & Energetic Cosmos Laboratory,

Nazarbayev University, Kabanbay Batyr, Nur-Sultan, 010000, Kazakhstan.

2Scuola Superiore Meridionale (SSM), Largo San Marcellino, Napoli, 80138 Italy.

3Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Napoli, 80126, Italy.

4School of Science and Technology, University of Camerino,

Via Madonna delle Carceri, Camerino, 62032, Italy.

5Dipartimento di Matematica, Universit`a di Pisa, Largo B. Pontecorvo, Pisa, 56127, Italy.

6Institute of Experimental and Theoretical Physics,

Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan.

7Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Perugia, Perugia, 06123, Italy.

We investigate the evaporation of an uncharged and non-rotating black hole (BH) in vacuum,

by taking into account the eﬀects given by the shrinking of the horizon area. These include the

back-reaction on the metric and other smaller contributions arising from quantum ﬁelds in curved

spacetime. Our approach is facilitated by the use of an analog accelerating moving mirror. We

study the consequences of this modiﬁed evaporation on the BH entropy. Insights are provided on

the amount of information obtained from a BH by considering non-equilibrium thermodynamics and

the non-thermal part of Hawking radiation.

PACS numbers: 04.62.+v, 04.70.-s, 04.70.Dy

Keywords: moving mirrors, black holes, non-equilibrium thermodynamics, quantum ﬁelds in curved space.

CONTENTS

I. Introduction 1

II. Black holes from mirror analogy 2

A. Black hole radiation 2

B. Mirrors and black hole analogy 3

III. Black hole evaporation 4

A. Modeling BH evaporation with mirrors 4

B. Evaluating the mass evolution 5

C. Interpreting the critical time 6

D. Mass evolution at early times 7

E. Dynamical behavior of mirrors at

intermediate stages 7

IV. Mirror thermodynamics and black hole analogy 8

A. Non-thermality 9

B. Temperature and entropy of an evaporating

BH 9

C. Eﬀective temperature 9

D. Consequences on thermodynamics 10

E. Consequences on entropy 11

V. Outlooks 11

Acknowledgments 12

References 12

∗michael.good@nu.edu.kz

†alessio.lapponi-ssm@unina.it

‡orlando.luongo@unicam.it

§stefano.mancini@unicam.it

I. INTRODUCTION

Despite the impressive eﬀorts spent on black hole (BH)

thermodynamics [1–8], it is still a challenge to know how

a BH’s mass changes in time. In the latter scenario, BH

evaporation may cause a back-reaction on the underly-

ing spacetime metric. Consequently, several attempts to

describe BH metrics encompassing back-reaction eﬀects

have been recently developed [9–11], leading to no unan-

imous consensus on how back-reaction occurs.

Naively if a BH radiates, the horizon area shrinks and

thermal Hawking radiative power would increase. How-

ever, even the opposite perspective may be plausible, see

e.g. [12], as quantum gravitational eﬀects are not fully-

employed [13–16]. In addition, the modiﬁcation of BH

particle production is also associated with other eﬀects,

i.e., due to horizon shrinking [17], where, for instance, the

evolution in time of the background spacetime provides

a non-zero small particle count [18].

In these scenarios, analog systems mimicking BHs,

namely BH mimickers [19–21], are helpful to overcome

the mathematical diﬃculties related to time-dependent

thermodynamic quantities, e.g., mass, temperature, en-

tropy and so forth. Among all possibilities, perfectly re-

ﬂecting moving mirrors in (1+1)-dimensional ﬂat space-

time, characterized by a given trajectory, see e.g. [22–25],

can reproduce thermal Hawking radiation1.

A net advantage of mirrors consists in studying BH

radiation properties, e.g. Hawking radiation, thermody-

namics, etc., without considering an underlying space-

1On the other hand, semitransparent moving mirrors may exhibit

quite diﬀerent energy emission and particle creation [22,25–35].

arXiv:2210.09744v1 [gr-qc] 18 Oct 2022

time associated with the BH itself2, as analog systems.

As a consequence, by using mirrors, one can deal with

BH radiation models without having a precise descrip-

tion of the (apparent) horizon area and/or of the BH

surface gravity3.

In this work, we investigate the thermodynamic prop-

erties of a mass-varying BH adopting the BH analog pro-

vided by a thermal moving mirror. We focus in particular

on the mass evolution of an evaporating BH in vacuum.

