Late-time symmetry near black hole horizons

Kentaro Tanabe,1 Tetsuya Shiromizu,2 and Shunichiro Kinoshita2

1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan2Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(Received 8 December 2011; published 27 January 2012)

It is expected that black holes are formed dynamically under gravitational collapses and approach

stationary states. In this paper, we show that the asymptotic Killing vector at late time should exist on the

horizon and then that it can be extended outside of black holes under the assumption of the analyticity of

spacetimes. This fact implies that if there is another asymptotic Killing vector which becomes a stationary

Killing at a far region and spacelike in the ‘‘ergoregion,’’ the rotating black holes may have the

asymptotically axisymmetric Killing vector at late time. Thus, we may expect that the asymptotic rigidity

of the black holes holds.

DOI: 10.1103/PhysRevD.85.024048 PACS numbers: 04.20.�q, 04.20.Ha

I. INTRODUCTION

Black holes in our Universe are expected to be formedunder gravitational collapses, and to finally approach sta-tionary and vacuum states by radiating and absorbingenergy, momentum, and angular momentum. Then theuniqueness theorem [1] guarantees that the black holecandidates in our Universe are the Kerr black holes. Thekey ingredient for the proof of the uniqueness theorem isthe rigidity theorem [2–5]. The rigidity theorem shows thatthe stationary rotating black holes have axisymmetricKilling vectors. The outline of the proof is as follows:The stationarity of spacetime implies that there are nogravitational waves around a black hole. Then the expan-sion and shear of the event horizon must vanish by virtue ofthe Raychaudhuri equation and stationarity. Using theEinstein equations, then, we can find that the null geodesicgenerator of the event horizon is a Killing vector. If theblack hole is rotating, this new Killing vector may deviatefrom the stationary Killing vector which becomes space-like in the ergoregion. This means that the stationary blackholes might have two Killing vectors, that is, the stationaryand axisymmetric Killing vectors. Hence the stationaryblack holes should rotate rigidly.

However, the late-time phase of black holes produced bygravitational collapses would not be exactly stationary butnearly stationary. ‘‘Nearly stationary’’ means that the blackholes are surrounded by the gravitational waves at late time.Then we cannot apply the rigidity theorem to such blackholes because the expansion and shear of the null geodesicgenerator on the event horizon do not vanish due to thepresence of the gravitational waves on the event horizon.Note that the late-time behaviors of the perturbationsaround the Schwarzschild and Kerr black holes were exam-ined and it was shown that the perturbations decay at latetime both on the horizon and null infinity at the same rate(for examples, see Refs. [6–8]). In the dynamical processesin gravitational collapses, however, it is quite nontrivialwhether the formed black hole approaches the Kerr black

hole. In this paper, we show that the asymptotic Killingvector, which will asymptotically approach a Killing vectorat late time, should exist on the horizon without assumingany symmetries and then that it can be extended outside ofthe event horizon using the Einstein equations. This indi-cates that if there is another asymptotic Killing vectorwhich will be an asymptotically stationary Killing vectorat a far region, the rotating black hole may have theasymptotically axisymmetric Killing vector at late time.This paper is organized as follows. In the next section we

provide the Einstein equations near the future event hori-zons. In Sec. III, we investigate the late-time behaviors ofthe metric on the event horizon and find the late-timesymmetry. Then, under the assumption of the analyticity,we show that this late-time symmetry can be extendedoutside of the event horizon using the Einstein equation.In Sec. IV, we provide a summary and discussion. InAppendix A we perform the (nþ 1) decompositions ofthe Einstein equation. Using them, we provide the explicitform of the Einstein equation in the current form ofthe metric in Appendix B. Almost a similar formulationis developed in the study on null infinity [9]. InAppendix C, we present the details of the discussionassociated with gauge freedom.

II. BONDI-LIKE COORDINATEAND EINSTEIN EQUATIONS

In this section, introducing the Bondi-like coordinatenear the event horizon, we investigate the initialvalue problem. For the details of these derivations, seeAppendixes A and B.

A. Bondi-like coordinate near event horizon

We consider a dynamical black hole and investigate thelate-time behavior of the near horizon geometry at latetime. The black hole is defined as the region which is notcontained in J�ðIþÞ where Iþ is the future null infinity.J�ðSÞ is the chronological past connecting a set S by causal

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curves from S. The event horizon is the boundary of theblack hole and it is a null hypersurface. See Ref. [2] for theprecise definitions. Then, we introduce the Bondi-like(Gaussian null) coordinates xA ¼ ðu; r; xaÞ near the eventhorizon as

ds2¼gABdxAdxB

¼�Adu2þ2dudrþhabðdxaþUaduÞðdxbþUbduÞ;(1)

where u is a time coordinate. In this coordinate, the horizonposition is taken to be r ¼ 0.We assume that a cross sectionof the event horizon is compact in the u ¼ constant hyper-surface and its topology is S2. xa are coordinates on S2.Since the event horizon is a null hypersurface, guu mustvanish on the event horizon and l ¼ @=@u is the null geo-desic generator on the event horizon. Furthermore we canchoose the coordinate u so that la ¼ 0 on the event horizon.Then, we have

