Quantum Anomalies at Horizon and Hawking Radiations in Myers-Perry Black Holes

0.18 MB·English

by  Satoshi Iso

KEK-TH-1127, OIQP-06-20
Quantum Anomalies at Horizon
and
Hawking Radiations in Myers-Perry Black Holes
8
0
0 a a b
Satoshi Iso , Takeshi Morita and Hiroshi Umetsu
2 ∗ † ‡

n
a
J
a Institute of Particle and Nuclear Studies, High Energy Accelerator Research
0
1 Organization(KEK), Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan
b Okayama Institute for Quantum Physics, Kyoyama 1-9-1, Okayama 700-0015, Japan
3
v
6
8
2
2
1 Abstract
6
0 A new method has been developed recently to derive Hawking radiations from
/
h black holes based on considerations of gravitational and gauge anomalies at the
-t horizon [1][2]. In this paper, we apply the method to Myers-Perry black holes with
p
multipleangularmomentainvariousdimensionsbyusingthedimensionalreduction
e
h technique adopted in the case of four-dimensional rotating black holes [3].
:
v
i
X
r
a
∗E-mail address: [email protected]
†E-mail address: [email protected]
‡E-mail address: hiroshi [email protected]
1 Introduction
Hawking radiation is the quantum effect to arise for quantum fields in a background
space-time with an event horizon. There are many different derivations, from the original
calculation based on Bogoliubov transformations [4] to Euclidean approaches [5], and
all of them universally give the same answer. The universality tells us that the Hawking
radiationmust bedeterminedonlybysomeuniversal quantumeffectsjust onthehorizon.∗
In the seminal paper by Robinson and Wilczek [1], it was proposed that the flux
of Hawking radiation can be fixed by the amount of the gravitational anomaly at the
horizon. The method was then generalized in [2] to charged black holes by using the
gauge anomaly in addition to the gravitational anomaly and further applied to rotating
black holes [3][7] and others [8]. The essential observation in [1] is that quantum fields
near the horizon behave as an infinite set of two-dimensional fields and ingoing modes at
the horizon can be considered as left moving modes while outgoing modes as right moving
modes. Once the ingoing modes fall into the black hole, they never come out classically
and cannot affect the physics outside the black hole. Quantum mechanically, however,
they cannot be neglected because, without the ingoing modes, the theory becomes chiral
at the horizon, which makes the effective theory anomalous under general coordinate or
gauge transformations. In this sense, the ingoing modes at the horizon only affect the
exterior region through quantum anomalies. This is the basic idea of the method but
there is a slight difference between the original calculation in [1] and that in [2]. In this
paper, we adopt thecalculation used in[2]. Wewill explain the difference intheappendix.
This approach is similar to the beautiful derivation of the Hawking flux based on con-
formal anomalies [6], in which the Hawking flux was obtained by solving the conservation
law of the energy-momentum tensor with the information of conformal anomaly. In our
method, instead of conformal anomaly, we use gravitational or gauge anomalies. Both
anomalies arequantum effects but therearethe following differences. The gravitationalor
gauge anomalies arise only for chiral theories while conformal anomaly can arise even for
vector-like theories. The reason that the gravitational and gauge anomalies are relevant
to the Hawking radiation is due to the chiral decomposition property near the horizon.
Namely, quantum fields near the horizon can be decomposed into the left (ingoing) and
right(outgoing)modesandtheleftmodesarecausallydecoupledfromtheexteriorphysics
classically. Also the gravitational and gauge anomalies are independent of the details of
quantum fields and can be universally determined. Another advantage to use the gauge
anomalies is that we can derive the fluxes of charges and angular momenta in addition to
the energy flux.
In this paper, we apply the method to higher-dimensional Myers-Perry (MP) black
∗ The effect of scatterings away from the horizon changes the spectrum to gray. But this is not the
universal part of the Hawking radiation and we do not discuss it in the present paper.
