Hawking flux of 4D Schwarzschild black hole
with supertransition correction to second-order

Shingo Takeuchi

Phenikaa Institute for Advanced Study and Faculty of Basic Science,

Phenikaa University, Hanoi 100000, Vietnam

## Abstract

Former part of this article is the proceedings for my talk [1] on [2], which
is a report on the issue in the title of this article. Later part is the
detailed description of [2].

###### Contents

  1. 1 Motivation for my analyzing Hawking flux
  2. 2 What’s supertranslation and its NG boson fields
  3. 3 Fun in the asymptotic symmetry
  4. 4 Our 2D effective action with supertranslation correction
  5. 5 Result of Hawking flux with supertranslation correction
  6. 6 Conclusion
  7. A Metrices with supertranslation correction
    1. A.1 Introduction of supertranslation
    2. A.2 Isotropic coordinates with supertranslations
    3. A.3 Rewriting from Schwarzschild to isotropic coordinates
    4. A.4 Rewriting from isotropic to Schwarzschild coordinates
    5. A.5 Metrices with correction to $\varepsilon^{2}$-order in 4D Schwarzschild coord.
    6. A.6 Comment on $c_{1,2}$
  8. B 2D effective near-horizon action
    1. B.1 Near-horizon metrices
    2. B.2 Hawking temperature in the original 4D
    3. B.3 Near-horizon action
    4. B.4 Integrate out of ($\theta$, $\phi$)
    5. B.5 2D effective near-horizon metrices
  9. C Hawking Temperature in the effective 2D
  10. D Hawking flux by anomaly cancellation
    1. D.1 Set up of the radial direction
    2. D.2 Hawking flux of the energy-momentum tensors
  11. E Comment on result in terms of the information paradox

##  1 Motivation for my analyzing Hawking flux

We will start with a 4D spacetime, then impose some falloff condition to its
spatially infinite region111 Talk slide is in the homepage [1]., which means
the spacetime asymptotes to the flat spacetime according to that. Then, we
will consider the diffeomorphism at its spatially infinite region in the range
of the falloff condition (at this time, gauge conditions to eliminate local
diffeomorphism ambiguities are also preserved). Transformation of the
spacetime by this diffeomorphism is called asymptotic (or BMS) symmetry [3,
4]. I list several facts known for this.

  * •

Diffeomorphism of the asymptotic symmetry is an infinite dimensional group,
which contains the Poincarè group as its subgroup.

  * •

There are so-called supertranslation and superrotation in the asymptotic
symmetry, which respectively contains the global translation and Lorentz
transformation.

  * •

Diffeomorphism of the asymptotic symmetry maps a configuration of an
asymptotically flat spacetime as a solution to an other physically different
asymptotically flat spacetime as a solution in the range of the falloff
condition.

  * •

Asymptotically flat spacetime is infinitely degenerated in the range of
asymptotic symmetry, and the symmetry of theory at that asymptotic region is
not Poincarè symmetry but the one associated with supertranslation and
superrotation.

  * •

Infinite number of conserved charges for supertranslation and superrotation
can be defined respectively. These can respectively generate these
diffeomorphism, however these are given by 2 parts: so-called hard- and soft-
parts, which will become creation and annihilation operators for soft-
gravitons at quantum level. For this, the quantum state for the infinite far
region of the asymptotically flat spacetime becomes another one of
asymptotically flat spacetime when these charges act on these, and soft-
gravitons can be considered as a kind of Goldstone boson. In this sense, the
asymptotic symmetry is some kinds of spontaneously broken symmetry [5].

  * •

Supertranslated spacetimes are normal [6], therefore considering asymptotic
symmetry is meaningful realistically.

In [2], I obtain the Schwarzschild black hole spacetime with supertranslation
correction to the second-order, which I sketch as

$\displaystyle ds^{2}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-(1-{2m}/{r}+\cdots+{\cal
O}(\varepsilon^{3}))\,dt_{s}^{2}+((1-{2m}/{r})^{-1}+\cdots+{\cal
O}(\varepsilon^{3}))\,dr_{s}^{2}$ (1) $\displaystyle+(r^{2}+\cdots+{\cal
O}(\varepsilon^{3}))\,d\theta_{s}^{2}+(r^{2}\sin^{2}\theta+\cdots+{\cal
O}(\varepsilon^{3}))\,d\phi_{s}^{2}$
$\displaystyle+2((\cdots)\varepsilon+\cdots+{\cal
O}(\varepsilon^{3}))\,dr_{s}d\theta_{s},$

where $\varepsilon$ mean the order of supertranslation correction and the full
expressions of metrices are given $j^{(2)}_{\mu\nu}$ in (45a)-(49). We can
obtain the position of the horizon from this as

$\displaystyle
r_{h,4D}=2m-\frac{15m\sin^{2}(2\theta)}{8\pi}\varepsilon^{2}+O(\varepsilon^{3}).$
(2)

The correction of supertranslation enters from the second-order, which is the
motivation for our analysis to the second-order. Here, as this is not
constant, there may be a concern for the zeroth law of the black hole
thermodynamics. It is no problem since the Hawking temperature is constant in
the range of our analysis’s order, $\varepsilon^{2}$, as shown below.

As for the problem with Birkhoff’s theorem, considering the point that the
information is encoded in the asymptotic region, it is inferred that the
deformed near-horizon geometry is meta-stable and will get settled down to
just a Schwarzschild finally, while that in the asymptotic region is the
stable.

Since the position of the horizon is displaced for supertranslation
correction, it is interesting to check how the Hawking temperature is. Its
result is

$\displaystyle T_{H}={1}/{8\pi m}+O(\varepsilon^{3}),$ (3)

which is no difference from just the Schwarzschild (reason is written in
Sec.B.2).

Since Hawking temperature can be calculated from Hawking flux, if the Hawking
temperature were preserved the Hawking flux would be expected to be preserved.
However I have considered a possibility that supertranslation corrections may
be involved in the Hawking flux but would be canceled out in the Hawking
temperature. This is one of my motivations for my computing the Hawking flux
in [2].

There is another motivation, which is that as a result of involving the
supertranslation corrections as in (1), it becomes obscure whether field
theories can reduce to free 2D or not in the near-horizon. Originally it
should be so for the strongly gravitational force at the horizon, and if not,
it would be physically abnormal. Although it can be shown in Sec.B.5 that the
scalar theory can reduce to free 2D, whether it is possible or not is unclear
before trying (I comment on the key for the feasibility of this in the last of
Sec.B.5).

There would be many works analyzing the Hawking radiation in some
supertranslated situations. In these, as the works relating with this study, I
take [7, 8, 9, 10] in the range I know. From the research situation mentioned
in the following, the analysis in this study would be worthwhile.

In [7], considering Vaidya spacetime with linear order supertranslation
correction, it is shown that Hawking radiation depends on supertranslation
correction if the mass depends on the advanced time $v$. Conversely, if the
mass is a constant, $M^{\prime}=0$, there is no correction in Hawking
radiation. This is consistent with the result in this study, however my
analysis is performed to second-order.

Result in [8] is that Hawking radiation in the asymptotically flat spacetime
given by Bondi coordinates does not get corrected, however the position of the
horizon in their analysis is assumed to be $2GM$. The analysis in this study
is performed taking the correction to the position of the horizon into
account.

[9] performs linear order analysis regarding general diffeomorphisms including
supertranslation expressed by $F$ (see (7)), then it is concluded the
proportional coefficient in the relation between the surface gravity and
Hawking temperature, $1/2\pi$ in $T_{H}=\kappa/2\pi$, gets some correction
(see 2 points: “However, macroscopic $\cdots$ transformation.” in P.2 and “as
expected $\cdots$ and Hawking [68].” in P.9, and (54)). This is against to the
conservative results in [7], [8] and this study.

In [10], a Hawking radiation in a 4D supertranslated Schwarzschild black hole
is analyzed by some expansion around large mass, and its result depends on the
angular. This interferes with the zeroth law of the black hole thermodynamics,
and does not agree to [7, 8, 9] and this study. However, remember it is the
laws for the stable solutions and the shape of the asymptotic region is the
key in the information paradox. Then, the interference with the black hole
thermodynamics would not be any problems if the deformation of the near-
horizon geometry is meta-stable, while that in the asymptotic region is stable
(as mentioned between (2) and (3)). As for the disagreement of results, we
could not say anything immediately, since the kind of the expansion is
different and cannot compare.

##  2 What’s supertranslation and its NG boson fields

We start with an expression of general 4D spacetime by the Bondi coordinates
$(u,r,\Theta^{A})$ ($u=t-r$ and $\Theta^{A}$ are the spherical coordinates
$(z,\bar{z})$ on the $S^{2}$) as

$\displaystyle
ds^{2}=-Udu^{2}-e^{2\beta}dudr+g_{AB}(d\Theta^{A}+\frac{1}{2}U^{A}du)(d\Theta^{B}+\frac{1}{2}U^{B}du),$
(4)

where the Bondi gauge is imposed to fix the local diffeomorphisms, which is
$g_{rr}=0$, $g_{rA}=0$ and $\partial_{r}\det(g_{AB}/r^{2})=0$. Then, supposing
that the spacetime will asymptote to the flat spacetime, let us consider to
describe the neighborhood of ${\mathcal{T}}^{+}$. At this time we need to
impose a falloff condition to the metrices, however there is no systematic
ways to determine the falloff condition, and various falloff conditions can be
considered. Typically, it is chosen so that physical solutions can exist and
unphysical solutions do not exist.

As an expansion of (4) to $r^{-1}$, the following one is conventionally
adopted [3, 4]:

$\displaystyle ds^{2}=$ $\displaystyle-
du^{2}-2dudr+2r^{2}\gamma_{z\bar{z}}dzd\bar{z}$
$\displaystyle+{2m_{B}}/{r}\,du^{2}+rC_{zz}dz^{2}+rC_{\bar{z}\bar{z}}d\bar{z}^{2}+D^{z}C_{zz}dudz$
$\displaystyle+\frac{1}{r}\,(\frac{4}{3}(N_{z}+u\partial_{z}m_{B})-\frac{1}{4}(C_{zz}C^{zz}))dudz+\text{c.c.}+O(r^{-2})\,,$
(5)

where $D_{z}$ is the covariant derivative with respect to $\gamma_{\bar{z}z}$.
It is usual that the structure to $r^{-1}$ is important. $C_{zz}$,
$C_{\bar{z}\bar{z}}$, $m_{B}$ and $N_{z}$ are functions of $(u,z,\bar{z})$ but
not of $r$, and

  * •

$m_{B}$ is the Bondi mass aspect. $\int_{S^{2}}dzd\bar{z}m_{B}$ gives the
Bondi mass, which can be ADM mass in the cases of black hole spacetimes.

  * •

$N_{z}$ is the angular momentum aspect. $\int_{S^{2}}dzd\bar{z}N_{z}V^{z}$
gives the total angular momentum, which is ADM angular momentum in the black
hole spacetimes.

  * •

$C_{zz}$ and $C_{\bar{z}\bar{z}}$ play the role of potential for gravitational
wave (akin to vector potential for electromagnetic field), and $N_{zz}$ is the
Bondi news given as $\partial_{u}C_{zz}$ ($N_{\bar{z}\bar{z}}$ is likewise).