A¼̂0; (2)

and

Ua¼̂0; (3)

where ¼̂ means the evaluation on the event horizon r ¼ 0.To solve the vacuum Einstein equations, we formulate

the initial value problem in the Bondi coordinates. For theconvenience of our study in the following sections, wesolve the Einstein equations as the evolution equations inthe direction of r. Thus, the initial value of the metricshould be set on the r ¼ constant surface. In this paperwe take the event horizon r ¼ 0 as the initial surface.The evolution equations are given by

Rrr ¼ �12ðloghÞ00 � 1

4hachbdðhabÞ0ðhcdÞ0 ¼ 0; (4)

RrBhaB ¼ 1

2Ua00 þ 1

2hach0bcU

b0 þ 14ðloghÞ0Ua0

þ 12h

ab �Dc½h0bc � hbcðloghÞ0� ¼ 0; (5)

and

RABhAah

Bb ¼ � 1

2Ah00ab �

1

2A0h0ab � ð _habÞ0 þ ðhÞRab þ

A

2hcdh0ach0bd þ

1

2hcdðh0ac _hbd þ h0bd _hacÞ þLUh

0ab

� 1

2h0acð �DbU

c þ �DcUbÞ � 1

2h0bcð �DaU

c þ �DcUaÞ � 1

2hachbdU

c0Ud0 � 1

4½ _ðloghÞ � 2 �DaU

a�h0ab� 1

4ðloghÞ0ðAh0ab þ _hab � �DaUb � �DbUaÞ þ 1

2ðhbc �DaU

c0 þ hac �DbUc0 Þ ¼ 0; (6)

where the prime and dot denote the r and u derivative,respectively, �Da is a covariant derivative with hab, and h ¼dethab. Also,

ðhÞRab is the Ricci tensor with respect to hab.The evolutions of the metric functions A, Ua, and hab aredetermined by Eqs. (4)–(6) completely. Note that the tracepart of Eq. (6) gives us

A0ðloghÞ0 ¼2ðhÞR�A

2½ðloghÞ0�2�½ _ðloghÞ�2 �DaU

a�ðloghÞ0

�habUa0Ub0 �AðloghÞ00 �2 _ðloghÞ0

þ2ð �DaUaÞ0þUa �DaðloghÞ0; (7)

which is used as the evolution equation for A.The other components of the Einstein equations, Ruu ¼

0, Rur ¼ 0, and Rua ¼ 0, are related to the above evolutionequations by the Bianchi identities. Therefore, once theyare satisfied at the initial surface r ¼ 0, we do not need tosolve them any more. In fact, if the evolution equations aresatisfied, the Bianchi identities lead to

ð ffiffiffih

pRuaÞ0 ¼

ffiffiffih

pRur; ð ffiffiffi

hp

RruÞ0 ¼� ffiffiffi

hp

�DaRau� _ð ffiffiffi

hp ÞRur;

ðloghÞ0Rur¼0; (8)

where Rru ¼ Ruu þ ARur �UaRua and Ra

u ¼ �UaRurþhabRub. Thus, the evolution equations guarantee that

Ruu ¼ 0, Rur ¼ 0, and Rua ¼ 0 will always be satisfiedat r � 0 if Ruu¼̂0, Rur¼̂0, and Rua¼̂0 at the initial surfacer ¼ 0. For convenience, at r ¼ 0 we will use the followingconstraint equations:

Ruu¼̂ � 1

2ð €loghÞ þ A0

4_ðloghÞ � 1

4hachbd _hab _hcd ¼ 0; (9)

and

Rra¼̂ � 12ð _UaÞ0 � 1

4ð _loghÞUa0 � 12h

abUc0 _hbc � 12�DaA0

þ 12h

ab �Dc _hbc � 12�Dað _loghÞ ¼ 0 (10)

[see Eqs. (A30) and (A31)]. Moreover, since A shouldvanish on the initial surface r ¼ 0 [see Eq. (2)], theevolution equations Eq. (6) become singular on r ¼ 0.Analyticity of hab on r ¼ 0 gives us the regularitycondition

� 12A

0h0ab � _h0ab þ ðhÞRab þ 12h

cdðh0ac _hbd þ h0bd _hacÞ� 1

2hachbdUc0Ud0 � 1

4ð _loghÞh0ab � 14ðloghÞ0 _hab

þ 12ðhbc �DaU

c0 þ hac �DbUc0 Þ¼̂0: (11)

If we give habjr¼0 on r ¼ 0, we can determine h0abjr¼0,

Ua0jr¼0, and A0jr¼0 by solving Eqs. (9)–(11). As a result,

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we will solve the evolution equations (4)–(6) using theabove initial values on r ¼ 0.

B. Some explicit solutions near event horizon

In this subsection, it is shown that we explicitly solve theconstraint equations and evolution equations near the eventhorizon using power series expansion around r ¼ 0. To dothis, we expand the metric functions with r near the eventhorizon as

A ¼ rAð1Þ þXi�2

riAðiÞ; (12)

Ua ¼ rUð1Þa þXi�2

riUðiÞa; (13)

and

hab ¼ hð0Þab þ rhð1Þab þXi�2

rihðiÞab; (14)

where from the gauge conditions Eqs. (2) and (3), theexpansions of A and Ua start from the first order of r. In

the following, the tensor index of hðiÞab and UðiÞa is raised

and lowered by hð0Þab . The trace and traceless parts are also

taken by hð0Þab .

First let us solve the constraint equations and regularityconditions in order to determine initial values. On the

initial surface r ¼ 0, hð0Þab , hð1Þab , A

ð1Þ, and Uð1Þa should be

set on r ¼ 0 as initial values. The constraint equations forthese initial values are Ruu¼̂0 and Rra¼̂0. Now Ruu¼̂0[Eq. (9)] is rewritten as

€h ð0Þ � 12A

ð1Þ _hð0Þ þ 12h

ð0Þachð0Þbd _hð0Þab_hð0Þcd ¼ 0; (15)

where hð0Þ ¼ dethð0Þab . Thus, Að1Þ should be given to satisfy

this equation for given hð0Þab . Also, Rra¼̂0 [Eq. (10)]

becomes

_Uð1Þa ¼ �DaAð1Þ þ hð0ÞacDb _hð0Þbc �Da _ðloghð0ÞÞ� hð0ÞacUð1Þb _hð0Þbc � 1

2Uð1Þa _ðloghð0ÞÞ; (16)

where Da is the covariant derivative with respect to hð0Þab .

Then the initial value Uð1Þa is given to satisfy the above.The regularity condition Eq. (11) becomes

� _hð1Þab � 12A

ð1Þhð1Þab þ ðhð0ÞÞRab þ 12h

ð0Þcdðhð1Þac_hð0Þbd þ hð1Þbd

_hð0Þac Þ� 1

2hð0Þac h

ð0ÞbdU

ð1ÞcUð1Þd � 14

_ðloghð0ÞÞhð1Þab � 14h

ð0Þcdhð1Þcd_hð0Þab

þ 12ðDaU

ð1Þb þDbU

ð1Þa Þ ¼ 0; (17)

where ðhð0ÞÞRab is the Ricci tensor of hð0Þab . We obtain hð1Þab

satisfying this equation. Hence, if we give hð0Þab on the initial

surface, the constraint equations and the regularity condi-

tion yield hð1Þab , Að1Þ, and Uð1Þa.

Next we solve the evolution equations. Equation (4)become near the event horizon as

Rrr ¼ �hð0Þabhð2Þab þ 14ðhð1ÞabÞ2 þOðrÞ ¼ 0: (18)

This equation gives us the trace part of hð2Þab as

hð0Þabhð2Þab ¼ 14ðhð1ÞabÞ2: (19)

The evolution equation of Ua as RrBhaB ¼ 0 [Eq. (5)] can

be expanded as

RrBhaB ¼ Uð2Þa þ 1

2hð0Þachð1ÞbcU

ð1Þb þ 14U

ð1Þahð0Þbchð1Þbc

þ 12h

ð1ÞacDbðhð1Þbc � hð0Þbchð0Þdehð1Þde Þ þOðrÞ: (20)

Then Uð2Þa is given by

Uð2Þa ¼ �12h

ð0Þachð1ÞbcUð1Þb � 1

4Uð1Þahð0Þbchð1Þbc

� 12h

ð1ÞacDbðhð1Þbc � hð0Þbchð0Þdehð1Þde Þ: (21)

In the same way, expanding the evolution equations Eq. (6)near the event horizon, we can obtain the traceless part

hð2Þhabi and Að2Þ in terms of hð0Þab , h

ð1Þab ,U

ð1Þa, and Að1Þ. Note thatAð2Þ is given by the trace part of Eq. (6), namely, Eq. (7).However, we do not provide its explicit form because itsform is very cumbersome.

Hence, we can determine hð2Þab , Uð2Þa, and Að2Þ using the

evolution equations. To determine the higher order quan-

tities, hðiÞab, UðiÞa, and AðiÞ (i > 2), we have to repeat the

same procedure.

III. LATE-TIME SYMMETRYON AND NEAR EVENT HORIZON

In this section, we show that there is late-time symmetryon the event horizon. Then we will extend it outside ofblack hole regions.

A. Late-time behaviors on event horizon

To investigate late-time behaviors of the event horizon,we introduce the expansion and shear of the null geodesicgenerator l ¼ @=@u of the event horizon. The expansion �and shear �ab are defined as

�ab þ 12�h

ð0Þab¼̂ha

AhbBrAlB¼̂1

2_hð0Þab; (22)

where �ab is the traceless part of _hð0Þab with respect to hð0Þab .

Then we can rewrite Eq. (9), one of the constraint equa-tions, using � and �ab as

_�� 12A

ð1Þ� ¼ �12�

2 � �ab�ab: (23)

We can regard Að1Þ=2 as a surface gravity of the blackhole with respect to the time coordinate u because

lArAlB¼̂Að1Þ

2lB (24)

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holds. Using the affine parameter w defined by

dw

du¼ exp

�Z u Að1Þ

2du0

�; (25)

we can obtain the Raychaudhuri equation

@w�ðwÞ ¼ �12�

2ðwÞ � �ðwÞab�ab

ðwÞ; (26)

where �ðwÞ and �ðwÞab are expansion and shear with respectto w. We can see the relation between �, �ab and �ðwÞ,�ðwÞab as

� ¼ �ðwÞ exp�Z u Að1Þ

2du0

�;

�ab ¼ �ðwÞab exp�Z u Að1Þ

2du0

�: (27)

Here we remember that the area law of the event horizonholds for spacetimes satisfying the null energy condition,that is, � � 0 and �ðwÞ � 0. Since �ðwÞab�ab

ðwÞ � 0, Eq. (26)

implies the inequality

@w�ðwÞ þ 12�

2ðwÞ � 0: (28)

Then the integration over w gives us

�ðwÞ � 1

1=�ð0Þ þ ðw� w0Þ=2 ! 0 ðas w ! 1Þ; (29)

where we used the fact of �ð0Þ ¼ �ðwÞðw ¼ w0Þ � 0. Inaddition, Eq. (26) shows that the shear �ðwÞab should also

vanish as w ! 1. This is shown as a part of the proof ofanother theorem [10]. From now on, we assume thatw ! 1 corresponds to u ! 1. Then, we see that � and�ab should also vanish as u ! 1. It is natural to assumethat the cross section of the event horizon is compact. Thenthe vanishing of the expansion implies that the horizon areaapproaches a constant and finite value.

Altogether we see the behavior of the metric at latetime as

LlgABjhorizon ! 0 ðu ! 1Þ: (30)

Here we impose the following decaying condition on theevent horizon for the metric:

LlgAB¼̂O

�1

un

�; (31)

which explicitly means _hð0Þab ¼ Oðu�nÞ. This equation

means that the null geodesic generator of the event horizonl should be an asymptotic Killing vector at late time (u !1). Thus there is a late-time symmetry on the event horizon.

Since we consider the dynamics only near the eventhorizon, we cannot determine the decaying rate of themetric. However, the details of the decaying propertiesare not important for our argument; that is, our resultdoes not depend on n. Note that it will be determined by

the boundary conditions at a far region from the blackholes as in Refs. [6–8].It should be remembered that the horizon which satisfies

this condition is called a slowly evolving horizon inRefs. [11,12]. If the right-hand side of Eq. (31) vanishes,the event horizon will be identical to the isolated horizon[13].

B. Extension of late-time symmetry

The purpose of this subsection is to show that the decay-ing condition of Eq. (31) can be extended outside of theevent horizon as

LlgAB ¼ O

�1

un

�: (32)

In the following we assume the analyticity of gAB.Under the presence of the analyticity of spacetimes, the

above is equivalent with

ðLnÞmLlgAB¼̂O

�1

un

�; (33)

where n ¼ @=@r and m ¼ 0; 1; 2; � � � . Note that ‘‘¼̂’’means the evaluation on the event horizon (r ¼ 0) again.First we will show the m ¼ 1 case:

LnLlgAB¼̂O

�1

un

�: (34)

Substituting the explicit form of gAB to the above, werewrite Eq. (34) as

�LnLlðA�UaUaÞðduÞAðduÞB þ 2LnLlUaðduÞðAðdxaÞBÞ

þLnLlhabðdxaÞAðdxbÞB¼̂O

�1

un

�: (35)

Using Eqs. (12)–(14), the above will be equivalent with

LlAð1Þ ¼ O

�1

un

�; (36)

LlUð1Þa ¼ O

�1

un

�; (37)

and

L lhð1Þab ¼ O

�1

un

�: (38)

Let us examine these conditions. First, we focus onEq. (36). As a result, using the gauge freedom, we can

show that the slightly strong condition, LlAð1Þ ¼

Oð1=unþ1Þ, holds. To see this, we decompose Að1Þ intothe u-independent term and others as

Að1Þ ¼ Að1Þ0 ðxaÞ þ ~Að1Þðu; xaÞ: (39)

As shown in Ref. [14], we can choose the gauge so that Að1Þ0

is a constant. Furthermore, using the residual gauge, we

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can take ~Að1Þ ¼ Oð1=unÞ. Then, using the gauge freedom in

our coordinate, we can make Að1Þ satisfy stronger decayingproperty as

L lAð1Þ ¼ O

�1

unþ1

�: (40)

For the details, see Appendix C. Of course, Eq. (36) issatisfied.

Now Uð1Þa should satisfy the constraint equation ofEq. (10) as

LlUð1Þa ¼ �Da ~Að1Þ þ hð0ÞacDb _hð0Þbc �Da _ðloghð0ÞÞ

� hð0ÞacUð1Þb _hð0Þbc � 12U

ð1Þa _ðloghð0ÞÞ: (41)

Together with Eqs. (31) and (40), we can see that Eq. (37)holds from the above.

Furthermore, hð1Þab satisfies the following equation [see

Eq. (17)] as a regularity condition:

_hð1Þab þ 12A

ð1Þhð1Þab ¼ ðhð0ÞÞRab þ 12h

ð0Þcdðhð1Þac_hð0Þbd þ hð1Þbd

_hð0Þac Þ� 1

2hð0Þach

ð0ÞbdU

ð1ÞcUð1Þd � 14

_ðloghð0ÞÞhð1Þab

� 14h

ð0Þcdhð1Þcd_hð0Þab þ 1

2ðDaUð1Þb þDbU

ð1Þa Þ:(42)

Then multiplying Ll to the above and using Eq. (16), wecan see

12A

ð1ÞLlhð1Þab ¼ Lðhð0ÞÞ

l Rab þOðu�nÞ (43)

holds. In the above, the higher-order terms like L2l h

ð1Þab ,

ðLlhð0ÞabÞ2, ðLlh

ð1ÞabÞ2, Llh

ð0ÞabLlh

ð1Þab , and so on are contained

in Oðu�nÞ. Thus Eq. (31) tells us that Llhð1Þab ¼ Oðu�nÞ

holds. As a consequence, we can show the m ¼ 1 case ofEq. (33).

For the m> 1 cases of Eq. (33), we perform the sameprocedure inductively. Letm> 1 be an integer and assumethat the metric satisfies

ðLnÞkLlgAB¼̂O

�1

un

�(44)

for all kð<mÞ. Then ðLnÞmLlgAB¼̂Oðu�nÞ is equivalentwith

LlAðmÞ ¼ O

�1

un

�; (45)

LlUðmÞa ¼ O

�1

un

�; (46)

and

L lhðmÞab ¼ O

�1

un

�: (47)

Since UðmÞa is written as Uð1Þa, Að1Þ; hð0Þab; � � � ; hðm�1Þab in

Eq. (5) like Eq. (21), the induction assumption

LlhðiÞab ¼ O

�1

un

�ði � m� 1Þ (48)

immediately shows us that Eq. (46) holds.

From Eq. (7), AðmÞ is written asUð1Þ, Að1Þ, hðiÞabði < mÞ andthe trace part of hðmÞ

ab . Since the trace part of hðmÞab is written

as Uð1Þ, Að1Þ, and hðiÞab (i < m) through Eq. (4), AðmÞ is

written only as Uð1Þ, Að1Þ, hðiÞabði < mÞ in the end. Then, by

the assumption of the induction, Eq. (45) holds. Next, the

evolution equation for hðmÞab is described by Eq. (6).

Expanding Eq. (6) near the event horizon and multiplying,ðLnÞm�1, Eq. (6) becomes the following form on the eventhorizon:

_h ðmÞab þ 1

2Að1ÞhðmÞ

ab ¼ Fab; (49)

where Fab is a function of hðiÞabði � mÞ, _hðiÞabði < mÞ, and so

on. Applying Ll to Eq. (49), then, we can see that

ð1=2ÞAð1Þ _hðmÞab ¼ _Fab � _hðjÞab ¼ Oð1=unÞ for j < m. Thus

Eq. (47) holds.Now we can confirm that the induction loop is closed.

Then we can show that Eq. (33), equivalently Eq. (32),holds if the spacetime is real analytic. Therefore we canshow that the null geodesic generator l of the event horizonis the asymptotic Killing vector at late time in the sense ofEq. (32).

IV. SUMMARYAND DISCUSSION

We have confirmed that the expansion and shear of theevent horizon should decay at late time in the vacuumspacetimes. Then, assuming the compactness of the crosssections of the event horizon, the null geodesic generatorson the horizon give us an asymptotic Killing vector l at latetime. This means that the horizon has late-time symmetry.By solving the Einstein equations, then, we have found thatthis late-time symmetry can be extended outside of theblack holes. Therefore, at late time, there is asymptoticsymmetry outside of black holes.If the black hole rotates and there is another asymptotic

Killing vector at late time, k, which will be a stationaryKilling vector at a far distance and spacelike near thehorizon, k� l is also an asymptotic Killing vector ex-pected to correspond to axisymmetry. In this sense, wewould expect that the rigidity holds in gravitational col-lapse at late time. In these discussions, we assume thecompactness of the event horizon. Thus this result cannotbe applied to other null hypersurfaces which do not have acompact cross section.There is a remaining issue. In the proof of the rigidity

theorem, the exact stationarity does show that the nullgeodesic generators of the horizon are a Killing orbit. Onthe other hand, our argument could show us that the null

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geodesic generators of the horizon are a Killing orbitwithout assuming the presence of asymptotically station-ary Killing vectors. It is likely that this difference suggeststhe existence of important and new points in black holephysics.