1
holes [9]. Hawking flux from MP black holes with a single charge (angular momentum)
was obtained in [7] but they could not obtain the flux in cases with multiple angular
momenta. In this paper, by using similar dimensional reduction technique adopted in [3],
we show that the anomaly method can be similarly applied to MP black holes with more
than one charge, and that it reproduces the flux of each angular momentum F and that
ai
of energy-momentum tensor F associated with the Hawking radiation by each partial
M
wave,
∞ dω m n m a
i j j
F = m N (ω) N (ω) = ,
ai 2π i {mj} − {−mj} 2π r2 +a2
Z0 j=1 H j
(cid:0) (cid:1) X
∞ dω 1 n m a 2 π
i i
F = ω N (ω)+N (ω) = + . (1.1)
M 2π {mj} {−mj} 4π r2 +a2 12β2
Z0 i=1 H i!
(cid:0) (cid:1) X
Here N (ω) is the Planck distribution with an inverse-temperature β and chemical
{mi}
potentials a /(r2 +a2) for the angular momenta m (i = 1,2, ,n).† These fluxes are
i H i i ···
given by the sums of contributions from a particle with charge m and antiparticle with
i
{ }
charge m .
i
{− }
The organization of the paper is as follows. In section 2, we consider quantum fields in
the background of the MP black holes in various dimensions and show that they behave
as an infinite set of two-dimensional conformal fields near the event horizon. In section 3,
we consider symmetries for them and relate the conservation laws of the original energy-
momentum tensor to those in the dimensionally reduced theories. In section 4, we obtain
the Hawking fluxes based on two-dimensional gravitational and gauge anomalies. Section
4 is devoted to discussions. In appendix, we explain the difference between the original
calculation in [1] and that in [2].
2 Quantum fields in Myers-Perry black hole
In the D = 2n+1+ǫ (ǫ = 0 or 1) dimensions, the metric of the Myers-Perry black hole
is given by [9]
n µr2−ǫ n 2
ds2 = dt2 ǫr2dα2 (r2 +a2) dµ2 +µ2dφ2 dt a µ2dφ
− − i i i i − ΠF − i i i
!
i=1 i=1
X (cid:0) (cid:1) X
ΠF
dr2, (2.1)
−Π µr2−ǫ
−
† Here we used the Planck distribution for fermions in order to avoid the problem of superradiance
which is related to scatterings away from the horizon.
2
where
n a2µ2
F = 1 i i , (2.2)
− r2 +a2
i=1 i
X
n
Π = (r2 +a2). (2.3)
i
i=1
Y
The following constraint is imposed for µ (i = 1,2, ,n) and α,
i
···
n
µ2 +ǫα2 = 1, (0 µ 1, 1 α 1). (2.4)
i ≤ i ≤ − ≤ ≤
i=1
X
This metric describes a black hole space-time with the mass M = (D−2)AD−2µ and angular
16πG
momenta 2 Ma in the φ -directions, where A is the volume of SD−2. This black
D−2 i i D−2
hole is stationary and has U(1)n isometries with the Killing vectors ∂ . We assume the
φi
existence of horizons located at positive solutions of Π µr2−ǫ = 0. The inverse of metric
−
is given by
(Π µr2−ǫ)F +µr2−ǫ
gtt = − ,
(Π µr2−ǫ)F
−
µr2−ǫ a
gtφi = i ,
(Π µr2−ǫ)F · r2 +a2
− i
1 µr2−ǫ a a
gφiφj = δ + i j ,
−µ2(r2 +a2) ij (Π µr2−ǫ)F · (r2 +a2)(r2 +a2)
i i − i j
Π µr2−ǫ
grr = − ,
− ΠF
1 r2 µ µ
gµiµj = δ + i j , (2.5)
−r2 +a2 ij F · (r2 +a2)(r2 +a2)
i i j
and the determinant by
ΠF
g = √γ , (2.6)
| | r1−ǫ D−2
p
where √γ is the determinant of the metric of SD−2,
D−2
n−1
µ for D = 2n+1,
i

√γ = i=1 (2.7)
D−2  Y1 n
 µ for D = 2n+2.