The falloffs of the metrices in (2) are given as follows:

$\begin{split}g_{uu}=&~{}-1+{\cal O}(r^{-1}),\quad g_{ur}=-1+{\cal
O}(r^{-2}),\quad g_{uz}={\cal O}(1),\\\ g_{zz}=&~{}{\cal O}(r),\quad
g_{z\bar{z}}=r^{2}\gamma_{z\bar{z}}+{\cal O}(1),\quad
g_{rr}=g_{rz}=0.\end{split}$ (6)

Let us turn to the supertranslation. Displacement of metrices, $m_{B}$,
$N_{zz}$ and $C_{zz}$ in (2) by the diffeomorphism of the supertranslation is
given by the Lie derivatives as

$\displaystyle{\cal L}_{\xi}g_{ur}$
$\displaystyle=-\partial_{u}\zeta^{u}+{\cal O}(r^{-1}),$ $\displaystyle{\cal
L}_{\xi}g_{zr}$
$\displaystyle=r^{2}\gamma_{z\bar{z}}\partial_{r}\zeta^{\bar{z}}-\partial_{z}\zeta^{u}+{\cal
O}(r^{-1}),$ $\displaystyle{\cal L}_{\xi}g_{z\bar{z}}$
$\displaystyle=r\gamma_{z\bar{z}}(2\zeta^{r}+rD_{z}\zeta^{z}+rD_{\bar{z}}\zeta^{\bar{z}})+{\cal
O}(1),$ $\displaystyle{\cal L}_{\xi}g_{uu}$
$\displaystyle=-2\partial_{u}\zeta^{u}-2\partial_{u}\zeta^{r}+{\cal
O}(r^{-1}),$ (7) $\displaystyle{\cal L}_{\xi}m_{B}$
$\displaystyle=f\partial_{u}m_{B}+\frac{1}{4}(N^{zz}D_{z}^{2}f+2D_{z}N^{zz}D_{z}f+c.c.),$
$\displaystyle{\cal L}_{\xi}N_{zz}$ $\displaystyle=f\partial_{u}N_{zz},$
$\displaystyle{\cal L}_{\xi}C_{zz}$
$\displaystyle=f\partial_{u}C_{zz}-2D_{z}^{2}f,$

where the vector field proscribing the coordinate transformation in the Lie
derivatives above is given as

$\displaystyle\xi=f\partial_{u}+\frac{1}{r}(D^{z}f\partial_{z}+D^{\bar{z}}f\partial_{\bar{z}})+D^{z}D_{z}f\partial_{r}.$
(8)

$f$ is arbitrary function of $(z,\bar{z})$, and normally spherical harmonics
are taken. The field referred to as NG boson field is defined for one for
every $f$ as

$\displaystyle{\cal L}_{\xi}C(z,\bar{z})=f(z,\bar{z}).$ (9)

##  3 Fun in the asymptotic symmetry

First of all, what 4D Minkowski spacetime has not been an unique vacuum but
infinitely degenerated would be a surprisingly interesting fact. This had been
already found in 1962 [3, 4], however it is in just the last decade that hep-
th has recognized this problem [11, 12]. As interesting directions from the
study of the asymptotic symmetry, the following ones could be taken: 1)
gravitational memory effect, 2) link with soft theorems and holography, and 3)
information paradox.

1) is the variation in the relativistic position of two objects near the
future null infinity ${\cal T^{+}}$ for the passing of the gravitational wave,
which could be measured by the formalism of the asymptotic symmetry.

Consider the gravitational wave is turned on at $u=u_{i}$ and off at
$u=u_{f}$, and two objects near ${\cal T^{+}}$ are exposed it during the time
interval $\Delta u=u_{f}-u_{i}$. The Bondi news tensor and the energy momentum
tensors are zero at any time except for the time getting the gravitational
wave. Then, one can evaluate the displaced amount as

$\displaystyle\Delta s^{\bar{z}}={\gamma^{z\bar{z}}\over 2r}\Delta
C_{zz}s^{z},$ (10)

where $\Delta s^{A}=s^{A}|_{u=u_{f}}-s^{A}|_{u=u_{i}}$ ($s^{A}$ mean the
relativistic position of the two objects), and $\Delta
C_{AB}=C_{AB}|_{u=u_{f}}-C_{AB}|_{u=u_{i}}$.

Hence, the passing of gravitational wave is considered to arise the
displacement by the order $r^{-1}$. Now observation of the gravitational
memory effect is undergoing [13, 14, 15, 16, 17, 18].

Other types of memory effect are also considered: spin memory effect [19],
color memory effect [20], and electromagnetic memory effect [21, 22, 23].
Observing the soft graviton may be also planed, however it is so silent that
it is not caught in our current detection.

Regarding 2), the equation of the soft theorem can be obtained from the Ward
identity with regard to the asymptotic symmetry ([24, 25] and [26, 27, 28, 29,
30, 31] for gauge theories and gravity, respectively). Therefore,

$\displaystyle\textrm{asymptotic symmetry}\,\longleftrightarrow\,\textrm{soft
theorem.}$ (11)

Currently the correspondence between the $S$-matrix in 4D asymptotically flat
spacetime and 2D CFT are ongoing [32, 33, 34, 35, 36, 37].

Next, the DC shift (equation given in P.91 of [5]) and “the effect of
attaching one soft-graviton line to an arbitrary Feynman diagrams” can be
identical each other via Fourier transformation (with adjustment of some
notation’s conventions). From this fact, it is considered that the
gravitational wave from black holes and the soft particles from the elementary
particle’s collisions will show analogous behavior at the long distance in the
observation [5]. Thus, as the phenomena showing analogous behavior [38],

$\displaystyle\textrm{soft theorem}\,\longleftrightarrow\,\textrm{memory
effect}.$ (12)

Lastly, the gravitational wave at the long distance can be considered as a
kind of diffeomorphism of the asymptotic symmetry, and has a relation with
memory effect. Hence,

$\displaystyle\textrm{memory effect}\,\longleftrightarrow\,\textrm{asymptotic
symmetry}.$ (13)

It is very interesting that different theories and phenomena can get related
like the one above. Same relations can be obtained in the gauge field theories
[5]. Therefore, gravitational and gauge theories would be universal in the IR-
region.

Regarding 3), an initial configuration to form a star or black hole finally
leads to some deformed spacetimes by supertranslation (for an explicit
analysis for this, see [6]), and its phase space is infinite dimensional.
Hence, we can expect that the information of the initial configuration could
be preserved in the final shape of the spacetime, which is the scenario we can
highly expect as the solution to the information paradox [12, 39, 40].

##  4 Our 2D effective action with supertranslation correction

From here, I would like to talk on my study. What I want to do first is to
obtain the Schwarzschild black hole metric with the supertranslation
correction to the second order in the Schwarzschild black hole coordinates.
For this we will start with the Schwarzschild black hole spacetime given in
the isotropic coordinates:

$\displaystyle
ds^{2}=-\frac{(1-{m}/{2\rho_{s}})^{2}}{(1+{m}/{2\rho_{s}})^{2}}dt_{s}^{2}+(1+{m}/{2\rho_{s}})^{4}(d\rho_{s}^{2}+\rho_{s}^{2}d\Omega_{s}^{2}),$
(14)

where a flat three-dimensional space part,
$d\rho_{s}^{2}+\rho_{s}^{2}d\Omega_{s}^{2}$, is convenient to involve the
supertranslation correction according to [6].

Then, writing as
$d\rho_{s}^{2}+\rho_{s}^{2}d\Omega_{s}^{2}=dx_{s}^{2}+dy_{s}^{2}+dz_{s}^{2}$
and $\rho_{s}^{2}=x_{s}^{2}+y_{s}^{2}+z_{s}^{2}$, we involve the
supertranslation correction according to [6]:

$\displaystyle x_{s}$
$\displaystyle=(\rho-C)\sin\theta\cos\phi+\sin\phi\csc\theta\,\partial_{\phi}C-\cos\theta\cos\phi\,\partial_{\theta}C,$
(15a) $\displaystyle y_{s}$
$\displaystyle=(\rho-C)\sin\theta\sin\phi-\cos\phi\csc\theta\,\partial_{\phi}C-\cos\theta\sin\phi\,\partial_{\theta}C,$
(15b) $\displaystyle z_{s}$
$\displaystyle=(\rho-C)\cos\theta+\cos\theta\cos\phi\,\partial_{\theta}C,$
(15c)

where the function $C$ is the NG boson field for supertranslation, which we
will take as

$\displaystyle C=m\varepsilon\,Y_{2}^{0}(\theta,\phi).$ (16)

  * •

$\varepsilon$ is dimensionless, which we attach to measure the order of
supertranslations in our analysis. $m$ is that in (14), which we involve to
have $C$ have the same dimension with $\rho$ (now, $G/c^{2}=1$). The
correction of $\varepsilon$ appears from the second-order (see (2)) in the
position of the horizon, which is our motivation for the analysis to
$\varepsilon^{2}$-order.

  * •

Why we consider $Y_{2}^{0}$ that this mode is expected to be dominant in the
process forming a soft-hairy black hole (e.g. [41]). We have also performed
the analysis with $Y_{1}^{0}$ just in case. Although we have not performed the
calculation to the end, it has been seemed to be essentially same with what
will present in the following.

Involving (15) into the isotropic coordinates (14) to $\varepsilon^{2}$-order,
we will rewrite it into the Schwarzschild coordinates (for detail, see Sec.A),
and finally obtain like (1).

Then, with these 4D metrics, we consider a complex scalar field theory as

$\displaystyle S=\int
d^{4}x\sqrt{-g}\,g^{MN}\partial_{M}\phi^{*}\partial_{N}\phi.$ (17)

Writing $\phi(t,r,\theta,\phi)=\phi_{lm}(t,r)Y_{m}^{l}(\theta,\phi)$, and
taking near-horizon limit by writing $r=r_{h,4D}+\Delta r$, we can get the 4D
near-horizon action as

$\displaystyle S=$ $\displaystyle-\sum_{l,m}\sum_{k,n}\int
dtdr(2m)^{2}\Big{\\{}$ $\displaystyle\hskip
14.22636pt\phi_{lm}^{*}\Big{(}-\frac{2m}{r-2m}\Lambda_{lm,\,kn}-\frac{15m^{2}\varepsilon^{2}}{4\pi(r-2m)^{2}}\\!\int\\!d\Omega\sin^{2}(2\theta)(Y_{l}^{m})^{*}Y_{k}^{n}\Big{)}\partial_{t}\partial_{t}\phi_{kn}$
$\displaystyle+\phi_{lm}^{*}\partial_{r}\Big{(}\frac{r-2m}{2m}\Lambda_{lm,\,kn}-\frac{15m^{2}\varepsilon^{2}}{16\pi}\\!\int\\!d\Omega\sin^{2}(2\theta)(Y_{l}^{m})^{*}Y_{k}^{n}\Big{)}\partial_{r}\phi_{kn}\Big{\\}}\\!+O(\varepsilon^{3})\,,$
$\displaystyle\Lambda_{lm,\,kn}\equiv$ $\displaystyle\int
d\Omega\Big{\\{}1+\frac{3}{2}\sqrt{\frac{5}{\pi}}\left(1+3\cos(2\theta)\right)\varepsilon$
$\displaystyle+\frac{45}{2\pi}\Big{(}\frac{\sin(2\theta)}{4}-\cos^{2}(\theta)+3\cos^{2}(\theta)\cos(2\theta)\Big{)}\varepsilon^{2}\Big{\\}}\,(Y_{l}^{m})^{*}Y_{k}^{n}.$
(18b)

Integrating out ($\theta$, $\phi$), we obtain 2D near-horizon effective action
as (Sec.B.4 and B.5)

$\displaystyle S_{\textrm{2D eff}}=\sum_{l=0}^{l_{max}}\sum_{|m|=0}^{l}\int
d^{2}x\,\Phi_{lm}\big{(}(g_{\rm
eff})^{tt}_{lm}\,\partial_{t}\varphi^{*}_{lm}\partial_{t}\varphi_{lm}+(g_{\rm
eff})^{rr}_{lm}\,\partial_{r}\varphi^{*}_{lm}\partial_{r}\varphi_{lm}\big{)},$
(19) $\displaystyle\Phi_{lm}$ $\displaystyle=(2(m_{\rm eff})_{lm,\,lm})^{2},$
(20a) $\displaystyle(g_{\rm eff})^{tt}_{lm}$ $\displaystyle=-1/(g_{\rm
eff})^{rr}_{lm}=-\frac{2(m_{\rm eff})_{kn,\,lm}}{r-2(m_{\rm
eff})_{kn,\,lm}}+O(\varepsilon^{3}),$ (20b) $\displaystyle(m_{\rm
eff})_{kn,\,lm}$ $\displaystyle=m+\frac{15m}{8\pi r}{\cal
I}^{C}_{kn,\,lm}\varepsilon^{2}+O(\varepsilon^{3}).$ (20c)

(For $r$-dependence in $(m_{\rm eff})_{kn,\,lm}$, see (81)). Whether the 2D
near-horizon effective action can be obtained or not is non-trivial before
trying as mentioned in Sec.1, to check which is one of the motivations in this
study.

##  5 Result of Hawking flux with supertranslation correction

We obtain Hawking flux by anomaly cancellation method [42, 43], in which
reducing to 2D is crucial, because analysis is performed with the 2D anomaly.
For details, see Sec.D.

Anomaly cancellation will focus on the fact: 1) At the classical level, there
is no outgoing flux in the near-horizon region for the strong gravitational
effect, 2) however, at the quantum level, outgoing flux will arise by the
quantum tunneling [44]. Hence, the outgoing flux exists in the near-horizon
region finally. At this time, if one takes in the analysis as

$\displaystyle\textrm{amount of flux from tunneling}=\textrm{amount of lack of
flux at the classical level},$ (21)

the amount of the flux by the quantum tunneling can be identified as the
Hawking flux.

