Slowly evolving horizons in Einstein gravity and beyond Ayon Tarafdar Department of Physics University of Calcutta

2025-05-03 0 0 894.21KB 13 页 10玖币

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Slowly evolving horizons in Einstein gravity and beyond

Ayon Tarafdar∗

Department of Physics, University of Calcutta,

92, Acharya Prafulla Chandra Road,

Kolkata - 700009, India

Srijit Bhattacharjee†

Indian Institute of Information Technology (IIIT), Allahabad,

Deoghat, Jhalwa, Prayagraj- 211015, India

(Dated: February 1, 2023)

We study event horizon candidates for slowly evolving dynamical black holes in General Relativity

and Einstein-Gauss-Bonnet (EGB) gravity. Such a type of horizon candidate has been termed as

slowly evolving null surface (SENS). It signiﬁes a near-equilibrium state of a dynamic black hole.

We demonstrate the time evolution of such surfaces for three diﬀerent metrics. First, we locate

such a surface for a charged Vaidya metric and show that the parameter space of the black hole

gets constrained to allow a physically admissible slowly evolving null surface. We then consider a

supertranslated Vaidya solution that contains a non-spherical horizon and study the properties of

the SENS. This spacetime generates a non-vanishing shear at the SENS due to the presence of the

supertranslation ﬁeld. The SENS for a spherically symmetric Vaidya-like solution in EGB gravity

yields a bound on the accretion rate that depends on the size of the horizon. We also show that the

ﬁrst and second laws of black hole mechanics can be established for these slowly evolving surfaces.

I. Introduction

Black Holes (BH) are arguably the most useful

testing beds in the sky for uncovering innumerable facts

regarding gravitational physics. The event horizon of

a black hole is a null hypersurface that oﬀers most

of the interesting features of it. Further, the laws of

black hole mechanics established on the event horizon

intriguingly anticipate the laws of thermodynamics [1–

3]. This fascinating interconnection between quantum

physics and the geometric laws of black hole mechanics

is perhaps the strongest motivation behind the eﬀorts

of establishing a quantum theory of gravity. However,

ﬁnding the location of the event horizon requires a

complete knowledge of the future for a black hole

spacetime. Moreover the laws of black hole mechanics,

established on stationary event horizons, are not always

suited for practical situations where a black hole becomes

dynamic due to its interaction with the surroundings. To

understand the near-equilibrium situation or a scenario

when a black hole geometry evolves, one needs an

alternative set up where the response of a black hole’s

horizon can be studied without resorting to the global

aspects.

The non-equilibrium scenario of black holes (BH) has

been addressed successfully in a quasi-local set-up where

∗ayon.trf@gmail.com

†srijuster@gmail.com

one focuses on trapped surfaces. A trapped surface is a

closed, spacelike codimension-2 surface with the property

that all future-directed null normal congruences have

decreasing cross-sectional area. The boundary of such

a region serves as a useful way to characterize horizons

from a geometric point of view rather than a causal

one. Such a local deﬁnition of black hole horizons and

corresponding laws of mechanics were ﬁrst advocated

by Hayward [4] and extended by Ashtekar et al [5–7].

The boundaries of such surfaces are usually regarded as

apparent horizons or trapping horizons.Isolated horizons

and dynamical horizons were proposed for a BH in

equilibrium with its surroundings and one that is not,

respectively [7]. The laws of black hole mechanics have

been successfully extended to these quasi-local horizons

for general relativistic BH. These horizons are devoid

of the teleological nature of the event horizon as they

respond only when matter or radiation falls into them.

The quasi-local trapping horizons are still not very

useful in the case of asymptotically ﬂat BH. Marginally

trapped surfaces for which the expansion of the outgoing

null normal congruence vanishes - are situated within

the event horizon and disconnected from the exterior.

To avoid this diﬃculty one may rather choose a surface

which is evolving and suﬃciently close to the trapping

horizon but not trapped. This set-up is somewhat related

to the membrane paradigm that was developed for the

sake of probing the near horizon physics of BH [8].

Among the diﬀerent types of trapping horizons the

future outer trapping horizon (FOTH) turns out to be

arXiv:2210.15246v2 [gr-qc] 31 Jan 2023

important as it provides an equivalent local description

of an event horizon [4]. A slowly evolving FOTH

was introduced by Booth and Fairhurst in [9,10] that

can emulate near-equilibrium behaviour. In this article

we study the trapping horizons of BH in Einstein

gravity and beyond. Particularly, we focus on locating

aslowly evolving null surface (SENS), introduced in

[11,12], that behaves similar to a weakly perturbed

event horizon. The slowly evolving horizons are not only

useful to accommodate initial and ﬁnal states that are

not inﬁnitesimally separated, but they also provide a

simple and eﬀective way to characterize near-equilibrium

situations. Studying these horizons may shed useful

insight on the near-equilibrium phases of colliding BH

in the early or very late stage of merger [13].

The paper is organised as follows: in the next section,

we describe the notation and the set-up to be used

throughout the paper. In section III, we introduce

the FOTH of a charged Vaidya metric and discuss the

properties of it. We also introduce the SENS to the

reader. Location and properties of the SENS are detailed

thereafter. In the next section we study the SENS of a

supertranslated Vaidya black hole having a non-spherical

horizon. However, the spacetime still has a spherical

topology. The location of slowly evolving horizons in

this case are dependent on the angular variable θ. The

consequences are discussed with plots. The next section

contains the study of SENS for a Vaidya black hole in

EGB gravity in 5 dimensions. Next, the ﬁrst and second

law of black hole mechanics for the SENS are introduced.

The validity of these laws for the EGB BH is discussed.

Finally, we conclude by highlighting notable outcomes of

our study and indicate some future prospects.

II. Notation and set-up

The mathematics of perturbatively constructing the

spacetime around a foliated, codimension-1 hypersurface

(in this case the FOTH) is well-known and our immediate

references are [10,12]. In this section, we just highlight

the key geometric structures and results we need.

A manifold Mof dimension d+ 1 is endowed with

a metric gab and its corresponding connection ∇a.

We denote the spacetime indices by lower case initial

Latin letters (a, b, ···). The quantities on d-dimensional

hypersurfaces are denoted by (i, j, k, ···) and capital

letters (A, B, ···) are used to denote quantities on d−1

dimensional surfaces. We use (−,+,+,··· ,+) signature

for the metric. We also set G=c= 1.

Afuture outer trapping horizon (FOTH) is deﬁned

as a hypersurface Σ of dimension dwith the following

properties: i) They are foliated by a family of d−1

dimensional spacelike surfaces Svwith future-directed

null normals l(outgoing) and n(ingoing) on it. Here v

is just a coordinate that labels the spatial surfaces. The

expansion w.r.t the null vector lvanishes i.e. Θ(l)= 0 on

FOTH. ii) The expansion w.r.t the other null normal n

is negative, Θ(n)<0. iii) δnΘ(l)<0, where δnis the Lie

derivative £npulled-back on the d−1 surface.

The null normals are cross-normalized as l·n=−1.

Now we deﬁne a vector ﬁeld along which the leaves of

the foliation (Sv) evolve. It is tangent to the horizon but

normal to the d−1 surface as:

Va=la−Cna,(1)

where Cis a scalar ﬁeld. This choice allows us to ﬁx the

scaling freedom of the null vectors. Under the rescalings:

la→h(v)la, na→na/h(v), C →h(v)2C,

we have Va→h(v)Va, which amounts to a relabelling of

the foliation leaves. The norm of this vector is VaVa=

2C. When the dominant energy condition holds, it can be

shown using δnΘ(l)<0 that C≥0. A FOTH is spacelike

for C > 0 and null for C= 0. We will not consider the

timelike (C < 0) case here. The d−1 dimensional metric

is given by qab =gab+lanb+lbna. The pulled-back metric

on this surface is given by qAB =ea

Aeb

Bgab, where we

denote eas the pull-back maps. The extrinsic curvatures

of this codimension-2 surface are deﬁned as

K(l)

AB =ea

Aeb

B∇alb,K(n)

AB =ea

Aeb

B∇anb.(2)

The expansion Θ(l)/(n)and shear σ(l)/(n)

AB of the null

congruences are obtained as the trace and trace-free

parts of these extrinsic curvatures. The evolution of

the expansion along the null congruences is given by

the Raychaudhuri equation and they indicate how the

geometry of the d−1 surface evolves along the null

direction.

£lΘ(l)=κ(l)Θ(l)−Θ2

(l)

d−1−σ(l)

AB σAB

(l)− Rablalb,(3)

where κ(l)is the dynamical surface gravity deﬁned as

κ(l)=−lanb∇alb.

A similar expression for Θ(n)can also be obtained.

Now, on an FOTH δVΘ(l)= 0. Using this, we can obtain

a second order diﬀerential equation for C[10]:

d2C−2ωAdAC−δlΘ(l)+CδnΘ(l)= 0.(4)

where dAis the covariant derivative of the (d−1)-surface.

ωA=−ea

Anb∇albis the connection on the normal bundle.

For spherical symmetry, the (d−1)-surface derivatives

drop out and we’re left with an explicit expression for

Slowly evolving horizon: When one refers to a

slowly evolving horizon, one commonly refers to the

FOTH, satisfying the slowly evolving conditions. These

conditions are determined from the intrinsic and extrinsic

geometry of the FOTH, discussed at length in [12].

A slowly evolving FOTH can be characterized as a

hypersurface that is perturbatively non-isolated with

the surroundings. One of the key slowness conditions

is that the evolution parameter Chas to be small. To

obtain the SENS, we perturbatively expand around

the FOTH along radial null geodesics normal to the

spatial codimension-2 surface. In this regime, the SENS

coincides with the event horizon of an asymptotically

stationary spacetime [12]. Hence it can also be termed as

a ‘slowly evolving horizon’. For a BH in an equilibrium

state, the SENS naturally coincides with the FOTH,

which then becomes null.

Slowly Evolving Null Surface: We now formally

introduce the conditions an SENS must satisfy. A section

of a d-dimensional null surface with tangent vector laand

a characteristic scale RΣ(typically given by the areal

radius of the horizon) is a slowly evolving null surface

(SENS) if [12]:

1. 1

d−1Θ2

(l)σ(l)

AB σAB

(l)+Rablalb,

and lacan be scaled so that κ(l)is of order 1/RΣso that:

2. £lΘ(l)κ(l)Θ(l).

These conditions allow us to treat the SENS as a

surface that mimics a near-equilibrium situation of a

weakly perturbed event horizon or isolated horizon.

We start with a discussion of the charged Vaidya metric

and use it to introduce our procedure and additional

useful notation.

III. Charged Vaidya metric

The charged Vaidya metric or the Vaidya-Reissner-

Nordstr¨om (VRN) metric in 4-d is a spherically

symmetric metric which in ingoing Eddington-

Finkelstein coordinates is [14]:

ds2=−4(v, r)dv2+ 2dvdr +qAB dxAdxB,(5)

4(v, r)=1−2m(v)

r+q(v)2

r2·(6)

The spatial 2-surface is described by the metric of a

sphere:

qABdxAdxB=r2(dθ2+ sin2θ dφ2).

To determine the trapping horizon, we focus on the

outgoing and ingoing radial null geodesics, the tangent

vectors to which are laand narespectively. These are

la:=1,4

2,0,0, na:= (0,−1,0,0).(7)

The expansion of these null congruences are:

Θ(l)=4

r,Θ(n)=−2

The expansion of the ingoing null normal, as expected,

is everywhere negative.

1. The energy condition

The stress-energy tensor for this metric looks like

Tab =T(em)

ab +T(ex)

where the T(em)

ab is the electromagnetic part and the extra

term T(ex)

ab is

8πT (ex)

ab =2

r3( ˙mr −q˙q)δv

aδv

b(8)

(we’ve dropped the function arguments for brevity).

This satisﬁes the null energy condition (NEC)

Tablalb≥0 for the outgoing null vector laif

r˙m(v)−q(v) ˙q(v)≥0 or r≥q(v) ˙q(v)

˙m(v),

where we can term rcs :=q˙q/ ˙mas a “critical surface”. If

this critical surface exists (rcs >0), within this surface,

the NEC will be violated. In our case, the critical

surface needs to be within the FOTH. Otherwise, even

for accretion of charged null dust, the FOTH radius may

decrease [15]. Denoting it by r+, this means r+≥rcs.

A. Locating the FOTH

The trapping horizon is located where Θ(l)= 0 = 4,

which gives two roots:

r±(v) = m(v)±pm(v)2−q(v)2.(9)

We’re interested in the larger root r+(v), which is our

“outer” horizon, i.e. on this hypersurface, £nΘ(l)<0.

Computing this quantity for this metric we get

£nΘ(l)=−2

+pm2−q2.

Since m > q and r+>0, the RHS is always negative.

Moreover, since Θ(n)<0, it is of future type. Thus

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SlowlyevolvinghorizonsinEinsteingravityandbeyondAyonTarafdarDepartmentofPhysics,UniversityofCalcutta,92,AcharyaPrafullaChandraRoad,Kolkata-700009,IndiaSrijitBhattacharjeeyIndianInstituteofInformationTechnology(IIIT),Allahabad,Deoghat,Jhalwa,Prayagraj-211015,India(Dated:February1,2023)Westudyeventho...

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Slowly evolving horizons in Einstein gravity and beyond Ayon Tarafdar Department of Physics University of Calcutta

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