i
α
| | i=1
 Y


We consider a scalar field ϕ in the Myers-Perry black hole background. The action is
1
S = dDx g gµν∂ ϕ∂ ϕ+S , (2.8)
µ ν int
2 | |
Z
p
3
where S includes a mass term and interaction terms. Near the outer horizon, which is
int
located at r = r the largest root of Π µr2−ǫ = 0, the kinetic term gives a dominant
H
−
contribution to the action and thus we can ignore a mass and interaction terms S .
int
Hence the action becomes near the outer horizon as
1 Π n a 2 Π µr2−ǫ
i
S = dtdrdΩ (µr) ϕ ∂ + ∂ ∂ − ∂ ϕ.
−2 D−2 Π µr2−ǫ t r2 +a2 φi − r Π r
Z − i=1 i !
X
 (2.9)
Note that the terms including ∂ ϕ in the kinetic part are also suppressed near the
µi
horizon compared to the above terms. The scalar field ϕ can be expanded by the
spherical harmonics YR (µ ,φ ) on SD−2, where R is a label of representations of
m1···mn i i
SO(D 1) and i∂ YR (µ ,φ ) = m YR (µ ,φ ). Performing the expansion, ϕ =
− φi m1···mn i i i m1···mn i i
ϕR (t,r)YR (µ ,φ ), the physics near the horizon can be effectively de-
R,mi m1···mn m1···mn i i
scribed by an infinite collection of massless (1+1)-dimensional fields with the following
P
action,
Π n im a 2 Π µr2−ǫ
S = dtdr(µr)(ϕR )∗ ∂ + i i ∂ − ∂ ϕR .
− m1···mn Π µr2−ǫ t r2 +a2 − r Π r m1···mn
Z − i=1 i !
X
  (2.10)
From this action we find that ϕR can be considered as a (1 + 1)-dimensional
m1···mn
complex scalar field in the background of the dilaton Φ, metric g and U(1) gauge fields
µν
A(φi),
µ
Π µr2−ǫ 1
Φ = µr, g = − f(r), g = , g = 0,
tt rr tr
Π ≡ −f(r)
a
A(φi) = i , A(φi) = 0. (2.11)
t −r2 +a2 r
i
The partial wave ϕR has the U(1) charge m for each gauge field A(φi).
m1···mn i µ
3 Symmetries and conservation laws
When we derive Hawking radiations from anomalies at the horizon in the next section, we
need to use various Ward-Takahashi identities in the two-dimensional effective theories.
In this section, we derive them by reinterpreting the original conservation laws associated
with general coordinate transformations in D = 2n+1+ǫ dimensions.
Following the general procedure in the Kaluza-Klein compactification, we write the
D-dimensional metric g (A,B = t,r,µ ,φ ) for the Meyers-Perry black holes in terms
AB i i
4
of a d-dimensional metric (d = n+1+ǫ), U(1)n gauge fields and dilatons,
g +h A(φi)A(φj) h A(φk)
(g ) = αβ φiφj α β φjφk α , (3.1)
AB hφiφkA(βφk) hφiφj !