The amount of the outgoing flux is represented by the integral constant
obtained from the formulas of the 2D anomaly:

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu,\,lm}=\,-\frac{\partial_{\nu}\Phi_{lm}}{\sqrt{-(g_{\rm
eff})_{lm}}}\frac{\delta
S_{\textrm{2D}}}{\delta_{L}\Phi_{lm}}+\textrm{both/either
$\mathscr{A}_{\nu,\,lm}^{\pm}$},$ (22)

which can be fixed by the condition that the system is symmetric, which is at
the point where the variation of the action vanishes:

$\displaystyle(\delta S_{\textrm{2D}})_{lm}=-\int d^{2}x\sqrt{-(g_{\rm
eff})_{lm}}\,\eta^{\nu}\nabla_{\mu,\,lm}T^{\mu}{}_{\nu,\,lm}.$ (23)

The Hawking flux we have obtained has been (127), $\pi T_{H}^{2}/12$, which is
the same with just a Schwarzschild. The reason of this is written in Sec.D.2.

##  6 Conclusion

Although the position of the horizon has been displaced and whether the near-
horizon field theories can reduce to free 2D has been non-trivial, Hawking
temperature and flux have been obtained without changes. This no changes in
Hawking temperature had been already clear when near-horizon metrices are
obtained, however whether Hawking flux can be obtained without any changes or
not is unclear for the reason written in Sec.1.

Furthermore, although there are works relating with this study as mentioned in
Sec.1, for the research situation mentioned there, the analysis in this study
would be worthwhile.

The value of the Hawking flux would be always (127) as long as the function
$C$ is 1) to the second-order, and 2) independent of $\phi$. The reason for
this is as follows.

First, if the correction is to $\varepsilon^{2}$ but $C$ is some one other
than (16) independent of $\phi$,

  * •

highly complicated terms of $\theta$ will be newly involved into the each
coefficient of $\varepsilon^{1,2}$ in (18). At this time, the feasibility of
the integrate out for ($\theta$, $\phi$) is the problem, however it would be
no problem by using the following formula and (77):

$Y^{m_{1}}_{l_{1}}Y^{m_{2}}_{l_{2}}=\sum_{L,M}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{4\pi(2L+1)}}\langle
l_{1}\,0\,l_{2}\,0|L\,0\rangle\langle l_{1}m_{1}\,l_{2}m_{2}|LM\rangle
Y^{M}_{L},$ (24)

  * •

coefficients of $\varepsilon^{1,2}$ in (20c) will get highly involved
concerning $\theta$, however the structure of (20b) as the function $f$ would
be no changed (see the last of Sec.D.2), since $C$ depends only on $\theta$
and $\phi$ by definition.

Therefore, if the two conditions above are satisfied, one could always get the
2D near-horizon effective action with the $f$ same with (19) as the structure.
On the other hand,

  * •

if the correction of $\varepsilon$ were involved more than 3rd-order, the
feasibility of the analysis to get the free 2D theory as in Sec.B.5 gets
unclear. See the last line in Sec.B.5. Namely, if the same behavior with (91)
were not held, the analysis to get the free 2D theory would be impossible.

  * •

If $\phi$-dependence were mixed in the $C$, the formulas (24) might get
unavailable, and we could not get the 2D action like (19).

This study has considered only $Y_{2}^{0}$ for the NG boson field of the
supertranslation, then given a conclusion considering the contribution of
other modes would be qualitatively same. Therefore this study should be
careful on whether other mode’s contributions are qualitatively the same or
not, which point is cared as above.

It is considered from our result, no changes, that Hawking temperature and
flux may be the conserved quantities under the asymptotic symmetry.

## Acknowledgment

Upon writing this article, many discussions with Prof.Feng-Li Lin, the
collaborator in [2], were very helpful. Using this opportunity, I would like
to offer my thanks to Prof.Hai-Qing Zhang who could financially support the
conference.

Appendix

##  Appendix A Metrices with supertranslation correction

This appendix is the detailed description of [2], and in this section, we
obtain the metrices for a 4D Schwarzschild black hole spacetime with
supertranslations to the second-order.

###  A.1 Introduction of supertranslation

We start with the following coordinate system for a 4D Schwarzschild black
hole spacetime:

$\displaystyle
ds^{2}=-(1-{2m}/{r_{s}})dt_{s}^{2}+(1-{2m}/{r_{s}})^{-1}dr_{s}^{2}+r_{s}^{2}d\Omega_{s}^{2}.$
(25)

We refer to this type of coordinate system as the “Schwarzschild coordinates”.
In order to involve the supertranslations, we rewrite (25) into (14), where
$r_{s}\\!=\\!\rho_{s}\left(1+{m}/{2\rho_{s}}\right)^{2}$. We refer to this
type of expression as “isotropic coordinates”.

Note that in this relation, two $\rho_{s}$ correspond to one $r_{s}$ as

$\displaystyle\rho_{s}=\big{(}-m+r_{s}\pm\sqrt{-2mr_{s}+r_{s}^{2}}\big{)}/2.$
(26)

See Fig.1. We can see 1) positions of horizon in isotropic and Schwarzschild
coordinates correspond each other, 2) isotropic coordinates do not cover the
inside of the horizon.

Figure 1: Plot for $r_{s}=\rho_{s}\left(1+{m}/{2\rho_{s}}\right)^{2}$ for
$m=1$.

We denote the supertranslated isotropic coordinates as $(t,\rho,\theta,\phi)$.
These and $(t_{s},x_{s},y_{s},z_{s})$ are related like (15), where

$\displaystyle\textrm{$d\rho_{s}^{2}+\rho_{s}^{2}d\Omega_{s}^{2}$ in
(\ref{vpoe})}=dx_{s}^{2}+dy_{s}^{2}+dz_{s}^{2},\quad\textrm{$t_{s}$ in
(\ref{vpoe})}=t.$ (27)

We take $C$ we consider as (16). Description here overlaps with the one under
(16) (but one comment; if we employ $Y_{1}^{0}$ as $C$, r.h.s. of (27) results
in just $d\rho^{2}+\rho^{2}d\theta^{2}+\rho^{2}\sin^{2}\theta d\phi^{2}$,
namely no supertranslation corrections, which get involved from (30),
supertranslated $\rho$.)

###  A.2 Isotropic coordinates with supertranslations

We now write (14) in terms of $(t,\rho,\theta,\phi)$. For the parts in (14),
we can write as

$\displaystyle\rho_{s}^{2}=$ $\displaystyle\,\,x_{s}^{2}+y_{s}^{2}+z_{s}^{2},$
(28a)
$\displaystyle(1+{m}/{2\rho_{s}})^{4}(d\rho_{s}^{2}+\rho_{s}^{2}d\Omega_{s}^{2})=$
$\displaystyle\,\,g_{\rho\rho}d\rho^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}.$
(28b)

We can evaluate $d\rho_{s}^{2}+d\rho_{s}^{2}d\Omega_{s}^{2}$ as (27), with
which we obtain $g_{MN}$ from now. We write

$g_{MN}$ for the metrices of the supertranslated isotropic coordinates (29a)
$j_{MN}$ for the metrices of the supertranslated Schwarzschild coordinates
(29b)

in what follows, where $M,N$ in $g_{MN}$ and $j_{MN}$ refer to
$(t,\rho,\theta,\phi)$ and $(t,r,\theta,\phi)$.

We can obtain $\rho_{s}$ by calculating (30) using (15) to
$\varepsilon^{2}$-order as

$\displaystyle\rho_{s}=\rho-\frac{1}{8}\sqrt{\frac{5}{\pi}}\varepsilon
m(3\cos(2\theta)+1)+\frac{45\varepsilon^{2}m^{2}\sin^{2}(2\theta)}{32\pi\rho}+O(\varepsilon^{3}).$
(30)

With this we can obtain $g_{MN}$ as

$\displaystyle\bullet\hskip 5.69054ptg_{tt}\\!=$
$\displaystyle\,-\frac{(m-2\rho)^{2}}{(m+2\rho)^{2}}-\sqrt{\frac{5}{\pi}}\varepsilon
m^{2}\frac{(m-2\rho)}{(m+2\rho)^{3}}(3\cos(2\theta)+1)-\frac{5\varepsilon^{2}m^{3}}{8\pi\rho(m+2\rho)^{4}}(22m\rho-9m^{2}$
$\displaystyle\,\,+14\rho^{2}+9\cos(4\theta)(m^{2}+2m\rho-6\rho^{2})+24\rho\cos(2\theta)(m-\rho))+O(\varepsilon^{3}),$
(31a) $\displaystyle\bullet\hskip 5.69054ptg_{\rho\rho}\\!=$
$\displaystyle\,\frac{(m+2\rho)^{4}}{16\rho^{4}}+\sqrt{\frac{5}{\pi}}\varepsilon
m^{2}\frac{(m+2\rho)^{3}}{32\rho^{5}}(3\cos(2\theta)+1)+\frac{5\varepsilon^{2}m^{3}(m+2\rho)^{2}}{1024\pi\rho^{6}}(19m-28\rho$
$\displaystyle\,\,+12\cos(2\theta)(5m+4\rho)+27\cos(4\theta)(3m+4\rho))+O(\varepsilon^{3}),$
(31b) $\displaystyle\bullet\hskip 5.69054ptg_{\theta\theta}\\!=$
$\displaystyle\,\frac{(m+2\rho)^{4}}{16\rho^{2}}+\sqrt{\frac{5}{\pi}}\varepsilon
m(m+2\rho)^{3}\,\frac{3\cos(2\theta)(5m+6\rho)+m-2\rho}{64\rho^{3}}+\frac{5\varepsilon^{2}m^{2}(m+2\rho)^{2}}{2048\pi\rho^{4}}($
$\displaystyle\hskip
11.38109pt12\cos(2\theta)(39m^{2}+76m\rho+20\rho^{2})+9\cos(4\theta)(27m^{2}+44m\rho+4\rho^{2})$
$\displaystyle\,\,+249m^{2}+484m\rho+236\rho^{2})+O(\varepsilon^{3}),$ (31c)
$\displaystyle\bullet\hskip 5.69054ptg_{\phi\phi}\\!=$
$\displaystyle\,\frac{(m+2\rho)^{4}\sin^{2}(\theta)}{16\rho^{2}}+\sqrt{\frac{5}{\pi}}\varepsilon
m\frac{(m+2\rho)^{3}}{64\rho^{3}}\sin^{2}(\theta)(\cos(2\theta)(9m+6\rho)+7m+10\rho)$
$\displaystyle\,\,+\frac{5\varepsilon^{2}m^{2}\sin^{2}(\theta)(m+2\rho)^{2}}{2048\pi\rho^{4}}(249m^{2}+484m\rho+236\rho^{2}+12\cos(2\theta)(39m^{2}+76m\rho$
$\displaystyle\,\,+20\rho^{2})+9\cos(4\theta)(27m^{2}+44m\rho+4\rho^{2}))+O(\varepsilon^{3}).$
(31d)

###  A.3 Rewriting from Schwarzschild to isotropic coordinates

Since we have obtained the metrices in the supertranslated isometric
coordinates $(t,\rho,\theta,\phi)$, we will rewrite these to the following
Schwarzschild coordinates:

$\displaystyle
ds^{2}=-(1-{2\mu}/{r})dt^{2}+(1-{2\mu}/{r})^{-1}dr^{2}+j_{\theta\theta}d\phi^{2}+j_{\phi\phi}d\phi^{2}.$
(32)

Then, we will find the mass part $\mu$ cannot remain constant (if analysis is
to $\varepsilon^{1}$-order, it can be constant). Therefore, we treat $\mu$ as
$\mu(\rho)$. In what follows, we obtain 1) a relation between $r$ and $\rho$,
and 2) $\mu(\rho)$ as the solution, by solving the following relations:

$\displaystyle\bullet$
$\displaystyle\quad\\!\\!-(1-{2\mu(\rho)}/{r})=\,g_{tt},$ (33a)
$\displaystyle\bullet$
$\displaystyle\quad\\!\\!\frac{1}{1-{2\mu(\rho)}/{r}}\Big{(}\frac{dr}{d\rho}\Big{)}^{2}=\,g_{\rho\rho}.$
(33b)

The argument in $\mu(\rho)$ should be $\rho$. If we express $\mu(\rho)$ in
terms of $r$, (37) is plugged in.

We can obtain the $r$ satisfying (33a) to $\varepsilon^{2}$-order as

$\displaystyle\\!\\!r\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{\mu(\rho)(m+2\rho)^{2}}{4m\rho}+\frac{\varepsilon}{32\rho^{2}}\sqrt{\frac{5}{\pi}}(3\cos(2\theta)+1)\mu(\rho)(m^{2}-4\rho^{2})$
(34)
$\displaystyle\\!\\!\\!+\frac{5\varepsilon^{2}m\mu(\rho)}{512\pi\rho^{3}}(9\cos(4\theta)(3m^{2}-8\rho^{2})+12m^{2}\cos(2\theta)-7m^{2}+72\rho^{2})+O(\varepsilon^{3}).$

Let us obtain the $\mu(\rho)$. For this, look (33b), then plugging (34) into
the $r$, solve it for $\mu(\rho)$ order by order to $\varepsilon^{2}$-order.
As a result we can obtain

$\displaystyle\mu(\rho)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!m+\frac{c_{1}\rho\varepsilon}{(m+2\rho)^{2}}+\frac{\varepsilon^{2}}{64\pi\rho^{2}(m+2\rho)^{3}}\\{-24\sqrt{5\pi}c_{1}m\rho^{2}\cos(2\theta)(m-2\rho)$
(35) $\displaystyle\\!\\!\\!+\,8\rho^{2}(8\pi
c_{2}\rho(m+2\rho)-\sqrt{5\pi}c_{1}m(m-2\rho))$
$\displaystyle\\!\\!\\!+\,30m^{3}\sin^{2}(2\theta)(m+2\rho)(m^{2}-12\rho^{2})\\}+O(\varepsilon^{3}),$

where we took the integral constant at $\varepsilon^{0}$-order so that
$\varepsilon^{0}$-order becomes $m$. $c_{1,2}$ are the integral constants at
$\varepsilon^{1,2}$-orders (these can depend on ($\theta$, $\phi$) and
determined at (41)).

Now we have obtained the relation “$r=\cdots$” as in (34), with which,
rewriting the Schwarzschild to the isotropic coordinates to
$\varepsilon^{2}$-order is possible:

$\displaystyle-(1-{2\mu(\rho)}/{r})dt^{2}+(1-{2\mu(\rho)}/{r})^{-1}dr^{2}+j_{\theta\theta}d\theta^{2}+j_{\phi\phi}d\phi^{2}$
$\displaystyle\rightarrow
g_{tt}dt^{2}+g_{\rho\rho}d\rho^{2}+\Big{(}j_{\theta\theta}+\frac{1}{1-\frac{2\mu(\rho)}{r}}\Big{(}\frac{\partial
r}{\partial\theta}\Big{)}^{2}\Big{)}d\theta^{2}+\frac{2}{1-\frac{2\mu(\rho)}{r}}\frac{\partial
r}{\partial\rho}\frac{\partial r}{\partial\theta}d\rho
d\theta+j_{\phi\phi}d\phi^{2}\,.$ (36)

However what is needed is rewriting from the isotropic to Schwarzschild
coordinates.

###  A.4 Rewriting from isotropic to Schwarzschild coordinates

We will obtain the relation between $\rho$ and $r$ in the form “$\rho=\cdots$”
to $\varepsilon^{2}$-order to become possible to rewrite (A.3) in the opposite
direction. For this, there are two ways: to solve 1) (33b) or 2) (34). As a
result of our try, if we solve to $\varepsilon^{1}$-order, we can get the same
$\rho$ from either of them (we checked this sameness numerically). However, if
we try to obtain to $\varepsilon^{2}$-order, we can obtain only from 2) (for
some technical reason of mathematica).

Writing what we did, plugging $\mu(\rho)$ in (35) into the $\mu$ in (34), then
expanding it to $\varepsilon^{2}$-order, we can obtain $\rho$ order by order.
As a result, four solutions are obtained. At this time, the
$\varepsilon^{0}$-order in the two of these do not agree with (26), while
those of the rest two can agree with (26). Therefore, we employ the latter
two, which are

$\displaystyle\rho^{(1,2)}(r)\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{1}{2}(r-m\mp\sqrt{r(r-2m)})+\frac{\varepsilon}{8\pi
mr(2m-r)}\\{\pi c_{1}r(r-2m)\pm\pi c_{1}m\sqrt{r(r-2m)}$ (37)
$\displaystyle\\!\\!\\!\\!\mp\pi
c_{1}r\sqrt{r(r-2m)}-\sqrt{5\pi}m^{2}r(3\cos(2\theta)+1)(r-2m)\\}-\frac{\varepsilon^{2}}{128\pi
mr^{2}(r-2m)^{2}}[$ $\displaystyle\\!\\!\\!\\!16\pi
c_{2}(r^{4}\mp\sqrt{r^{7}(r-2m)})-120m^{4}r(4r\pm\sqrt{r(r-2m)})$
$\displaystyle\\!\\!\\!\\!+60m^{3}(16r^{3}\pm
9\sqrt{r^{5}(r-2m)})-8m^{2}\\{(60\sqrt{r^{7}(r-2m)}\mp 8\pi
c_{2}r^{2})+r^{4}\pm 75$ $\displaystyle\\!\\!\\!\\!+4\pi
c_{2}r\sqrt{r(r-2m)}\\}+m\\{120r^{5}\pm 120\sqrt{r^{9}(r-2m)}\mp 2\pi
c_{1}{}^{2}\sqrt{r(r-2m)}$ $\displaystyle\\!\\!\\!\\!-64\pi c_{2}r^{3}\pm
48\pi c_{2}\sqrt{r^{5}(r-2m)}\\}+60mr(2m-r)\cos(4\theta)\\{$
$\displaystyle\\!\\!\\!\\!m^{2}(4r\pm\sqrt{r(r-2m)})+2(r^{3}\pm\sqrt{r^{5}(r-2m)})-2mr(3r\pm
2\sqrt{r(r-2m)})\\}$ $\displaystyle\\!\\!\\!\\!]+O(\varepsilon^{3}),$

where the $1$ and $2$ in the $\rho^{(1,2)}(r)$ correspond as

(1,2) $\to$ $(+,-)$ of $\pm$ and $(-,+)$ of $\mp$. (38)

Let us determine which $\rho^{(1,2)}(r)$ we employ and determine $c_{1,2}$.
For this, plugging $\rho^{(1,2)}$ in (37) into $\mu(\rho)$ in (35), write it
in terms of $r$ to $\varepsilon^{2}$-order as

$\displaystyle\mu^{(1,2)}\big{(}\rho^{(1,2)}(r)\big{)}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!m+\frac{c_{1}\varepsilon}{4r}+\frac{\varepsilon^{2}}{8\pi
mr(\sqrt{r(r-2m)}\mp r)^{3}(\sqrt{r(r-2m)}\pm m\mp r)^{2}}[$ (39)
$\displaystyle\\!\\!\\!\\!\pi\\{-4c_{2}m^{4}r\sqrt{r(r-2m)}+48c_{2}m^{3}\sqrt{r^{5}(r-2m)}-c_{1}{}^{2}m^{3}\sqrt{r(r-2m)}$
$\displaystyle\\!\\!\\!\\!-80c_{2}m^{2}\sqrt{r^{7}(r-2m)}+12c_{1}{}^{2}m^{2}r\sqrt{r(r-2m)}+32c_{2}m\sqrt{r^{9}(r-2m)}$
$\displaystyle\\!\\!\\!\\!+r(\pm 7m^{3}\mp 28m^{2}r\pm 28mr^{2}\mp
8r^{3})(4c_{2}mr+c_{1}{}^{2})+8c_{1}{}^{2}\sqrt{r^{7}(r-2m)}$
$\displaystyle\\!\\!\\!\\!-20c_{1}{}^{2}m\sqrt{r^{5}(r-2m)}\\}-30m^{4}r\sin^{2}(2\theta)\\{m^{2}(\sqrt{r(r-2m)}\mp
7r)$ $\displaystyle\\!\\!\\!\\!\pm 3mr(5r\mp 3\sqrt{r(r-2m)})+6(\mp
r^{3}+\sqrt{r^{5}(r-2m)})\\}]+O(\varepsilon^{3}).$

Behavior of these at the distant region is given as

$\displaystyle\mu^{(1)}\big{(}\rho^{(1)}(r)\big{)}\\!=$
$\displaystyle\,m+\varepsilon\big{(}\frac{c_{1}}{4r}+O\left(r^{-2}\right)\big{)}+\varepsilon^{2}\Big{(}\frac{15r^{2}\sin^{2}(2\theta)}{2\pi
m}-\frac{45\sin^{2}(2\theta)r}{2\pi}+\frac{45m\sin^{2}(2\theta)}{4\pi}$
$\displaystyle+\frac{15m^{2}\sin^{2}(2\theta)/\pi+c_{2}}{4r}+O\left(r^{-2}\right)\Big{)}+O(\varepsilon^{3}),$
(40a) $\displaystyle\mu^{(2)}\big{(}\rho^{(2)}(r)\big{)}\\!=$
$\displaystyle\,m+\varepsilon\big{(}\frac{c_{1}}{4r}+O\left(r^{-6}\right)\big{)}+\frac{\varepsilon^{2}}{4}\Big{(}\frac{c_{2}}{r}+\frac{2c_{1}{}^{2}-45m^{4}\sin^{2}(2\theta)/\pi}{8mr^{2}}-\frac{45m^{4}\sin^{2}(2\theta)}{8\pi
r^{3}}$ $\displaystyle-\frac{105m^{5}\sin^{2}(2\theta)}{16\pi
r^{4}}-\frac{135m^{6}\sin^{2}(2\theta)}{16\pi
r^{5}}+O\left(r^{-6}\right)\Big{)}+O\left(\varepsilon^{3}\right).$ (40b)

It can be seen from the above we should discard $\mu^{(1)}$ by the reason: It
is always diverged at the distant region irrelevantly of $c_{1,2}$ for the
terms, $\frac{15r^{2}\sin^{2}(2\theta)}{2\pi
m}-\frac{45r\sin^{2}(2\theta)}{2\pi}$. Thus, it is enough only with
$\mu^{(2)}$ in the following, but we proceed with both just in case.

Now we determine $c_{1,2}$. Since these are integral constants, we are allowed
to take these arbitrarily. However in this study, by the reason written in
what follows we will take as

$\displaystyle c_{1,2}=0.$ (41)

Looking $\mu^{(2)}(\rho^{(2)}(r))$, we can find that it diverges at $m=0$
unless $c_{1}$ is zero for the term $\frac{2c_{1}{}^{2}}{32mr^{2}}$ at its
$\varepsilon^{2}$-order. Hence we take $c_{1}$ to $0$.

As for our logic for $c_{2}$, 1) consider starting with just a flat spacetime
patched by Schwarzschild coordinates with the zero mass, therefore $\mu^{(2)}$
at the starting stage is zero. 2) Suppose changing it to the isotropic
coordinates, involve the supertranslations. Then, back the coordinates to the
Schwarzschild. At this time, the expanded $\mu^{(2)}$ is given by (40). 3) At
this time, the mass should be zero, therefore $\mu^{(2)}$ should be zero.
However, if $c_{2}$ is not zero, we can see $\mu^{(2)}$ is not zero for the
terms $\frac{c_{1}}{4r}$ at the $\varepsilon^{1}$-order and $\frac{c_{2}}{4r}$
at the $\varepsilon^{2}$-order. 4) As $c_{1}$ has been taken to zero in the
above, we take $c_{2}$ to zero.

Above, we have considered in terms of the supertranslation toward the flat
spacetime and based on the consideration that mass in the spacetime should not
be changed by the supertranslation. The same issue is taken up in Sec.24.2 in
[45]. There, again mass is not introduced, though $C_{zz}$ and
$C_{\bar{z}\bar{z}}$ are introduced.

With (41), (37) and (39) are fixed as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\rho^{(1,2)}=\frac{1}{2}(\mp\sqrt{r(r-2m)}-m+r)+\frac{1}{8}\sqrt{\frac{5}{\pi}}\varepsilon
m(3\cos(2\theta)+1)-\frac{15\varepsilon^{2}\sin^{2}(2\theta)}{16\pi
r^{2}(r-2m)^{2}}\\{$ $\displaystyle\hskip 2.84526pt-10mr^{4}\mp
8m\sqrt{r^{7}(r-2m)}+2(r^{5}\pm\sqrt{r^{9}(r-2m)})-2m^{3}r(4r\pm\sqrt{r(r-2m)})$
$\displaystyle\hskip 2.84526pt+m^{2}(16r^{3}\pm
9\sqrt{r^{5}(r-2m)})\\}+O(\varepsilon^{3}),$ (42)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mu^{(1,2)}(\rho^{(1,2)}(r))=m+\frac{15\varepsilon^{2}m^{3}\sin^{2}(2\theta)(m^{2}-3(\sqrt{r(r-2m)}\pm
m\mp r)^{2})}{8\pi(r\mp\sqrt{r(r-2m)})^{2}(\sqrt{r(r-2m)}\pm m\mp
r)^{2}}+O(\varepsilon^{3}).$ (43)

Using these we can rewrite the isotropic to the Schwarzschild coordinates as

$\displaystyle\\!\\!\\!\\!g_{tt}dt^{2}+g_{\rho\rho}d\rho^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$
(44) $\displaystyle\rightarrow$
$\displaystyle\\!\\!\\!\\!-\Big{(}1-\frac{2\mu(\rho)}{r}\Big{)}dt^{2}+\Big{(}1-\frac{2\mu(\rho)}{r}\Big{)}^{-1}dr^{2}+\Big{(}g_{\theta\theta}+g_{\rho\rho}\Big{(}\frac{\partial\rho}{\partial\theta}\Big{)}^{2}\Big{)}d\theta^{2}$
$\displaystyle\\!\\!\\!\\!+\,2g_{\rho\rho}\frac{\partial\rho}{\partial
r}\frac{\partial\rho}{\partial\theta}d\rho d\theta+g_{\phi\phi}d\phi^{2}$
$\displaystyle\\!\\!\\!\\!\equiv
j_{tt}dt^{2}+j_{rr}dr^{2}+j_{\theta\theta}d\theta^{2}+2j_{r\theta}drd\theta+j_{\phi\phi}d\phi^{2},$

where $\rho=\rho(r,\theta)$ and $g_{MN}$ are in (31). We give the expressions
of $j_{MN}$ in the next.

###  A.5 Metrices with correction to $\varepsilon^{2}$-order in 4D
Schwarzschild coord.

We give the expression of $j_{MN}$ in (44) in the case of (41).

$\displaystyle\bullet\hskip 5.69054ptj^{(1,2)}_{tt}\\!=$
$\displaystyle\,-(1-\frac{2m}{r})+\frac{15\varepsilon^{2}\sin^{2}(2\theta)(m^{5}-3m^{3}(\sqrt{r(r-2m)}\pm
m\mp r)^{2})}{4\pi r(r\mp\sqrt{r(r-2m)})^{2}(\sqrt{r(r-2m)}\pm m\mp
r)^{2}}+O(\varepsilon^{3}),$ (45a) $\displaystyle\bullet\hskip
5.69054ptj^{(1,2)}_{rr}\\!=$
$\displaystyle\,(1-\frac{2m}{r})^{-1}+\frac{15\varepsilon^{2}\sin^{2}(2\theta)}{4\pi
m(r-2m)^{2}}\\{m^{3}+3m^{2}r-6mr^{2}\pm 2(\sqrt{r^{5}(r-2m)}\pm r^{3})$
$\displaystyle\mp 4mr\sqrt{r(r-2m)}\\}+O(\varepsilon^{3}),$ (46a)
$\displaystyle\bullet\hskip 5.69054ptj^{(1)}_{\theta\theta}\\!=$
$\displaystyle\,r^{2}-\frac{3\sqrt{\frac{5}{\pi}}\varepsilon
m\cos(2\theta)(r-\sqrt{r(r-2m)})^{4}}{2(\sqrt{r(r-2m)}+m-r)^{3}}+\frac{15\varepsilon^{2}m^{2}}{8\pi
r(r-2m)^{2}(\sqrt{r(r-2m)}+m-r)^{3}}$
$\displaystyle\times\\{r^{2}(r-2m)^{2}(\cos(4\theta)(m^{2}+6mr-9r^{2})-(m^{2}-18mr+15r^{2}))$
$\displaystyle+12m^{3}\sqrt{r^{5}(r-2m)}(\cos(4\theta)-1)+m^{2}\sqrt{r^{7}(r-2m)}(24\cos(4\theta)+72)$
$\displaystyle-m\sqrt{r^{9}(r-2m)}(33\cos(4\theta)+63)+\sqrt{r^{11}(r-2m)}(9\cos(4\theta)+15)\\}+O(\varepsilon^{3}),$
(47a) $\displaystyle j^{(2)}_{\theta\theta}\\!=$
$\displaystyle\,r^{2}+\frac{3\sqrt{\frac{5}{\pi}}\varepsilon
m\cos(2\theta)(\sqrt{r(r-2m)}+r)^{4}}{2(\sqrt{r(r-2m)}-m+r)^{3}}$
$\displaystyle+\frac{}{8\pi
r(r-2m)^{2}\\{m^{4}-4m^{3}(\sqrt{r(r-2m)}+4r)+20m^{2}r(\sqrt{r(r-2m)}+2r)}$
$\displaystyle\frac{15\varepsilon^{2}m^{2}}{-24m\sqrt{r^{5}(r-2m)}-32mr^{3}+8(\sqrt{r^{7}(r-2m)}+r^{4})\\}}[$
$\displaystyle-4m^{5}r^{2}+8m^{4}(2\sqrt{r^{5}(r-2m)}+13r^{3})-5m^{3}(32\sqrt{r^{7}(r-2m)}+73r^{4})$
$\displaystyle+m^{2}(268\sqrt{r^{9}(r-2m)}+409r^{5})+\cos(4\theta)\\{4m^{5}r^{2}-8m^{4}(2\sqrt{r^{5}(r-2m)}+r^{3})$
$\displaystyle+m^{3}(-32\sqrt{r^{7}(r-2m)}-115r^{4})+m^{2}(116\sqrt{r^{9}(r-2m)}+191r^{5})$
$\displaystyle-84m\sqrt{r^{11}(r-2m)}-102mr^{6}+18(\sqrt{r^{13}(r-2m)}+r^{7})\\}$
$\displaystyle+30(\sqrt{r^{13}(r-2m)}+r^{7})-6m(26\sqrt{r^{11}(r-2m)}+31r^{6})]+O(\varepsilon^{3}),$
(47b) $\displaystyle\bullet\hskip 5.69054ptj^{(1,2)}_{r\theta}\\!=$
$\displaystyle\,-\frac{3\sqrt{\frac{5}{\pi}}\varepsilon
m\sin(2\theta)(r\mp\sqrt{r(r-2m)})^{4}}{8\sqrt{r(r-2m)}(\sqrt{r(r-2m)}\pm m\mp
r)^{3}}\pm\frac{15\varepsilon^{2}m^{4}r\sin(4\theta)}{4\pi(2m-r)(\sqrt{r(r-2m)}\pm
m\mp r)^{3}}$ $\displaystyle+O(\varepsilon^{3}),$ (48a)
$\displaystyle\bullet\hskip 5.69054ptj^{(1)}_{\phi\phi}\\!=$
$\displaystyle\,r^{2}\sin^{2}(\theta)-\frac{3\sqrt{\frac{5}{\pi}}\varepsilon
m\sin^{2}(2\theta)(r-\sqrt{r(r-2m)})^{4}}{8(\sqrt{r(r-2m)}+m-r)^{3}}$
$\displaystyle+\frac{15\varepsilon^{2}m^{2}\sin^{2}(2\theta)}{8\pi
r(r-2m)^{2}(\sqrt{r(r-2m)}+m-r)^{3}}\\{12m^{3}\cos(2\theta)\sqrt{r^{5}(r-2m)}$
$\displaystyle-12m^{3}\sqrt{r^{5}(r-2m)}+24m^{2}\sqrt{r^{7}(r-2m)}+r^{2}\cos(2\theta)(r-2m)^{2}(m^{2}-3r^{2})$
$\displaystyle-r^{2}(r-2m)^{2}(m^{2}-6mr+3r^{2})+3\cos(2\theta)\sqrt{r^{11}(r-2m)}+3\sqrt{r^{11}(r-2m)}$
$\displaystyle-9m\cos(2\theta)\sqrt{r^{9}(r-2m)}-15m\sqrt{r^{9}(r-2m)}\\}+O(\varepsilon^{3}),$
(49a) $\displaystyle j^{(2)}_{\phi\phi}\\!=$
$\displaystyle\,r^{2}\sin^{2}(\theta)+\frac{3\sqrt{\frac{5}{\pi}}\varepsilon
m\sin^{2}(2\theta)(\sqrt{r(r-2m)}+r)^{4}}{8(\sqrt{r(r-2m)}-m+r)^{3}}$
$\displaystyle+\frac{15\varepsilon^{2}m^{2}\sin^{2}(2\theta)}{4\pi
r^{2}(r-2m)^{2}(\sqrt{r(r-2m)}-m+r)^{4}}[-4m^{5}r^{3}+8m^{4}(2\sqrt{r^{7}(r-2m)}+7r^{4})$
$\displaystyle-m^{3}(64\sqrt{r^{9}(r-2m)}+125r^{5})+m^{2}(76\sqrt{r^{11}(r-2m)}+109r^{6})$
$\displaystyle+\cos(2\theta)\\{4m^{5}r^{3}-16m^{4}(\sqrt{r^{7}(r-2m)}+2r^{4})+m^{3}(16\sqrt{r^{9}(r-2m)}+5r^{5})$
$\displaystyle+m^{2}(20\sqrt{r^{11}(r-2m)}+41r^{6})+6(\sqrt{r^{15}(r-2m)}+r^{8})$
$\displaystyle-6m(4\sqrt{r^{13}(r-2m)}+5r^{7})\\}+6(\sqrt{r^{15}(r-2m)}+r^{8})$
$\displaystyle-6m(6\sqrt{r^{13}(r-2m)}+7r^{7})]+O(\varepsilon^{3}).$ (49b)

1) numbers in the superscripts mean those $j_{MN}$ are associated with which
of $\rho^{(1,2)}$ in (A.4) with (38). (Origin of $j^{(1,2)}_{MN}$ is (37),
then it turns out above (41) that $\mu^{(1)}$ is unphysical and $\mu^{(2)}$ is
physical. $j^{(2)}_{MN}$ is associated with $\mu^{(2)}$.) 2) Killing vector in
the system above is $\xi^{M}=(1,0,0,1)$ as well. With either $j^{(1,2)}_{MN}$,
3) Einstein eq. is satisfied to $\varepsilon^{2}$-order. Also from either
$j^{(1,2)}_{MN}$, 4) (31) can be obtained using following one, (34) with (35)
and (41):

$\displaystyle r\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{m^{2}}{4\rho}+m+\rho+\frac{\sqrt{\frac{5}{\pi}}m(m^{2}-4\rho^{2})(3\cos(2\theta)+1)}{32\rho^{2}}\varepsilon-\frac{5m^{4}(1-12\cos(2\theta)-21\cos(4\theta))}{512\pi\rho^{3}}\varepsilon^{2}$
(50) $\displaystyle\\!\\!\\!\\!+O(\varepsilon^{3}).$

###  A.6 Comment on $c_{1,2}$

Positions of the horizon in supertranslated isotropic and Schwarzschild
coordinates are

$\displaystyle\rho_{h,4D}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{m}{2}+\frac{1}{8}\sqrt{\frac{5}{\pi}}m\varepsilon(3\cos(2\theta)+1)-\frac{45}{16\pi}\varepsilon^{2}m\sin^{2}(2\theta)+O(\varepsilon^{3}),$
(51) $\displaystyle r_{h,4D}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!2m-\frac{15}{8\pi}\varepsilon^{2}m\sin^{2}(2\theta)+O(\varepsilon^{3}),$
(52)

where $r_{h,4D}$ gets to this above regardless of $j^{(1,2)}_{MN}$. Then it
turns out $\rho_{h,4D}$ can be transformed to $r_{h,4D}$ through (34)
($\mu(\rho)$ is replaced by (35)), but $r_{h,4D}$ is transformed to

$\displaystyle\frac{m}{2}+\sqrt{\frac{5}{\pi}}m\varepsilon(\frac{3\cos(2\theta)+1}{8}-\frac{1}{16}\sqrt{6(-1+\cos(4\theta))})-\frac{45\varepsilon^{2}m\sin^{2}(2\theta)}{16\pi}+O(\varepsilon^{3}).$
(53)

through (A.4).

Since $j_{MN}$ can be transformed to $g_{MN}$ by (A.4), $r$-coordinate
corresponds to $\rho$-coordinate by (A.4). However, as mentioned above (41),
there is freedom for how to take $c_{1,2}$. Moreover, as the problem of how
the coordinates are patched toward a spacetime, there is ambiguity up to
$c_{1,2}$ for the mapping of each point in $r$-coordinate to each point in
$\rho$-coordinate through (A.4), and vice verse through (34).

Actually, the position of horizon in the Schwarzschild coordinate is obtained
if one proceeds calculation with unfixed $c_{1,2}$ as

$\displaystyle
2m+\frac{c_{1}\varepsilon}{4m}+\varepsilon^{2}(\frac{c_{2}}{4m}-\frac{15m\sin^{2}(2\theta)}{8\pi})+O(\varepsilon^{3}).$
(54)

Here, the position of horizon in the Schwarzschild coordinate obtained from
$\rho_{h,4D}$ through (34) (this (34) is given with unfixed $c_{1,2}$) is
(54). Therefore, the position of horizon in the isotropic coordinate is always
mapped to that in the Schwarzschild coordinate.

Toward (54), if we take as $c_{1}=0$ and
$c_{2}=\frac{15m^{2}\sin^{2}(2\theta)}{2\pi}$, $\rho_{h,4D}$ can be obtained
through (A.4) (this (A.4) is given with these $c_{1,2}$). However, the
$r_{h,4D}$ at that time is $2m+O(\varepsilon^{3})$.

One may consider to determine $c_{1,2}$ based on agreement of the positions of
horizon. However these should be zero for the reason under (41), therefore the
positions of horizon determine to those obtained from $c_{1,2}=0$, (51) and
(52).

##  Appendix B 2D effective near-horizon action

We have obtained the metrices with the supertranslation correction to the
second-order in the Schwarzschild coordinates. In this section, obtaining the
near-horizon expression of these, we consider the scalar field theory. Then,
expanding the field by the spherical harmonics, we integrate out its
($\theta$, $\phi$). We will finally obtain 2D effective near-horizon action.
The scalar field theory we consider is

$\displaystyle S_{\rm scalar}=\frac{1}{2}\int
d^{4}x\,\sqrt{-g}\,j^{MN}\partial_{M}\phi^{*}\partial_{N}\phi,$ (55)

where $M,N=t,r,\theta$, $\phi$, and $j^{MN}$ mean $j^{(2)MN}$. We do not
include the mass and interaction terms, since these are ignorable in the near-
horizon [46]222 It is considered that theories effectively become 2D free
massless in the near-horizon at the classical level as the particles
effectively fall freely and these longitudinal motions get dominant. .

###  B.1 Near-horizon metrices

To obtain the near-horizon expression of (55), we first obtain the 4D metrices
$j^{(2)}_{MN}$ in Sec.A.5 in the near-horizon. For this, we replace $r$ in
those $j^{(2)}_{MN}$ with $r_{h,4D}+\Delta r$ ($\Delta r=r-r_{h,4D}$ and
$r_{h,4D}$ is given in (52))333 We give the Jacobian and partial derivatives
when we change from $r$ to $\Delta r$. We denote the old and new coordinates
as ($r$, $\theta$) to ($\Delta r$, $\tilde{\theta}$). The relations between
these are $\Delta r=r-r_{h}(\theta)$ and $\tilde{\theta}=\theta$. Then,
$drd\theta=\bigg{|}\begin{array}[]{cc}\frac{\partial r}{\partial(\Delta
r)}&\frac{\partial r}{\partial\tilde{\theta}}\\\
\frac{\partial\theta}{\partial(\Delta
r)}&\frac{\partial\theta}{\partial\tilde{\theta}}\\\
\end{array}\bigg{|}d(\Delta r)d\tilde{\theta}=d(\Delta r)d\tilde{\theta}$.
Further, $\frac{\partial}{\partial r}=\frac{\partial(\Delta r)}{\partial
r}\frac{\partial}{\partial(\Delta r)}+\frac{\partial\tilde{\theta}}{\partial
r}\frac{\partial}{\partial\tilde{\theta}}=\frac{\partial}{\partial(\Delta r)}$
and $\frac{\partial}{\partial\theta}=\frac{\partial(\Delta
r)}{\partial\theta}\frac{\partial}{\partial(\Delta
r)}+\frac{\partial\tilde{\theta}}{\partial\theta}\frac{\partial}{\partial\tilde{\theta}}=-\frac{\partial
r_{h}(\theta)}{\partial\theta}\frac{\partial}{\partial(\Delta
r)}+\frac{\partial}{\partial\tilde{\theta}}.$ , then expand around $\Delta
r=0$. Writing these as $t_{MN}$,

$\displaystyle t_{tt}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{(}-\frac{\Delta r}{2m}+\frac{\Delta
r^{2}}{4m^{2}}+O\left(\Delta
r^{3}\right)\Big{)}+\varepsilon^{2}\frac{15\sin^{2}(2\theta)}{4\pi}\Big{\\{}\frac{3\Delta
r}{4m}-\frac{\sqrt{2}\Delta r^{3/2}}{m^{3/2}}+\frac{3\Delta r^{2}}{4m^{2}}$
(56) $\displaystyle+\frac{\Delta r^{5/2}}{2\sqrt{2}m^{5/2}}+O\left(\Delta
r^{3}\right)\Big{\\}}+O(\varepsilon^{3}),$ $\displaystyle t_{rr}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\Big{(}\frac{2m}{\Delta
r}+1+O\left(\Delta
r^{3}\right)\Big{)}+\varepsilon^{2}\frac{15\sin^{2}(2\theta)}{\pi}\Big{\\{}\frac{3m}{4\Delta
r}-\frac{\sqrt{2m}}{\sqrt{\Delta r}}+\frac{6}{4}-\frac{3\sqrt{\Delta
r}}{2\sqrt{2m}}+\frac{2\Delta r}{4m}$ (57) $\displaystyle-\frac{3\Delta
r^{3/2}}{16\sqrt{2}m^{3/2}}+O\left(\Delta
r^{5/2}\right)\Big{\\}}+O(\varepsilon^{3}),$ $\displaystyle
t_{\theta\theta}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{(}4m^{2}+4\Delta rm+\Delta r^{2}+O\left(\Delta
r^{3}\right)\Big{)}+\varepsilon
24\sqrt{\frac{5}{\pi}}\cos(2\theta)\Big{\\{}m^{2}-m^{3/2}\sqrt{2}\sqrt{\Delta
r}$ (58) $\displaystyle+2m\Delta r-\frac{5}{4}\sqrt{2m}\Delta
r^{3/2}+\frac{5}{4}\Delta r^{2}+O\left(\Delta
r^{5/2}\right)\Big{\\}}+\frac{90\varepsilon^{2}}{\pi}\Big{\\{}2m^{2}\cos^{2}(2\theta)$
$\displaystyle-4\sqrt{2}m^{3/2}\cos^{2}(2\theta)\sqrt{\Delta
r}+\frac{m(41\cos(4\theta)+39)\Delta
r}{8}-\frac{\sqrt{2m}}{3}(14\cos(4\theta)+13)\Delta r^{3/2}$
$\displaystyle+\frac{5(21\cos(4\theta)+19)\Delta r^{2}}{16}+O\left(\Delta
r^{5/2}\right)\Big{\\}}+O\left(\varepsilon^{3}\right),$ $\displaystyle
t_{\phi\phi}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sin^{2}(\theta)\left(4m^{2}+4\Delta rm+\Delta
r^{2}+O\left(\Delta
r^{3}\right)\right)+3\varepsilon\sqrt{\frac{5}{\pi}}\sin^{2}(2\theta)\Big{\\{}2m^{2}-2\sqrt{2}m^{3/2}\sqrt{\Delta
r}$ (59) $\displaystyle+4m\Delta r-5\sqrt{m}\Delta r^{3/2}+\frac{3}{2}\Delta
r^{2}+O\left(\Delta
r^{5/2}\right)\Big{\\}}+\varepsilon^{2}\frac{45\sin^{2}(2\theta)}{\pi}\Big{\\{}m^{2}\cos(2\theta)$
$\displaystyle-2\sqrt{2}m^{3/2}\cos(2\theta)\sqrt{\Delta
r}+\frac{m}{4}(21\cos(2\theta)-1)\Delta r+\frac{5(11\cos(2\theta)-1)\Delta
r^{2}}{8}$ $\displaystyle+\frac{675}{2}\sqrt{2m}(2-29\cos(2\theta))\Delta
r^{3/2}+O\left(\Delta r^{5/2}\right)\Big{\\}}+O(\varepsilon^{3}),$
$\displaystyle t_{r\theta}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-3\varepsilon\sqrt{\frac{5}{\pi}}\sin(2\theta)\Big{\\{}\frac{m^{3/2}\sqrt{2}}{\sqrt{\Delta
r}}-2m+\frac{7}{2}\sqrt{\frac{m}{2}}\sqrt{\Delta r}-6\Delta r+\frac{27\Delta
r^{3/2}}{16\sqrt{2m}}-\frac{\Delta r^{2}}{2m}$ (60)
$\displaystyle+O\left(\Delta
r^{5/2}\right)\Big{\\}}+\varepsilon^{2}\frac{15\sin(4\theta)}{\pi}\frac{m^{2}}{2\Delta
r}-\frac{3m^{3/2}}{\sqrt{2}\sqrt{\Delta
r}}+\frac{19m}{4}-\frac{41\sqrt{m}\sqrt{\Delta r}}{4\sqrt{2}}$
$\displaystyle+\frac{33\Delta r}{4}-\frac{329\Delta
r^{3/2}}{32\sqrt{2m}}+\frac{5\Delta r^{2}}{m}+O\left(\Delta
r^{5/2}\right)\Big{\\}}+O(\varepsilon^{3}).$

Contravariant metrices toward these are obtained as

$\displaystyle t^{tt}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{(}-\frac{2m}{\Delta r}-1-\frac{\Delta
r}{2m}+O\left(\Delta
r^{2}\right)\Big{)}-\varepsilon^{2}\frac{45\sin^{2}(2\theta)}{2\pi}\Big{(}\frac{m}{2\Delta
r}-\frac{2\sqrt{2m}}{3\sqrt{\Delta r}}+1-\frac{\sqrt{\Delta r}}{\sqrt{2m}}$
(61) $\displaystyle+\frac{7\Delta r}{8m}+O\left(\Delta
r^{3/2}\right)\Big{)}+O(\varepsilon^{3}),$ $\displaystyle t^{rr}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!\Big{(}\frac{\Delta
r}{2m}+O\left(\Delta
r^{2}\right)\Big{)}+\varepsilon^{2}\frac{15\sin^{2}(2\theta)}{\pi}\Big{(}\frac{3\Delta
r}{16m}-\frac{\Delta r^{3/2}}{\sqrt{2}m^{3/2}}+O\left(\Delta
r^{2}\right)\Big{)}+O\left(\varepsilon^{3}\right),$ (62) $\displaystyle
t^{\theta\theta}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\Big{(}\frac{1}{4m^{2}}-\frac{\Delta
r}{4m^{3}}+O\left(\Delta
r^{2}\right)\Big{)}+3\varepsilon\sqrt{\frac{5}{\pi}}\cos(2\theta)\Big{(}-\frac{1}{2m^{2}}+\frac{\sqrt{\Delta
r}}{\sqrt{2}m^{5/2}}+O\left(\Delta r^{3/2}\right)\Big{)}$ (63)
$\displaystyle+\varepsilon^{2}\frac{45}{8\pi}\Big{(}\frac{11\cos(4\theta)+13}{4m^{2}}-\frac{(11\cos(4\theta)+13)\sqrt{\Delta
r}}{\sqrt{2}m^{5/2}}+\frac{(65\cos(4\theta)+79)\Delta r}{8m^{3}}$
$\displaystyle+O\left(\Delta r^{3/2}\right)\Big{)}+O(\varepsilon^{3}),$
$\displaystyle t^{r\theta}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\varepsilon\frac{3}{4}\sqrt{\frac{5}{\pi}}\sin(2\theta)\Big{(}\frac{\sqrt{\Delta
r}}{\sqrt{2}m^{3/2}}-\frac{\Delta r}{m^{2}}+O\left(\Delta
r^{3/2}\right)\Big{)}$ (64)
$\displaystyle-\varepsilon^{2}\frac{15\sin(4\theta)}{16\pi}\Big{(}\frac{1}{m}+\frac{6\sqrt{\Delta
r}}{\sqrt{2}m^{3/2}}-\frac{16\Delta r}{m^{2}}+O\left(\Delta
r^{3/2}\right)\Big{)}+O(\varepsilon^{3}),$ $\displaystyle
t^{\phi\phi}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{\csc^{2}\theta}{4}\Big{(}\frac{1}{m^{2}}-\frac{\Delta
r}{m^{3}}+O\left(\Delta
r^{2}\right)\Big{)}+3\varepsilon\sqrt{\frac{5}{2\pi}}\cot^{2}\theta\Big{(}-\frac{1}{\sqrt{2}m^{2}}+\frac{\sqrt{\Delta
r}}{m^{5/2}}+O\left(\Delta r^{3/2}\right)\Big{)}$ (65)
$\displaystyle+\varepsilon^{2}\frac{45\cot^{2}(\theta)}{\pi}\Big{(}\frac{\cos(2\theta)+2}{4m^{2}}-\frac{(\cos(2\theta)+2)\sqrt{\Delta
r}}{\sqrt{2}m^{5/2}}+\frac{(11\cos(2\theta)+25)\Delta r}{16m^{3}}$
$\displaystyle+O\left(\Delta r^{3/2}\right)\Big{)}+O(\varepsilon^{3}).$

We can check these are the inverse each other in the range of
$\varepsilon^{2}$. (Leading of these are the same with just a Schwarzschild,
which is the technical reason for our result, no change.)

###  B.2 Hawking temperature in the original 4D

We have given the Killing vector in (A.5), and obtained the position of the
horizon in (52) and the near-horizon metrices. With these and the formula:
$\kappa^{2}=-\frac{1}{2}D^{M}\xi^{N}D_{M}\xi_{N}$, the Hawking temperature in
the original 4D spacetime can be obtained as

$\displaystyle T_{H}={1}/{8\pi m}+O(\varepsilon^{3}).$ (66)

This is the same with the one in just the Schwarzschild. We can understand
this as follows.

Generally,
$T_{H}=\frac{1}{4\pi}\big{|}\partial_{r}f(r)\big{|}_{r=r_{h}}\big{|}$ for
$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+\cdots$, where $f(r)|_{r=r_{h}}=0$ (these
$f(r)$ and $r_{h}$ are irrelevant with this study). However, our $f(r)$
behaves same with just the Schwarzschild at $\Delta r=0$ ($r=r_{h,4D}$) as in
(56).

Our Hawking temperature might have been expected to depend on the angular
directions, which breaks the zeroth law of black hole thermodynamics. However,
we could expect from the result above that it would be always out of the
analysis’s order.

###  B.3 Near-horizon action

Having obtained 4D near-horizon metrices, let us obtain near-horizon action.
For this, we write down (55) term by term, then express each line order by
order as to $\varepsilon^{2}$-order as

$\displaystyle\textrm{$\cal L$ of (\ref{sacf})}=-$
$\displaystyle\phi^{*}\partial_{t}\big{\\{}\big{(}\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\big{)}\big{(}t^{tt}{}^{(0)}+t^{tt}{}^{(2)}\big{)}\partial_{t}\big{\\}}\phi$
(67a) $\displaystyle-$
$\displaystyle\phi^{*}\partial_{r}\big{\\{}\big{(}\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\big{)}\big{(}t^{rr}{}^{(0)}+t^{rr}{}^{(2)}\big{)}\partial_{r}\big{\\}}\phi$
(67b) $\displaystyle-$
$\displaystyle\phi^{*}\partial_{r}\big{\\{}\big{(}\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\big{)}\big{(}t^{r\theta}{}^{(1)}+t^{r\theta}{}^{(2)}\big{)}\partial_{\theta}\big{\\}}\phi$
(67c) $\displaystyle-$
$\displaystyle\phi^{*}\partial_{\theta}\big{\\{}\big{(}\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\big{)}\big{(}t^{\theta\theta}{}^{(0)}+t^{\theta\theta}{}^{(1)}+t^{\theta\theta}{}^{(2)}\big{)}\partial_{\theta}\big{\\}}\phi$
(67d) $\displaystyle-$
$\displaystyle\phi^{*}\partial_{\theta}\big{\\{}\big{(}\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\big{)}\big{(}t^{r\theta}{}^{(1)}+t^{r\theta}{}^{(2)}\big{)}\partial_{r}\big{\\}}\phi$
(67e) $\displaystyle-$
$\displaystyle\phi^{*}\partial_{\phi}\big{\\{}\big{(}\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\big{)}\big{(}t^{\phi\phi}{}^{(0)}+t^{\phi\phi}{}^{(1)}+t^{\phi\phi}{}^{(2)}\big{)}\partial_{\phi}\big{\\}}\phi,$
(67f)

where the numbers in the superscripts mean the part of that quantity at that
order when that quantity is expanded with regard to $\varepsilon$ 444 E.g.,
$\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}$ means the first three terms
writing as
$\sqrt{-t}=(\cdots)+(\cdots)\varepsilon+(\cdots)\varepsilon^{2}+\cdots$. .

We write the order behavior of the ingredients in (67a)-(67f) based on
(61)-(65) as

  * •

$\displaystyle\sqrt{-t}^{(0)}+\sqrt{-t}^{(1)}+\sqrt{-t}^{(2)}\sim(1+\Delta
r)+(1+\sqrt{\Delta r})\varepsilon+(1+\sqrt{\Delta r})\varepsilon^{2}$,

  * •

$\displaystyle t^{tt}{}^{(0)}+t^{tt}{}^{(2)}\sim(\Delta
r)^{-1}+\varepsilon^{2}(\Delta r)^{-1}$,

  * •

$\displaystyle t^{rr}{}^{(0)}+t^{rr}{}^{(2)}\sim\Delta r+\Delta
r\varepsilon^{2}$,

  * •

$\displaystyle t^{r\theta}{}^{(1)}+t^{r\theta}{}^{(2)}\sim\sqrt{\Delta
r}\varepsilon+(1+\sqrt{\Delta r})\varepsilon^{2}$,

  * •

$\displaystyle
t^{\theta\theta}{}^{(0)}+t^{\theta\theta}{}^{(1)}+t^{\theta\theta}{}^{(2)}\sim(1+\Delta
r)+(1+\sqrt{\Delta r})\varepsilon+(1+\sqrt{\Delta r})\varepsilon^{2}$,

  * •

$\displaystyle
t^{\phi\phi}{}^{(0)}+t^{\phi\phi}{}^{(1)}+t^{\phi\phi}{}^{(2)}\sim(1+\Delta
r)+(1+\sqrt{\Delta r})\varepsilon+(1+\sqrt{\Delta r})\varepsilon^{2}$.

With these, we can get the order behavior of the each line (67a)-(67f) as

(67a) $\displaystyle\sim$ $\displaystyle\\!\\!\\!\\!\Big{(}\frac{1}{\Delta
r}+1+O\left(\Delta r^{2}\right)\Big{)}+\varepsilon\Big{(}\frac{1}{\Delta
r}+\frac{1}{\sqrt{\Delta r}}+O\left(\Delta r^{3/2}\right)\Big{)}$
$\displaystyle+\varepsilon^{2}\Big{(}\frac{2}{\Delta r}+\frac{1}{\sqrt{\Delta
r}}+1+O\left(\Delta r^{3/2}\right)\Big{)}+O(\varepsilon^{3}),$ (67b)
$\displaystyle\sim$ $\displaystyle\\!\\!\\!\\!\Big{(}1+2\Delta r+O\left(\Delta
r^{2}\right)\Big{)}+\varepsilon\Big{(}1+\frac{3\sqrt{\Delta
r}}{2}+O\left(\Delta r^{3/2}\right)\Big{)}$
$\displaystyle+\varepsilon^{2}\Big{(}2+\frac{3\sqrt{\Delta r}}{2}+2\Delta
r+O\left(\Delta r^{3/2}\right)\Big{)}+O(\varepsilon^{3}),$ (67c)
$\displaystyle\sim$
$\displaystyle\\!\\!\\!\\!\varepsilon\Big{(}\frac{1}{2\sqrt{\Delta
r}}+\frac{3\sqrt{\Delta r}}{2}+O\left(\Delta
r^{3/2}\right)\Big{)}+\varepsilon^{2}\Big{(}\frac{1}{\sqrt{\Delta
r}}+2+\frac{3\sqrt{\Delta r}}{2}+O\left(\Delta
r^{3/2}\right)\Big{)}+O(\varepsilon^{3}),$ (67d) $\displaystyle\sim$
$\displaystyle\\!\\!\\!\\!\big{(}1+2\Delta r+O\big{(}\Delta
r^{2}\big{)}\big{)}+\varepsilon\big{(}2+2\sqrt{\Delta r}+2\Delta
r+O\big{(}\Delta r^{3/2}\big{)}\big{)}$
$\displaystyle+\varepsilon^{2}\big{(}3+4\sqrt{\Delta r}+3\Delta
r+O\big{(}\Delta r^{3/2}\big{)}\big{)}+O\big{(}\varepsilon^{3}\big{)},$ (67e)
$\displaystyle\sim$ $\displaystyle\\!\\!\\!\\!\varepsilon\big{(}\sqrt{\Delta
r}+O\big{(}\Delta r^{3/2}\big{)}\big{)}+\varepsilon^{2}\big{(}1+2\sqrt{\Delta
r}+2\Delta r+O\big{(}\Delta
r^{3/2}\big{)}\big{)}+O\big{(}\varepsilon^{3}\big{)},$ (67f)
$\displaystyle\sim$ $\displaystyle\\!\\!\\!\\!\big{(}1+2\Delta
r+O\big{(}\Delta r^{2}\big{)}\big{)}+\varepsilon\big{(}2+2\sqrt{\Delta
r}+2\Delta r+O\big{(}\Delta r^{3/2}\big{)}\big{)}$
$\displaystyle+\varepsilon^{2}\big{(}3+4\sqrt{\Delta r}+3\Delta
r+O\big{(}\Delta r^{3/2}\big{)}\big{)}+O\big{(}\varepsilon^{3}\big{)}.$

We find (67a) is dominant and others are vanish or ignorable compared with
(67a) at $\Delta r\to 0$. Therefore, from the viewpoint of which parts remain
at $\Delta r\to 0$, we may remain only (67a). However, since the ($t$,
$r$)-space is crucial in the analysis of Hawking temperature and flux, the
parts on and tangling with the ($t$, $r$)-space are indispensable in our
analysis. (Also note that (67b) makes the action at $\Delta r\to 0$ finite.)
Therefore, remaining (67b), (67c) and (67e) in addition to (67a), we consider
the following action:

(67) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sqrt{-t}(t^{tt}\partial_{t}\phi^{*}\partial_{t}\phi-t^{rr}\partial_{r}\phi^{*}\partial_{r}\phi-t^{r\theta}\partial_{r}\phi^{*}\partial_{\theta}\phi-t^{\theta
r}\partial_{\theta}\phi^{*}\partial_{r}\phi)+\cdots$ (68) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sqrt{-t}\big{(}t^{tt}\partial_{t}\phi^{*}\partial_{t}\phi+t^{rr}\big{(}\partial_{r}\phi^{*}+\frac{t^{\theta
r}}{t^{rr}}\partial_{\theta}\phi^{*}\big{)}\big{(}\partial_{r}\phi+\frac{t^{\theta
r}}{t^{rr}}\partial_{\theta}\phi\big{)}-\frac{(t^{\theta
r})^{2}}{t^{rr}}\partial_{\theta}\phi^{*}\partial_{\theta}\phi\big{)},$

as the near-horizon action, where “$\cdots$” is (67d), (67f) (the terms
vanishing at $\Delta r\to 0$ and irrelevant for the ($t$, $r$)-space) and
terms under $\varepsilon^{3}$-order.