The mathematical simpliﬁcation of moving mirrors eas-

ily describes the mass evolution through a diﬀerential

equation that can be numerically solved. We ﬁnd cor-

rections to Hawking radiation without postulating the

horizon area and/or the surface gravity. Those correc-

tions are related to the eﬀects that the evaporation is

expected to cause to the radiation, above all, mimick-

ing the back-reaction eﬀect on the metric. We compare

the results that we infer with those in Ref. [23], where

qualitative arguments for the evaporation have been dis-

cussed in view of mirrors. We debate how the expected

small corrections to Hawking radiation, obtained as the

BH evaporates, are of primary importance to help under-

stand BH information loss [41–43]. Indeed, if BH radia-

tion is not precisely thermal, then it carries some infor-

mation from inside to outside the event horizon. Hence,

non-thermality of BH radiation represents a landscape for

the information paradox4. We work out the hypothesis of

quasi-static processes to approximate the ﬁrst thermody-

namics principle by means of an eﬀective non-equilibrium

temperature. In this respect, we show that the devia-

tions from Hawking radiation is initially small, becoming

larger as BHs evaporate. This causes a decrease of a BH’s

lifetime by a factor ∼3/8. Thus, since the eﬀects of BH

evaporation drastically aﬀects Hawking radiation, mir-

rors may conﬁrm quantum tunneling models for Hawk-

ing radiation [13,14], showing the emitted radiation to

be less entropic than the one predicted in the literature

[2,49–53]. This may be interpreted by assuming part

of the information can be transmitted by BH radiation.

Furthermore, we emphasize in our treatment, it is pos-

sible to construct an argument for the BH age from its

mass and Hawking radiation.

The paper is organized as follows. In Sec. II we explain

how moving mirror radiation emulates BHs. In Sec. III

we use this analogy to study BH radiation and its mass

2Attempts towards investigating metrics considering the variation

of its mass can be found in [36–39]

3For evaporating BHs, there is no Killing horizon and the concept

of surface gravity is controversial. The deﬁnition of a surface

gravity in these contexts is an ongoing subject of study (see e.g.

[40]).

4In particular, by considering BH evaporation eﬀects, quantum

tunneling models for Hawking radiation provide a signiﬁcant de-

viation from the thermal spectrum [13–16], which cause a re-

duction of the total BH entropy similar to the one predicted

by quantum gravity [14,16,44–46]. Moreover, there exist a

model-independent argument proving that the non-thermal part

of Hawking radiation cannot be omitted [47,48].

evolution from its creation to its complete evaporation.

In Sec. IV we study the non-equilibrium thermodynam-

ics of BH evaporation, adopting the quasi-static approx-

imation. Finally, Sec. Vis devoted to conclusions and

perspectives of our scheme. Throughout the paper, we

use Planck units c=G=~=kB= 1.

II. BLACK HOLES FROM MIRROR ANALOGY

Here we brieﬂy review the radiation emitted by BHs

and by moving mirrors. We conﬁrm that a trajectory for

a (1+1)D mirror exactly reproduces Hawking radiation

emitted by a (3+1)D BH. We limit our analysis to the

emission of scalar massless particles. The discussion is

split into two subsections focusing on BHs ﬁrst and then

the moving mirror analog.

A. Black hole radiation

By quantum ﬁeld theory in curved spacetime, parti-

cle creation occurs whenever the background spacetime

evolves in time [26]. This particle production is easy to

quantify when a spacetime is ﬂat in the inﬁnite past and

inﬁnite future. Indeed, in this case, the normal modes of

the scalar ﬁeld in the inﬁnite future (or output modes)

{φout

ω}ωcould be obtained from the ones in the inﬁnite

past (or input modes){φin

ω}through the following Bo-

goliubov transformation:

φout

ω=Z∞

0αωω0φin

ω0+βωω0φin∗

ω0dω0.(1)

The non-trivial Bogoliubov coeﬃcients βωω0are not zero,

indicating that particle creation occurs from the vacuum5

[26,56]. The spectrum of particles produced is given by:

Nω=Z∞

0|βωω0|2dω0.(2)

Hawking calculated [1] the Bogoliubov coeﬃcients rela-

tive to a spacetime where a star collapses into a black

hole. In this context, Eq. (1) holds by considering the

output modes {φout

ω}ωas the modes outgoing from the

collapsing star and the input modes {φin

ω}ωas the modes

ingoing towards it. The Bogoliubov coeﬃcient βωω0aris-

ing from a collapsing star with mass Mreads:

βωω0=rω0

ωΓ(1 −4iMω)(iω0)−1+4iMω ,(3)

where Γ is the Euler gamma function. By applying the

modulus square of Eq. (3) we obtain:

|βωω0|2=2M

πω0

e8πMω −1,(4)

5For relevant cosmological applications see e.g. [54,55].

leading to the known thermal spectrum with tempera-

ture:

T=1

8πM .(5)

The spectrum of particles radiated, obtained from

Eq. (2), is divergent because BH mass evaporation is not

considered, so that the BH continues to emit forever.

This is the model for what is now called an eternal BH

with exactly thermal emission. Nevertheless, by making

use of wave packets, we can localize the input and out-

put modes in a ﬁnite range of time and frequencies. In

this way, Hawking proved that [1], in a ﬁnite range of

time, the collapsing star emits a ﬁnite number of par-

ticles, following a thermal spectrum with temperature

1/8πM. The astonishing result is that this radiation is

always constant in time, with exact Planck-distributed

particles originating from the collapsed star.