ACKNOWLEDGMENTS

K.T. is supported by JSPS Grant-in-Aid for ScientificResearch (No. 21-2105). T. S. is supported by Grant-in-Aidfor Scientific Research from the Ministry of Education,Science, Sports and Culture of Japan (No. 21244033,No. 21111006, No. 20540258, and No. 19GS0219). Thiswork is also supported by the Grant-in-Aid for the GlobalCOE Program ‘‘The Next Generation of Physics, Spunfrom Universality and Emergence’’ from the Ministry ofEducation, Culture, Sports, Science and Technology(MEXT) of Japan.

APPENDIX A: EINSTEIN EQUATIONSNEAR EVENT HORIZON

In Appendix A, using the suitable variables, we willshow the derivations of the Einstein equations near theevent horizon.

1. (nþ 1) decomposition

First we describe the formula of the (nþ 1) decompo-sition. The metric can be written as

gAB ¼ �nAnB þ �AB; (A1)

where �AB is an n-dimensional induced metric. nA is theunit normal vector with nAn

A ¼ � ¼ þ1 (nA: spacelike) or�1 (nA: timelike).

We define the extrinsic curvature as

KAB ¼ 12Ln�AB: (A2)

Now nA can be written as nA ¼ �Nðd�ÞA, where � is afunction which describes the hypersurface as � ¼ const,and N is the ‘‘lapse’’ function. Then the Riemann tensor isdecomposed into

REFGH�AE�B

F�CG�D

H¼ð�ÞRABCD��KACKBDþ�KADKBC;

(A3)

REFGD�AE�B

F�CGnD ¼ rAKBC �rBKAC; (A4)

RACBDnCnD¼�LnKABþKACKB

C��1

NrArBN; (A5)

where rA denotes the covariant derivative with respect to�AB.

The Ricci tensor is decomposed into

RABnAnB ¼ �LnK � KABK

AB � �1

Nr2N; (A6)

RACnA�B

C ¼ rAKAB �rBK; (A7)

RCD�AC�B

D ¼ ð�ÞRAB � �LnKAB � �KKAB

þ 2�KACKBC � 1

NrArBN: (A8)

The Ricci scalar is written as

R ¼ ð�ÞR� 2�LnK � �K2 � �KABKAB � 2

Nr2N

¼ ð�ÞRþ �K2 � �KABKAB � 2

Nr2N � 2�rAðKnAÞ:

(A9)

The components of the Einstein tensor are

GABnAnB ¼ 1

2ð��ð�ÞRþ K2 � KABKABÞ; (A10)

GACnA�B

C ¼ rAKAB �rBK; (A11)

GCD�AC�B

D ¼ ð�ÞGAB � �KKAB þ 2�KACKBC

þ �

2�ABðKCDK

CD þ K2Þ � �LnKAB

þ ��ABLnK � 1

NrArBN þ 1

N�ABr2N:

(A12)

2. Einstein equations

We apply the (nþ 1) decomposition presented in theprevious section to the r-constant surface in our currentfour-dimensional metric form

ds2 ¼ �Adu2 þ 2dudrþ habðdxa þUaduÞðdxb þUbduÞ:(A13)

We express the above in the following form:

ds2 ¼ N2dr2 þ q��ðdx� þ N�drÞðdx� þ N�drÞ; (A14)

where

N2 ¼ 1

A; (A15)

Nu ¼ � 1

A; (A16)

Na ¼ 1

AUa: (A17)

q�� is the induced metric on the r-const. hypersurface as

q�� ¼ �AþUaUa Ub

Ua hab

!: (A18)

Note that the Latin indices a; b; . . . and the Greekindices �; �; . . . are raised and lowered by hab and q��,

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respectively. The unit normal vector to the r-const. hyper-surface is given by mA ¼ NðdrÞA and mA ¼ N�1ð@r �N�@�ÞA.

The extrinsic curvature on the r-const. hypersurface isdefined as

K��¼1

2Lmq��¼ 1

2Nð@rq���D�N��D�N�Þ; (A19)

where D� is the covariant derivative with respect to q��.

We rewrite the induced metric as

q��dx�dx� ¼ ��2du2 þ habðdxa þ �aduÞðdxb þ �bduÞ;

(A20)

where

�2 ¼ A; �a ¼ Ua: (A21)

The timelike unit vector to the u-const. surface is given byu ¼ ��du and u� ¼ ��1ð@u � �a@aÞ�.

The extrinsic curvature on the u-const. surface is givenby

kab ¼ 1

2Luhab ¼ 1

2�ð@uhab � �Da�b � �Db�aÞ; (A22)

where �Da is the covariant derivative with respect to hab.We introduce the following quantities for later conve-

nience:

� � Ku�u�u� ¼ � 1

N@rðlog�Þ �Lu logN; (A23)

a � Ka�u

� ¼ 12@r�

a þ �Da log�; (A24)

ab � Kcdhachb

d ¼ �

2@rhab þ kab; (A25)

and

¼ abhab ¼ �

2ðloghÞ0 þ k; (A26)

where �u� ¼ 0 ¼ ��u

�, h ¼ dethab, and k ¼ kabhab.

The prime denotes the r differentiation. Then K�� can be

written as

K�� ¼ �u�u� � 2ð�u�Þ þ ��: (A27)

Using these quantities, we can decompose the four-dimensional Ricci tensor RAB into the quantities on two-dimensional space:

RABmAmB ¼ 1

Nð�� Þ0 þLuð�� Þ ��2

þ 2aa � ab

ab þ 1

NðLuLuN

þ kLuN � �D2N � �DaN �Da log�Þ; (A28)

RABmAqBCu

C ¼ �Luþ �Daa � kabab

þ 2a �Da log���k; (A29)

RABqACq

BDu

CuD¼� 1

N�0 �Lu�þ�2���2aa

�2a �Da logN

�þ �Da log� �Da logN

þ 1

��D2��Luk�kabk

ab� 1

NLuLuN;

(A30)

RABmAhBa¼� �Da log��2bk

ab�kaþ �Dbab� �Da

þ �Da�þab �Db log�� 1

�ð@ua�L�

aÞ;(A31)

RABqACu

ChBb¼ �Dakab� �Dbk�b�2a

ab� 1

N�DbLuN

þkab �Da logN� 1

NðbÞ0�� �Db log

N

�

�ab �Da logN

�� 1

�ð@ub�L�

bÞ;(A32)

and

RABhAah

Bb¼ðhÞRabþLukabþkkab�2kackb

c� 1

��Da

�Db�

� 1

N0ab�

1

�_abþ 1

�L�ab�2ða �DbÞ log

N

�

þð��Þab�2abþ2accb�

1

N�Da

�DbN

þkabLu logN; (A33)

where the dot denotes @u. The vacuum Einstein equation isgiven by RAB ¼ 0.

APPENDIX B: EXPLICIT FORMOF EINSTEIN EQUATIONS

In Appendix B, we describe the components of theEinstein equations in terms of the metric functionsexplicitly.Using

u ¼ ��1ðl�Ua@aÞ ¼ ��1ð@u �Ua@aÞ; (B1)

� ¼ �ðA1=2Þ0 þ 12Lu logA; (B2)

a ¼ 12U

a0 þ 12�Da logA; (B3)

and

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ab ¼ A1=2

2h0ab þ

1

2A1=2ð _hab � �DaUb � �DbUaÞ; (B4)

we can rewrite the Einstein equation RAB ¼ 0 in terms of our metric form. We will not provide all components of theEinstein equation explicitly.

For example, RABhAah

Bb ¼ 0 becomes

Llh0ab þ

1

2A0h0ab ¼ ðhÞRab þ

A

2hcdh0ach0bd þ

1

2hcdðh0ac _hbd þ h0bd _hacÞ � 1

2h0acð �DbU

c þ �DcUbÞ þLUh0ab

� 1

2h0bcð �DaU

c þ �DcUaÞ � 1

2hachbdU

c0Ud0 � 1

4½ð _loghÞ � 2 �DaU

a�h0ab� 1

4ðloghÞ0ðAh0ab þ _hab � �DaUb � �DbUaÞ � 1

2Ah00ab þ

1

2ðhbc �DaU

c0 þ hac �DbUc0 Þ: (B5)

This determines the evolutions of hab. For RABmAhBa ¼ 0, we have

LlUa0 ¼ � �DaA0 � hacUb0 ð _hbc � �DaUb � �DbUaÞ þ hab �DcðAh0bc þ _hbc � �DbUc � �DcUbÞ

� A �DaðloghÞ0 � 12U

a0 ½ð _loghÞ � 2 �DaUa� � 1

2ðloghÞ0 �DaA� �Da½ð _loghÞ � 2 �DbUb� þLUU

a0 : (B6)

APPENDIX C: THE GAUGEISSUE FOR Að1Þ [EQ. (40)]

In Appendix C we will show the presence of the gaugewhere Eq. (40) is satisfied. In our coordinate xA ¼ðu; r; xaÞ, the metric can be written as

ds2¼�Adu2þ2dudrþhabðdxaþUaduÞðdxbþUbduÞ:(C1)

The components of the metric are expanded near the eventhorizon (r ¼ 0) as

A ¼ rAð1Þ þOðr2Þ; (C2)

Ua ¼ rUð1Þa þOðr2Þ; (C3)

and

hab ¼ hð0Þab þOðrÞ: (C4)

Here Að1Þ can be decomposed as

Að1Þ ¼ Að1Þ0 þ ~Að1Þðu; xaÞ; (C5)

where Að1Þ0 is set to be a constant as shown in Ref. [14].

When we consider the gauge transformation xA ! xA þ�A, the metric is transformed as

gAB ! gAB þL�gAB � gAB þ �gAB: (C6)

To keep our gauge, the following conditions will beimposed:

�gur ¼ 0; �grr ¼ 0; �gra ¼ 0; �guu ¼OðrÞ;�gua ¼OðrÞ; �gab ¼Oð1Þ: (C7)

From �grr ¼ 2@r�u ¼ 0, we have �u ¼ fðu; xaÞ. Since

�gra and �gur are given by

�gra ¼ @a�u þUa@r�

u þ hab@r�b; (C8)

�gur¼ð�AþUaUaÞ@r�uþ@r�rþ@u�

uþUa@r�a; (C9)

�gra ¼ 0 and �gur ¼ 0 lead to

�r ¼ �r@ufþ @afZ r

Uadr0;

�a ¼ �@bfZ r

habdr0: (C10)

Then �guu becomes

�guu ¼ r½�@uðfAð1ÞÞ � @2uf� þOðr2Þ; (C11)

where we used Eqs. (C2) and (C3). This means that Að1Þ istransformed under the gauge transformation as

Að1Þ ! Að1Þ þ @uðfAð1ÞÞ þ @2uf: (C12)

Thus if we choose f as

f ¼ � 1

Að1Þ0

Z udu0 ~Að1Þ; (C13)

Að1Þ is transformed as

Að1Þ ! �Að1Þ ¼ Að1Þ0 � 1

Að1Þ0

½@u ~Að1Þ þ @uðf ~Að1ÞÞ�: (C14)

Let assume that ~Að1Þ decays as u ! 1. For the moment,

we write ~Að1Þ ¼ Oð1=umÞ, where m is an integer. If m � n,Eq. (40) is already satisfied. Therefore, we suppose that mis smaller than n.

In the current gauge transformation, the transformed �Að1Þsatisfies

�A ð1Þ ¼ Að1Þ0 þOð1=umþ1Þ: (C15)

Repeating this procedure, we can always choose the gaugesatisfying

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LlAð1Þ ¼ O

�1

unþ1

�: (C16)

If one wishes, one can continue the same procedure and then achieve an arbitrary faster decaying rate. But the above isenough for our current purpose.

[1] W. Israel, Phys. Rev. 164, 1776 (1967); B. Carter, Phys.Rev. Lett. 26, 331 (1971); S.W. Hawking, Commun.Math. Phys. 25, 152 (1972); D. C. Robinson, Phys. Rev.Lett. 34, 905 (1975); P. O. Mazur, J. Phys. A 15, 3173(1982); for a review, see M. Heusler, Black HoleUniqueness Theorems (Cambridge University Press,Cambridge, England, 1996); P. O. Mazur, arXiv:hep-th/0101012; G. L. Bunting, Ph. D. thesis, University of NewEngland, Armidale, 1983.

[2] S.W. Hawking and G. F. R. Ellis, The Large ScaleStructure of Space-Time (Cambridge University Press,Cambridge, England, 1973).

[3] S.W. Hawking, Commun. Math. Phys. 25, 152 (1972).[4] H. Friedrich, I. Racz, and R.M. Wald, Commun. Math.

Phys. 204, 691 (1999).[5] I. Racz, Classical Quantum Gravity 17, 153 (2000).

[6] C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49,883 (1994).

[7] L. Barack and A. Ori, Phys. Rev. Lett. 82, 4388 (1999).[8] L. Barack and A. Ori, Phys. Rev. D 60, 124005 (1999).[9] K. Tanabe, S. Kinoshita, and T. Shiromizu, Phys. Rev. D

84, 044055 (2011).[10] S. A. Hayward, T. Shiromizu, and K.-i. Nakao, Phys. Rev.

D 49, 5080 (1994).[11] I. Booth and S. Fairhurst, Phys. Rev. Lett. 92, 011102

(2004).[12] I. Booth and S. Fairhurst, Phys. Rev. D 75, 084019 (2007).[13] A. Ashtekar and B. Krishnan, Living Rev. Relativity 7, 10

(2004) [http://relativity.livingreviews.org/Articles/lrr-2004-10/].

[14] S. Hollands, A. Ishibashi, and R.M. Wald, Commun.Math. Phys. 271, 699 (2007).

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