where the indices α and β denote d-dimensional coordinates (t,r,µ ) and φ ’s are the
i i
angular coordinates. Note that since the metric g does not depend on these angu-
AB
lar coordinates there are U(1) isometries. Then the D-dimensional general coordinate
φi
transformation generated by a Killing vector ξφi(t,r,µ ) becomes U(1) gauge trans-
i (φi)
formation in the d dimensions. By using the Ward-Takahashi identity with respect to
this transformation, we can obtain a conservation law for U(1) gauge current. Under this
transformation, the fields change as
δA(φi) = ∂ ξφi, δg = δh = 0, (3.2)
α µ αβ φiφj
and the partition function in the above background changes as
δ
dDxδg lnZ[g] = i dDxξφi∂ g Tα
ABδg α | | φi
AB
Z Z (cid:16)p (cid:17) n−1+ǫ
= i dDxξφi ∂ g Tr + ∂ g Tµj = 0. (3.3)
r | | φi µj | | φi
" #
Z (cid:16)p (cid:17) Xi=j (cid:16)p (cid:17)
i δ
Here T = lnZ[g] is a D-dimensional energy-momentum tensor and de-
AB − g δgAB
| |
pendsonlyonr andµ duetotheisometries ofthebackground. Wedefineanr-component
i
p
of a U(1) gauge current in the two-dimensional field theory as
ΠF
Jr (r) dΩ Tr . (3.4)
(φi) ≡ − D−2 r1−ǫ φi
Z (cid:18) (cid:19)
We assume that on the right hand side in eq. (3.3), the second term is negligibly small
compared to the first term near the horizon of the MP black hole. This corresponds to
the fact that the terms including ∂ ϕ could be dropped in the discussion of the effective
µi
action near the horizon in the previous section. We then can see from eq.(3.3) that the
following conservation law holds near the horizon,
∂ Jr = 0. (3.5)
r (φi)
Namely, by studying the behavior of the two-dimensional U(1) gauge current, we can see
the flux of the D-dimensional angular momentum.
Next, by considering a D-dimensional general coordinate transformation generated by
a Killing vector ξt(t,r,µ ), we can obtain another conservation law. Variations of the
i
fields under it are
δg = g ξt +g ξt, δA(φj) = ξt A(φj) +A(φj) ξt,
αβ βt∇α αt∇β α ∇t α t ∇α
δh = ξt∂ h = 0, (3.6)
φiφj t φiφj
5
where is the covariant derivative associated with the d-dimensional metric g . Then
α αβ
∇
a change of the partition function is given by
δ
dDxδg lnZ[g]
AB
δg
AB
Z
n
= i dDxξt ∂ g Tα A(φi)Tα g F(φi)Tα
α | | t − t (φi) − | | tα φi
" #
Z (cid:16)p (cid:16) (cid:17)(cid:17) p Xi=1
n
= i dDxξt ∂ g Tr A(φi)Tr +∂ g Tµi A(φj)Tµi g F(φi)Tα
r | | t − t φi µi | | t − t φj − | | tα φi
" #
Z (cid:16)p (cid:16) (cid:17)(cid:17) (cid:16)p (cid:16) (cid:17)(cid:17) p Xi=1
= 0, (3.7)
where we usedtheisometries oftheMPbackgrounds. F(φi) aredefined by thecomponents
tα
of the field strengths of the U(1) gauge fields A(φi),
α
F(φi) = ∂ A(φi) ∂ A(φi), (3.8)
tα t α − α t
µr2−ǫ a
A(φi) = i , (3.9)
t −(Π µr2−ǫ)F +µr2−ǫ · r2 +a2
− i
A(φi) = A(φi) = 0. (3.10)
µj r
The equation (3.7) holds in the whole region outside the horizon but it can be simplified
near the horizon as follows. Near the horizon, because of Π µr2−ǫ = 0, F(φi) vanishes
− tµj
and F(φi) = ∂ A(φi) where the potential becomes
tr r t
−
a
A(φi) i , (r r ). (3.11)
t −→ −r2 +a2 −→ H
i
These gauge field backgrounds, of course, coincide with those in eq. (2.11) derived by
dimensional reduction of the action near the horizon. Furthermore we assume that in
eq. (3.7), ∂ g Tµi A(φj)Tµi is negligible compared to the other terms near the
µi | | t − t φj
horizon, by a(cid:16)spimilar(cid:16)reason to the o(cid:17)n(cid:17)e described below eq. (3.4). Defining a component
Tr (r) of the two-dimensional energy-momentum tensor as
t(2)
ΠF
Tr (r) dΩ Tr A(φi)Tr , (3.12)
t(2) ≡ D−2 r1−ǫ t − t φi
Z (cid:18) (cid:19)
(cid:16) (cid:17)
we find from eq. (3.7) the following conservation equation near the horizon,
n
∂ Tr F(φi)Jr = 0. (3.13)
r t(2) − rt (φi)
i=1
X
The energy flux in D dimensions is given by the two-dimensional energy current. The
second term in the above Ward-Takahashi identity can be interpreted as the dissipation
of energy due to interactions of the current Jr with the background electric field F(φi).
(φi) rt
6
4 Quantum anomalies and Hawking fluxes
Ingoing modes near the horizon are classically irrelevant to physics outside the horizon.
If we neglect the ingoing modes, in the two-dimensional effective theory near the horizon
gauge and diffeomorphism invariance are broken by quantum anomalies. The underlying
theory is, of course, invariant. Therefore these anomalies are cancelled by quantum effects
of ingoing modes which are irrelevant classically. In the following we show that the
condition for vanishing of anomalies at the horizon leads to the correct Hawking fluxes of
angular momenta and energy.
First we consider the U(1) currents Jr in the effective theory which is defined in
(φi)
r [r , ]. Each current corresponds to angular momentum in the φi-direction in the
H
∈ ∞
D-dimensional theory, as shown in the previous section. We divide the region outside the
horizon into two regions, r [r ,r +ǫ] near the horizon and r > r +ǫ. If we omit the
H H H
∈
ingoing modes near the horizon, U(1) current has gauge anomaly there. The consistent
form of the abelian gauge anomaly [10, 11, 12] in two dimensions is given by
m
Jµ = i ǫµν∂ , (4.1)
∇µ (φi) −4π√ g µAν
(2)
−
where ǫ01 = +1 and µ,ν run t and r. In our case, is a sum of n U(1) gauge fields
µ
A
multiplied by charges,
n
= m A(φj), (4.2)
Aµ j µ
j=1
X
n
m a
j j
= , = 0. (4.3)
At − r2 +a2 Ar
j=1 j
X
It is noted that the charge of the partial mode in the effective theory is m . The consistent
i
anomaly satisfies the Wess-Zumino condition. The consistent current is derived from the
quantum effective action and is not gauge covariant. We can define covariant current as
m
J˜µ Jµ i ǫµν , (4.4)
(φi) ≡ (φi) − 4π√ g Aν
(2)
−
which satisfies
m
J˜µ = i ǫµν . (4.5)
∇µ (φi) −4π√ g Fµν
(2)
−
Here is given by
µν
F
n
= ∂ ∂ = m F(φj). (4.6)
Fµν µAν − νAµ j µν
i=j
X
In the case we consider here, the consistent current Jµ in r [r ,r + ǫ] depends
(φi),H ∈ H H
only on r and satisfies the following anomalous equation with the gauge field background
7
(2.11),
m
∂ Jr = i∂ . (4.7)
r (φi),H 4π rAt
In the outside region r > r + ǫ, the current is conserved, ∂ Jr = 0. Hence we can
H r (φi),O
solve them in each region as
Jr = ci , (4.8)
(φi),O O
m
Jr = ci + i ( (r) (r )), (4.9)
(φi),H H 4π At −At H
where ci and ci are integration constants. ci is the value of the current at r and
O H O → ∞
ci is the value of the consistent current of the outgoing modes at the horizon r = r .
H H
The current is written as a sum in two regions
Jµ = Jµ Θ (r)+Jµ H(r), (4.10)
(φi) (φi),O + (φi),H
where Θ (r) = Θ(r r ǫ) and H(r) = 1 Θ (r) are step functions which are defined
+ H +
− − −
in the region r r . Note that this current is a part of the total current because we
H
≥
omitted the ingoing modes near the horizon. The total current including a contribution
of the ingoing modes is given by
Jµ = Jµ +Kµ , (4.11)
(φi)total (φi) (φi)
where
m
Kr = i (r)H(r). (4.12)
(φi) −4πAt
This contribution cancels the anomalous part of Jµ near the horizon.
(φi)
In the previous papers [1][2][3], the invariance of the quantum effective action under
gauge and general coordinate transformations was considered. Requiring the invariance
can determine the relation between the integration constants ci and ci . This method
O H
has revealed the significance of each contribution from the ingoing and outgoing modes,
but the relation itself can be easily obtained by imposing the conservation of the total
current, ∂ Jr = 0. This condition gives the relation,
r (φi)total
m
ci = ci i (r ). (4.13)
O H − 4πAt H
In order to fix the value of the current, we need a further boundary condition at
the horizon. Here we impose that the covariant form of the outgoing current should
vanish at the horizon. The outgoing modes can produce possible divergences for physical
quantities seen by infalling observers into black holes, as discussed in, for example, [14]
[6]. The physical vacuum in the black hole backgrounds should be chosen as such that
these unphysical divergences should vanish. In our case, this condition corresponds to the
above boundary condition. We discuss it in the appendix again. Another condition to
8
µ
specify the vacuum is the boundary condition for the ingoing currents K . In eq. (4.12)
(φi)
we have already chosen an integration constant such that it vanishes at r . This
→ ∞
condition corresponds to taking Unruh vacuum, in stead of Hartle-Hawking vacuum.
Since the covariant current is given by J˜r = Jr + mi (r)H(r), the condition
(φi) (φi) 4πAt
J˜r (r ) = 0 determines the value of the charge flux at r as
(φi) H → ∞
n
m m m a
ci = i (r ) = i j j . (4.14)
O −2πAt H 2π r2 +a2
j=1 H j
X
This coincides with the flux of angular momentum associated with the Hawking radiation.
Next we consider the flux of energy-momentum radiated from the Myers-Perry black
holes. Omitting the ingoing modes in the near horizon region r r r + ǫ, the
H H
≤ ≤
consistent energy-momentum tensor Tµ satisfies a modified conservation equation with
ν,H
the consistent anomalies. In two dimensions the energy-momentum tensor for right-hand
modes satisfies the following Ward-Takahashi identity with U(1) gauge fields A(φi) and
µ
dilaton Φ backgrounds,
n
∂ Φ δS
Tµ = F(φi)Jµ +A(φi) Jµ ν +A , (4.15)
∇µ ν µν (φi) ν ∇µ (φi) − √ g δΦ ν
(2)
Xi=1 (cid:16) (cid:17) −
where A is the consistent gravitational anomaly [13] which is given by
µ
1
A = ǫνρ∂ ∂ Γσ . (4.16)
µ 96π√ g ρ σ µν
(2)
−
This energy-momentum tensor is not covariant under general coordinate transformations.
On the other hand the covariant energy-momentum T˜µ satisfies the Ward-Takahashi iden-
ν
tity with the same form as the above but the anomaly term A is replaced with the
µ
covariant one A˜ = 1 ǫ νR.
µ −96π√−g µν∇
(2)
In the case considered here, the ν = t component of the Ward-Takahashi identity in
the consistent form becomes
n
∂ Tr = F(φi)Jr +A(φi) Jµ +∂ Nr(r), (4.17)
r t,H rt (φi) t ∇µ (φi) r t
Xi=1 (cid:16) (cid:17)
where background fields (2.11) are used. Nr(r) is defined by A = ∂ Nr,
t t r t
1
Nr(r) = f′2(r)+f(r)f′′(r) . (4.18)
t 192π
(cid:0) (cid:1)
Eq. (4.17) is the same Ward-Takahashi identity as eq. (3.13) if there are no anomalous
terms. See [2] for the Ward-Takahashi identity in presence of anomalies. In eq. (4.17),
the first and second terms in the right hand side are combined in terms of the covariant
9