The fact that a BH continues to emit even when the BH

is created is justiﬁed by the presence of the horizon when

considering vacuum ﬂuctuations near it [1,2]. Finally,

the renormalized stress energy tensor in presence of an

emitting BH was also calculated [28,57]. From it, one

can ﬁnd the ﬂux of energy (power) radiated by a BH as:

F=1

768πM2=π

12T2.(6)

The conclusion is that, following the ﬁrst quantum BH

model [1,2,28,57] an eternal BH emits as a 1Dblack

body6[59].

If we impose energy conservation, the ﬂux of energy

radiated by a BH should drain the BH mass, namely

M=−F. By considering the ﬂux (6) we have:

M=−1

768πM2,(7)

providing

M(t) = M3

0−t

256π1

,(8)

where M0=M(t= 0). Following this model, the BH

evaporates completely in a time

ev = 256πM3

0.(9)

B. Mirrors and black hole analogy

Another physical system providing particle production

is given by an accelerating mirror. In particular, the ra-

diation by perfectly reﬂecting accelerating mirrors comes

6To model the BH as an n-dimensional black body, it is suﬃ-

cient to modify the pre-factor 1

768πfrom Eq. (6) according to

the n-dimensional Stefan-Boltzmann constant [58] and appropri-

ate temperature scaling.

from the acceleration of the boundary condition imposed

by perfect reﬂection, providing the well-known dynami-

cal Casimir eﬀect [26–28,60]. Let us consider a perfectly

reﬂecting (1+1)D mirror with a generic trajectory z(t).

Each normal mode, reﬂected back by a mirror with fre-

quency ω, i.e. φout

ω, can be written as a combination

of the normal modes incoming to the mirror {φin

ω}ωas

Eq. (1).

The Bogoliubov coeﬃcient βωω0is [24,61]

βωω0=1

4π√ωω0Z+∞

−∞

exp (−i(ω+ω0)t+i(ω−ω0)z(t))

×((ω+ω0) ˙z(t)−ω+ω0)dt .

(10)

Using the renormalized stress energy tensor [28] one

can derive the ﬂux of energy radiated by the mirror, say

to its right, as:

...

z( ˙z2−1) −3 ˙z¨z2

12π( ˙z−1)4( ˙z+ 1)2

12π...

( ˙z−1)3( ˙z+ 1) −3˙z¨z2

( ˙z−1)4( ˙z+ 1)2.

(11)

The Carlitz-Willey trajectory corresponds to a (1+1)D

trajectory and represents a simple approach to model

thermal mirror trajectories. It reads

z(t) = −t−1

κW(e−2κt),(12)

where Wis the Lambert function and κa free constant

related to mirror acceleration [22]. If a mirror has this

trajectory, then by Eq. (10) we get

βωω0=−1

2πκrω

ω0e−πω

2κΓiω

κω0

κ−iω

,(13)

and its modulus square is

|βωω0|2=1

2πκω0

e2πω/κ −1.(14)

By computing from Eq. (11) the ﬂux of energy that a

mirror with trajectory (12) radiates to its right we obtain

F=κ2

48π.(15)

By comparing Eq. (4) with (14), they turn out to be

equivalent, putting κ=1

4M. In this case, also Eqs. (6)

and (15) are the same.

Hence, a (1 + 1)-dimensional mirror with a trajectory

given by Eq. (12) exactly emulates the Hawking radiation

from a (3 + 1)-dimensional Schwarzschild BH with mass

4κ: both in terms of particle produced and in terms of en-

ergy radiated. Considering an appropriate modiﬁcation

of the Carlitz-Willey trajectory (12), it is possible to ﬁnd

an analog mirror emulating the particle production prop-

erties of a Kerr BH [32], a Reissman-Nordstrom BH [29]

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ModelingblackholeevaporativemassevolutionviaradiationfrommovingmirrorsMichaelR.R.Good,1,AlessioLapponi,2,3,yOrlandoLuongo,4,5,6,7,zandStefanoMancini4,7,x1DepartmentofPhysics&EnergeticCosmosLaboratory,NazarbayevUniversity,KabanbayBatyr,Nur-Sultan,010000,Kazakhstan.2ScuolaSuperioreMeridionale(SSM),La...

展开>> 收起<<

Modeling black hole evaporative mass evolution via radiation from moving mirrors Michael R. R. Good1Alessio Lapponi2 3yOrlando Luongo4 5 6 7zand Stefano Mancini4 7x 1Department of Physics Energetic Cosmos Laboratory.pdf

共13页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Modeling black hole evaporative mass evolution via radiation from moving mirrors Michael R. R. Good1Alessio Lapponi2 3yOrlando Luongo4 5 6 7zand Stefano Mancini4 7x 1Department of Physics Energetic Cosmos Laboratory

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: