Harrison metrics for the Schwarzschild black hole

0.14 MB·English

by  Fang-Fang Yuan

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1 Harrison metrics for the Schwarzschild black hole
0
2

n
Fang-FangYuan
a
J Institute of Theoretical Physics, Beijing Universityof Technology, Beijing 100124, China

8 ff[email protected]
2

Yong-ChangHuang
]
h Institute of Theoretical Physics, Beijing Universityof Technology, Beijing 100124, China
t
- [email protected]
p
e
h
Basedonthehiddenconformalsymmetry,someauthorshaveproposedaHarrisonmetric
[
for the Schwarzschild black hole. We give a procedure which can generate a family of
1 Harrisonmetrics starting fromageneral set of SL(2,R)vector fields. Byanalogy with
v the subtracted geometry of the Kerr black hole, we find a new Harrison metric for
8 the Schwarzschild case. Its conformal generators are also investigated using the Killing
4 equations inthenear-horizonlimit.
5
6
. 1. Introduction
1
0 The Kerr/CFT correspondence1 has attracted a lot of attention in recent years.2
3
Compared to the extremal cases, the non-extremal black holes are still difficult
1
: to tackle with using the conformal field theory or related techniques. However, a
v
hidden conformal symmetry has been discovered in the near-region, low-frequency
i
X limit of the Kerr black hole.3 It is related to the scalar wave equation rather than
r the symmetry structure of spacetime geometry. This powerful approach has been
a
applied to many different black holes.
The hidden conformal symmetry of the Schwarzschild black hole was investi-
gated in Ref. 4. In contrast with most black holes, only one set of the SL(2,R)
generators could be defined. By applying an SU(2,1) transformation, i.e., a Harri-
sonor Kinnersley transformation,5,6 onthe Schwarzschildblack hole,a new metric
was found which has the same structure as the Bertotti-Robinson spacetime. Its
Killing vectors were shown to reproduce the hidden conformal symmetry genera-
tors. The quasinormal modes were also studied using the operator method in Ref.
7. For subsequent developments, see Refs. 8–13.
Through an interesting observation on the separability of the wave equations,
the authors of Refs. 14 and 15 have uncovered a possible geometric realization of
the hidden conformal symmetry. This subtracted geometry involves a change of
the warp factor in the metric. Although the asymptotic structure is modified, the
thermodynamicsoftheblackholeisstillpreserved(apreviousstudyinRef.16may
beusefulonthispoint).AsexplicitlyshowninRef.17,thesubtractedgeometrycan
1
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also be obtained through a Harrison transformation on the original metric. Recent
discussions of the related topics can be found in Refs. 18–22.
Inspiredbytheabovediscoveries,onemayattempttofindaconstructivemethod
which could realize the hidden conformal symmetry through the Harrison metrics.
The resulting metrics may also be interesting in their own right. As an elementary
investigation along this direction, we think it would be valuable and reasonable to
focus on the Schwarzschildblack hole at first.
The layout of this paper is as follows. In the next section, the relevant results
of the Harrison metric proposed in Ref. 4 are reviewed. In Sec. 3, based on a cor-
respondence between the expansion of the Killing vector and the hidden conformal
symmetrygenerators,wegiveanalternativeproceduretoobtainthismetric.InSec.
4,the subtractedgeometryofthe Kerrblackhole is invokedto find a new Harrison
metric for the Schwarzschild case which still preserves the black hole thermody-
namics. In Sec. 5, we discuss the conformal generators for this new metric through
the Killing (orKilling’s)equationsinthe near-horizonlimit. The conclusioncanbe
found in Sec. 6.
2. The Harrison metric from hidden conformal symmetry
In this section, we review the hidden conformal symmetry generators for
Schwarzschildblack hole and the Harrison metric obtained in Ref. 4.
The Schwarzschild black hole is given by the metric
∆ r2
ds2 = dt2+ dr2+r2(dθ2+sin2θdφ2), (1)
−r2 ∆
∆= r(r r ), (2)
+
−
where the horizon is at r =2M. One can easily find the Laplacian as follows
+
2 = 1 ∂ (√ ggµν∂ )
µ ν
∇ √ g −
−
1 1 1
= [ 2+ ∂ sinθ∂ + ∂2 ], (3)
r2 ∇ sinθ θ θ sin2θ φ
r4
2 =∂ ∆∂ ∂2. (4)
∇ r r − ∆ t
FollowingRef.3,toinvestigatethe hiddenconformalsymmetrywe haveto take
the near-region, low frequency limit rω 1, r ω 1. Here ω is the frequency in
+
≪ ≪
the wave function ansatz Φ(t,r,θ,φ) = e iωtR(r)Yl(θ,φ). Roughly speaking, all
− m
we need here is the correspondence ∂ iω.
t
→−
AsshowninRef.4,intheabovelimitonecanregardthe resultingradialLapla-
cian 2 as SL(2,R) quadratic Casimir. Explicitly, we have
∇
1
2 = (H H +H H ) H2
H 2 1 −1 −1 1 − 0
r4
= ∂ ∆∂ +∂2. (5)
r r− ∆ t
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Note that as in the literature the radialLaplacianhere refers to 2 while the warp
∇
factor part 1 has been discarded. This is essentially because the latter plays no
r2
role in the radial wave equation
2R=l(l+1)R. (6)
∇
Now we can define the following vector fields as hidden conformal symmetry
generators
H1 = ie4Mt √∆∂r 4M(r−M)∂t ,
− √∆
(cid:18) (cid:19)
H = i4M∂ , (7)
0 t
−
H 1 = ie−4Mt √∆∂r+ 4M(r−M)∂t ,
− − √∆
(cid:18) (cid:19)
which satisfy the SL(2,R) commutation relations
[H ,H ]= iH , [H ,H ]=2iH . (8)
1 0 1 1 1 0
± ± ± −
If one naively uses the general formulas in Ref. 23 and inserts the following
parameters for Schwarzschildblack hole
1
T = , T =0, (9)
L R
8πM
1
n = 0, n = , (10)
L R
−8M
δ = r 2M, δ =r, (11)
+
− −
1
A= n T n T = , (12)
L R− R L 64πM2
then one would also obtain the other set of generators
4M2
H =i √∆∂ + ∂ ,
1 r t
√∆
(cid:18) (cid:19)
H =0, (13)
0
4M2
H = i √∆∂ ∂ .
1 r t
− − − √∆
(cid:18) (cid:19)
They obviously do not obey the analogous commutation relations as in Eq. (8),
and the radialLaplacian could not be reproduced. In this paper, we will follow the
interpretation that the Schwarzschild black hole should be described by a chiral
conformal field theory (see Ref. 4).
By applying some Harrison transformations on the Schwarzschild metric, the
authors of Ref. 4 found a new metric. It has a similar structure to the AdS S2
2
×
Bertotti-Robinson spacetime, and the explicit form is
r(r 2M) M2
ds2 = −e−2x M−2 dt2+e2xr(r 2M)dr2+e2xM2(dθ2+sin2θdφ2).(14)
−
In order to retain the entropy of the original Schwarzschild black hole, one must
set ex = 2. As pointed out there, the resulting Killing vectors of the AdS factor
2
exactly reproduce the hidden conformal symmetry generators in Eq. (7).
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3. Harrison metric and Killing vector
With the intention to generalize the observation in Ref. 4 to other black holes,
we propose an alternative procedure which permits us to determine the Harrison
metric(s) from the hidden conformal symmetry generators.
Let us start with the following set of SL(2,R) generators
H =ieαt+βφ(A∂ +B∂ +C∂ ),
1 r t φ
i
H = ∂ , (15)
0 t
−α
H =ie αt βφ( A∂ +B∂ +C∂ ).
1 − − r t φ
− −
Here α,β are constants, A,B,C are functions of the parameter r, and they all
depend onthe blackhole charges.Note that forgeneralblack holesone wouldhave
another set of generators H ,H ,H , and H may acquire an extra term D∂ .
1 0 1 0 φ
{ − }
However, the calculation is quite similar as below.
We assume that the expansion of the Killing vector has a correspondence with
the conformal generators as follows
iξ =i(ξr∂ +ξt∂ +ξφ∂ )
r t φ
=aH +bH +cH , (16)
1 1 0
−
where a,b,c are some irrelevant constants. Then we can obtain
ξr =A(aM bN), (17)
−
c
ξt =B(aM +bN) , (18)
− α
ξφ =C(aM +bN). (19)
Here M eαt+βφ,N e αt βφ.
− −
≡ ≡
Using the Killing (or Killing’s) equation
ξρ∂ g +∂ ξρg +∂ ξρg =0, (20)
ρ µν µ ρν ν ρµ
and treating the metric components g ,g ,g ,g as unknown variables, we
tt rr φφ tφ
{ }
arrive at the following equation set
A∂ g +2αBg +2αCg =0, (21)
r tt tt tφ
A∂ g +2∂ Ag =0, (22)
r rr r rr
A∂ g +2βCg +2βBg =0, (23)
r φφ φφ tφ
A∂ g +(αB+βC)g +βBg +αCg =0, (24)
r tφ tφ tt φφ
αAg +∂ Bg +∂ Cg =0, (25)
rr r tt r tφ
βAg +∂ Cg +∂ Bg =0. (26)
rr r φφ r tφ
ThankstothebeautifulstructureofEq.(15),eachequationhereinvolvesacommon
factorwitha,bwhichhasbeendiscarded.IfoneusestheconformalKillingequation
instead, these unwanted constants will appear on both sides of the equations.
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Since the first partial derivatives of the metric components are involved, we
would have four integration constants, two of which can be fixed by the last two
equations. This means that we will obtain a family of Harrison metrics for any
specific black hole. By requiring the angular part of the metric structure to be
preserved, one can read off the component g . Furthermore, the thermodynamic
θθ
constraints will determine some integration constants. At the end, the resulting
Harrison metric may still not be unique.
Using the hidden conformal symmetry generators of the Schwarzschild black
hole given in Eq. (7), the Killing equations are obtained as
2(r M)
√∆∂ g − g = 0, (27)
r tt tt
− √∆
√∆∂ g +2∂ √∆g = 0, (28)
r rr r rr
∂ g = 0, (29)
r φφ
√∆ r M
g 4M∂ − g = 0. (30)
rr r tt
4M − √∆
The solutions can be easily found to be
K K
1 1
g = , g = ∆, g =K . (31)
rr ∆ tt −16M4 φφ 2
Note that one of the three integration constants has been fixed by the constraint
equation (30).
To preserve the angular part of the metric structure, we obtain a constraint as
g =g sin2θ.Ontheotherhand,forthetemperatureandentropytoberetained:
φφ θθ
T = 1 ,S = Area =4πM2, we must have
H 8πM 4
K =4M2, K =4M2sin2θ. (32)
1 2
Inthisspecialcase,therequirementabouttheentropyisequivalenttothecondition:
√ g =√ g . Here g is the determinant of the original metric in Eq. (1).
− − 0|r+ 0
Thus we have reproduced the Harrison metric in Eq. (14) with ex = 2. More
explicitly, this unique metric can be written as
r(r 2M) 4M2
ds2 = − dt2+ dr2+4M2(dθ2+sin2θdφ2). (33)
− 4M2 r(r 2M)
−
4. A new Harrison metric from subtracted geometry
Although the subtracted geometry of general asymptotically flat black holes in
four dimensions has been studied in Ref. 15, here we only need to extract the
corresponding results for the Kerr black hole.
Using the conventions similar to those in Ref. 15, the metric of the Kerr black
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hole can be written as follows
1 2Mr dr2 X
ds2 = G(dt+ asin2θdφ)2+ Σ ( +dθ2+ sin2θdφ2), (34)
0
−√Σ G X G
0
X =(r r )(r r )=r2 2Mr+a2p, (35)
+
− − − −
G=X a2sin2θ, (36)
−
Σ =(r2+a2cos2θ)2. (37)
0
From this, one can derive the Laplacian
2 = 1 [ 2+ 1 ∂ sinθ∂ + 1 ∂2 ], (38)
∇ √Σ ∇ sinθ θ θ sin2θ φ
0
(2Mr∂ +a∂ )2 (2Mr)2 Σ
2 = ∂ X∂ t φ + − 0∂2. (39)
∇ r r− X G t
The(minimally)subtractedgeometrycorrespondstothefollowingchangeofthe
warp factor
Σ Σ=4M2(2Mr a2cos2θ). (40)
0
→ −
Now one has the radial Laplacian given by
(2Mr∂ +a∂ )2
2 =∂ X∂ t φ +4M2∂2, (41)
∇ r r− X t
which is exactly the same as the original work on hidden conformal symmetry.3
If one sets the angular momentum a to zero, the Kerr metric (34) naturally
reduces to that of the Schwarzschildblack hole in Eq.(1). When the warpfactor is
changed as Σ Σ=8M3r, we obtain a new Harrison metric as follows
0
→
r(r 2M) 2M√2Mr
ds2 = − dt2+ dr2+2M√2Mr(dθ2+sin2θdφ2). (42)
−2M√2Mr r(r 2M)
−
The corresponding radial Laplacian is
8M3r
2 =∂ ∆∂ ∂2, (43)
∇ r r− ∆ t
where ∆=r(r 2M).
−
One can of course define ρ √2Mr, and write the metric (42) in another form
≡
ρ(ρ2 4M2) 8Mρ
ds2 = − dt2+ dρ2+2Mρ(dθ2+sin2θdφ2). (44)
− 8M3 ρ2 4M2
−
The temperature and entropy can be checked to be the same as the original
Schwarzschildblack hole.
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5. Conformal generators from Killing equations
In this section, we derive the (pseudo-)conformal generators for the new Harrison
metric giveninEq.(42). This heavilyreliesonthe relevantKillingequationsin the
near-horizonlimit.
The following procedure can be regarded as an updated or simplified versionof
that in Ref. 8. However, some comments on the method given there are in order.
Firstly,onecanapplytheKillingequationdirectlyratherthantheconformalKilling
equation. Secondly, it is more convenient to use the Killing vectors ξr,ξt rather
thanξ =g ξr,ξ =g ξt.Alsointhiswayonedoesnotneedtoinvokethespecific
r rr t tt
rescaling at the end of that paper.
We firstly use the Killing equationin Eq. (20) to obtainthe following equations
ξr∂ g +2∂ ξtg =0, (45)
r tt t tt
ξr∂ g +2∂ ξrg =0, (46)
r rr r rr
∂ ξrg +∂ ξtg =0. (47)
t rr r tt
Note that the angular part of the metric is irrelevant as in Ref. 8. Inserting the
parameters given in the metric (42), we find the explicit expressions as
4∆∂ ξt+(3r 2M)ξr = 0, (48)
t
−
4∆∂ ξr (3r 2M)ξr = 0, (49)
r
− −
∆2∂ ξt 8M3r∂ ξr = 0. (50)
r t
−
The integration of Eq. (49) leads to
ξr =A(t) r−14√∆. (51)
By inserting it in Eqs. (48) and (50), we have
3r 2M
∂ ξt = A − , (52)
t − 4r14√∆
∂ ξt = ∂ A 8M3r34. (53)
r t ∆32
With these twoequations combinedtogether andnoticing the fact∂ ∂ ξt =∂ ∂ ξt,
r t t r
we arrive at
∂2A λA=0, (54)
t −
3r2 4Mr+12M2
λ= e −128M3r . (55)
From Eq. (52), we can writee
3r 2M t
ξt =− 4r−14√∆ dt′A(t′)+B(r). (56)
Z
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ByinspectionofEq.(54),wehave tdtA(t)= ∂tA.Sotakingthepartialderivative
′ ′ e
λ
of the above equation leads to
R
3r2 4Mr+12M2∂ A
∂ ξt = − t +∂ B(r). (57)
r 16r41∆32 λ r
Recall Eqs. (53) and (55), then we observe that ∂ B(r)=0. This means
er
B(r)=K′, (58)
which is a constant.
Following Ref. 8, let us take the near-horizonlimit λ λ . Then the solution
≡ |r+
of Eq. (54) can be found to be
e
A(t)=αe√λt+βe−√λt, (59)
where α,β are integration constants, and √λ = 1 which is equal to the surface
4M
gravity of the Schwarzschild black hole. From the expressions in Eqs. (51), (56),
(58) and (59), we obtain the Killing vectors as follows
ξr =r−14√∆ (αe√λt+βe−√λt), (60)
3r 2M 1
ξt =− 4r−14√∆ √λ(αe√λt−βe−√λt)+K′. (61)
Using a correspondence similar to Eq. (16), we finally arrive at the following
pseudo-conformal generators for the new Harrison metric
H1 = ie4Mt r−41 √∆∂r M(3r−2M)∂t ,
− √∆
(cid:18) (cid:19)
H = K∂ = i4M∂ , (62)
0 t t
−
H 1 = ie−4Mt r14 √∆∂r+ M(3r−2M)∂t .
− − √∆
(cid:18) (cid:19)
The previous parameter K is related to K = i by another irrelevant constant.
′ −√λ
If we define ∆=√r(r 2M), they can be rewritten as
−
e t M(3r 2M)
H1 =ie4M ∆∂r − ∂t ,
− r∆ !
p
H0 = i4M∂t, e p (63)
−
e
t M(3r 2M)
H 1 = ie−4M ∆∂r+ − ∂t .
− − r∆ !
p
e p
These generators clearly have some resemblance withethose in Eq. (7).
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6. Conclusion
The authors of Ref. 4 made an interesting observationthat the Killing vectors of a
specific Harrison metric can reproduce the hidden conformal symmetry generators
of the Schwarzschild black hole. As shown in Sec. 3, by proposing a simple corre-
spondence between the expansion of the Killing vector and the generators, we can
treatthe metric componentsas unknownvariablesanduse the Killing equationsto
find a family of Harrison metrics.
Sincethesubtractedgeometrychangesthewarpfactorintheoriginalblackhole
and the resulting metric can be obtained through a Harrison transformation, one
can find a new Harrison metric for the Schwarzschildblack hole using the result of
Ref. 15. The explicit form has been given in Sec. 4. After that we have derived the
pseudo-conformal generators for this new metric by applying the Killing equations
in the near-horizonlimit. This procedure used in Sec. 5 can be seen as a simplified
version of the method in Ref. 8.
It wouldbe interesting to find a refined versionof the technique in Sec. 3 which
canbegeneralizedtootherblackholes.Thismayalsogiveaunifiedwaytodescribe
the minimally and non-minimally subtracted geometries. Note that in Ref. 15, the
subtracted geometry of general asymptotically flat black holes in four dimensions
has been connectedto the AdS space withthe metric componentg unaccounted
3 θθ
for. Our method also has this disadvantage. However, it may not be an issue since
onecanstillgetreasonableresultsbyresortingtothethermodynamicrequirements.
Alsothisisactuallyrelatedtothegeneralfeatureofthehiddenconformalsymmetry
generators.
ThenewHarrisonmetricinEq.(42)(orEq.(44))mayprovideabridgebetween
the Schwarzschild black hole and the Kerr black hole. The previous work in Ref.
24 may be relevant for this idea. On the other hand, the physical implications of
this metric deserve further investigation. For example, one can use the traditional
methodsinRef.25tofindthequasinormalmodes.However,toresolvethe problem
pointed out in Ref. 13 within the hidden conformal symmetry approach, one may
need to improve the operator method proposed in Ref. 7. Last but not the least,
one may follow the works in Refs. 19–22 to find more connections between the
subtracted geometry approach and other areas.
Acknowledgments
This work was supported by National Natural Science Foundation of China (No.
11173028and No. 10875009).
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