Let us look ${(t^{\theta r})^{2}}/{t^{rr}}$ and ${t^{\theta r}}/{t^{rr}}$ in
(68). Using (61)-(65), we can write these as

$\displaystyle\bullet\quad$ $\displaystyle\hskip 15.649pt\frac{t^{\theta
r}}{t^{rr}}=\varepsilon\frac{3}{2}\sqrt{\frac{5}{\pi}}\sin(2\theta)\Big{(}\frac{1}{\sqrt{2m}\sqrt{\Delta
r}}-\frac{1}{m}+O\left(\sqrt{\Delta r}\right)\Big{)}$ $\displaystyle\hskip
42.67912pt-\varepsilon^{2}\frac{15}{8\pi}\sin(4\theta)\Big{(}\frac{1}{\Delta
r}+\frac{6}{\sqrt{2m}\sqrt{\Delta r}}+O\left(\Delta
r^{0}\right)\Big{)}+O(\varepsilon^{3}),$ (69a) $\displaystyle\bullet\quad$
$\displaystyle\frac{(t^{\theta
r})^{2}}{t^{rr}}=\varepsilon^{2}\frac{45\sin^{2}(2\theta)}{16\pi}\Big{(}\frac{1}{m^{2}}-\frac{4\sqrt{\Delta
r}}{\sqrt{2}m^{5/2}}+O\left(\Delta r^{1}\right)\Big{)}+O(\varepsilon^{3}).$
(69b)

Thus, ${(t^{\theta r})^{2}}/{t^{rr}}$ is ignorable in the limit $\Delta r\to
0$, but ${t^{\theta r}}/{t^{rr}}$ is not. However ${t^{\theta r}}/{t^{rr}}$ is
ignorable finally in the analysis of Hawking temperature and flux for the
following facts:

  * •

We can regard $\frac{t^{\theta r}}{t^{rr}}\partial_{\theta}\phi$ as the
$r$-component of U(1) gauge field in the sense that we can evaluate the
anomalies and currents associated with it using the way to evaluate those for
U(1) gauge field. The point here is that it is composed of the $t$-independent
$r$-component only, therefore we can see by looking at (4) in [43] the gauge
anomalies do not arise from $\frac{t^{\theta
r}}{t^{rr}}\partial_{\theta}\phi$. Hence, $\frac{t^{\theta
r}}{t^{rr}}\partial_{\theta}\phi$ is irrelevant of this study.

  * •

Next, as for the gravitational anomalies, since $J^{\mu}$ is zero according to
(4) in [43], the second term in r.h.s. in (16) in [43] is some constants. The
first term in the (16) will be also zero, since our gauge field is composed of
only $t$-independent $r$-component.

Hence, since ${t^{\theta r}}/{t^{rr}}$ has nothing to do with gauge and
gravitational anomalies, we can ignore it in our analysis and are allowed to
write the near-horizon action we consider as

$\displaystyle S_{\rm nh}=\frac{1}{2}\int
dx^{4}\sqrt{-t}(t^{tt}\partial_{t}\phi^{*}\partial_{t}\phi+t^{rr}\partial_{r}\phi^{*}\partial_{r}\phi).$
(70)

Let us obtain the concrete expression for (70). For this we write as

$\displaystyle\textrm{$\cal L$ of (\ref{cncwra})}=$
$\displaystyle-\phi^{*}\\{(\sqrt{-t})^{(0)}t^{tt}{}^{(0)}\\}\partial_{t}\partial_{t}\phi-\phi^{*}\partial_{r}\\{(\sqrt{-t})^{(0)}t^{rr}{}^{(0)}\partial_{r}\phi\\}$
(71a)
$\displaystyle-\phi^{*}\\{(\sqrt{-t})^{(0)}t^{tt}{}^{(2)}+((\sqrt{-t})^{(1)}+(\sqrt{-t})^{(2)})g^{tt}{}^{(0)}\\}\partial_{t}\partial_{t}\phi$
(71b)
$\displaystyle-\phi^{*}\partial_{r}\\{((\sqrt{-t})^{(0)}t^{tt}{}^{(2)}+((\sqrt{-t})^{(1)}+(\sqrt{-t})^{(2)})t^{tt}{}^{(0)})\partial_{r}\phi\\},$
(71c)

where the meaning of the numbers in the superscripts are the same with (67).

We list the ingredients needed to calculate (71) as

$\displaystyle\bullet\quad(\sqrt{-t})^{(0)}=4m\sin(\theta)(\Delta r+m),$ (72a)
$\displaystyle\bullet\quad\sum_{i=1,2}(\sqrt{-t})^{(i)}=\frac{90\varepsilon^{2}m^{2}\sin(\theta)\cos^{2}(\theta)(3\cos(2\theta)-1)}{\pi}+6\sqrt{\frac{5}{\pi}}\varepsilon
m^{2}\sin(\theta)(3\cos(2\theta)+1),$ (72b) $\displaystyle\bullet\quad
t^{tt}{}^{(0)}=-{2m}/{\Delta r},\quad t^{rr}{}^{(0)}={\Delta r}/{2m},$ (72c)
$\displaystyle\bullet\quad
t^{tt}{}^{(2)}=-{45\varepsilon^{2}m\sin^{2}(2\theta)}/{4\pi\Delta r},\quad
t^{rr}{}^{(2)}={45\Delta r\varepsilon^{2}\sin^{2}(2\theta)}/{16\pi m}.$ (72d)

Using these, we can obtain the concrete expression of (70) to
$\varepsilon^{2}$-order as

(70) $\displaystyle=$ $\displaystyle\\!\\!\\!-\int
d^{4}x\,(2m)^{2}\sin(\theta)\Big{\\{}1+\frac{3}{2}\sqrt{\frac{5}{\pi}}\left(1+3\cos(2\theta)\right)\varepsilon+\frac{45}{2\pi}\Big{(}\frac{\sin(2\theta)}{4}-\cos^{2}(\theta)$
(73)
$\displaystyle+3\cos^{2}(\theta)\cos(2\theta)\Big{)}\varepsilon^{2}\Big{\\}}\phi^{*}(t^{tt}\partial_{t}\partial_{t}+\partial_{r}(t^{rr}\partial_{r}))\phi$
$\displaystyle(t^{tt},t^{rr})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\Big{(}-\frac{2m}{\Delta r},\,\frac{\Delta
r}{2m}\Big{)}$ (74) $\displaystyle=$
$\displaystyle\\!\\!\\!\Big{(}-\frac{2m}{r-2m}-\frac{15\varepsilon^{2}m^{2}\sin^{2}(2\theta)}{4\pi(r-2m)^{2}},\,\frac{r-2m}{2m}-\frac{15\varepsilon^{2}\sin^{2}(2\theta)}{16\pi}\Big{)}.$

Note that the mass parts in the denominator of $-\frac{2m}{\Delta r}$ and
numerator of $\frac{\Delta r}{2m}$ do not agree each other. We fix this point
in (80b) by defining the effective mass (80d).

###  B.4 Integrate out of ($\theta$, $\phi$)

We will obtain the 2D effective near-horizon action by integrating out
($\theta$, $\phi$) of (73). For this, we first expand $\phi$ by the spherical
harmonics as

$\displaystyle\phi(t,r,\theta,\phi)=\sum_{l,m}\phi_{lm}(t,r)Y_{l}^{m}(\theta,\phi).$
(75)

Then defining the following $\Lambda_{lm,\,kn}$ ($d\Omega=d\theta
d\phi\sin\theta$), we can write (73) as

$\displaystyle\Lambda_{lm,\,kn}=$ $\displaystyle\int
d\Omega\Big{\\{}1+\frac{3}{2}\sqrt{\frac{5}{\pi}}\left(1+3\cos(2\theta)\right)\varepsilon$
$\displaystyle+\frac{45}{2\pi}\Big{(}\frac{\sin(2\theta)}{4}-\cos^{2}(\theta)+3\cos^{2}(\theta)\cos(2\theta)\Big{)}\varepsilon^{2}\Big{\\}}\,(Y_{l}^{m})^{*}Y_{k}^{n},$
(76a) $\displaystyle{\rm(\ref{acale2})}=$
$\displaystyle-\sum_{l,m}\sum_{k,n}\int dtdr(2m)^{2}\Big{\\{}$
$\displaystyle\hskip
14.22636pt\phi_{lm}^{*}\\!\Big{(}-\frac{2m}{r-2m}\Lambda_{lm,\,kn}-\frac{15m^{2}\varepsilon^{2}}{4\pi(r-2m)^{2}}\\!\int\\!d\Omega\sin^{2}(2\theta)(Y_{l}^{m})^{*}Y_{k}^{n}\Big{)}\partial_{t}\partial_{t}\phi_{kn}$
$\displaystyle+\phi_{lm}^{*}\partial_{r}\\!\Big{(}\frac{r-2m}{2m}\Lambda_{lm,\,kn}-\frac{15m^{2}\varepsilon^{2}}{16\pi}\\!\int\\!d\Omega\sin^{2}(2\theta)(Y_{l}^{m})^{*}Y_{k}^{n}\Big{)}\partial_{r}\phi_{kn}\Big{\\}}.$
(76b)

We can evaluate all kinds of the ($\theta$, $\phi$)-integrals in (76) (totally
four) using (77) as555 Necessary formulas for the calculations (78)-(78):
$\begin{split}&(Y_{l}^{m})^{*}=(-)^{m}Y_{l}^{-m},\quad\int
d\Omega~{}(Y_{l_{1}}^{m_{1}})^{*}Y_{l_{2}}^{m_{2}}=\delta_{l_{1}l_{2}}\delta_{m_{1}m_{2}},\quad\int
d\Omega~{}\sin 2\theta(Y_{l_{1}}^{m_{1}})^{*}Y_{l_{2}}^{m_{2}}=0\quad{\rm
and}\\\ &\int
d\Omega~{}(Y_{l_{1}}^{m_{1}})^{*}(Y_{l_{2}}^{m_{2}})^{*}Y_{L}^{M}=\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{4\pi(2L+1)}}\langle
l_{1}\,0\,l_{2}\,0|L\,0\rangle\langle
l_{1}\,m_{1}~{}l_{2}\,m_{2}|L\,M\rangle,\end{split}$ (77) where $\langle
l_{1}m_{1}l_{2}m_{2}|LM\rangle$ mean Clebsch-Gordan coefficients [47]. We can
obtain (78)-(78) using (77) by rewriting these integrands into the form of the
3 products of spherical harmonics. To be concrete, express $\cos(2\theta)$,
$\cos^{2}\theta$, $\sin^{2}(2\theta)$ and $\cos^{2}\theta\cos(2\theta)$ by
$Y_{0}^{0}$, $Y_{2}^{0}$ and $Y_{4}^{0}$ (e.g.
$\cos(2\theta)=\frac{8}{3}\sqrt{\frac{\pi}{5}}Y_{2}^{0}-\frac{2\sqrt{\pi}}{3}Y_{0}^{0}$).

$\displaystyle\bullet\quad$ $\displaystyle\int d\Omega\cos
2\theta\,(Y_{k}^{n})^{*}Y_{l}^{m}$ $\displaystyle=$
$\displaystyle\,\,-\frac{4m^{2}-1}{4l^{2}+4l-3}\delta_{kl}\delta_{nm}+\frac{2(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+5)}}}{2l+3}\delta_{k-2,l}\delta_{nm}$
$\displaystyle\equiv$ $\displaystyle\,\,{\cal
I}^{A_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{A_{2}}_{lm}\,\delta_{k-2,l}\delta_{nm}\equiv{\cal I}^{A}_{lm,\,kn},$ (78a)
$\displaystyle\bullet\quad$ $\displaystyle\int
d\Omega\cos^{2}\theta\,(Y_{k}^{n})^{*}Y_{l}^{m}$ $\displaystyle=$
$\displaystyle\,\,\frac{2l^{2}+2l-2m^{2}-1}{4(l+1)^{2}-4(l+1)-3}\delta_{kl}\delta_{nm}+\frac{(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{(2l+1)(2l+5)}}}{2l+3}\delta_{k-2,l}\delta_{nm}$
$\displaystyle\equiv$ $\displaystyle\,\,{\cal
I}^{B_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{B_{2}}_{lm}\,\delta_{k-2,l}\delta_{nm}\equiv{\cal I}^{B}_{lm,\,kn},$ (78b)
$\displaystyle\bullet\quad$ $\displaystyle\int
d\Omega\sin^{2}(2\theta)\,(Y_{k}^{n})^{*}Y_{l}^{m}$ $\displaystyle=$
$\displaystyle\,\,\frac{8(-1)^{2m}(l(l+1)(l^{2}+l-5)+2l(l+1)m^{2}-3m^{4}+3)}{(2l-3)(2l-1)(2l+3)(2l+5)}\delta_{kl}\delta_{nm}$
$\displaystyle+4(-1)^{2m-2l}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{4l^{2}+12l+5}}$
$\displaystyle\times\frac{(-1)^{2l}(2l-1)(2l+7)-4(-1)^{2m}(l(l+3)-7m^{2})}{7(2l-1)(2l+3)(2l+7)}\delta_{k-2,l}\delta_{nm}$
$\displaystyle-4(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})((l+3)^{2}-m^{2})((l+4)^{2}-m^{2})}{(2l+1)(2l+3)^{2}(2l+5)^{2}(2l+7)^{2}(2l+9)}}\delta_{k-4,l}\delta_{nm}$
$\displaystyle\equiv$ $\displaystyle\,\,{\cal
I}^{C_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{C_{2}}_{lm}\,\delta_{nm}\delta_{k-2,l}+{\cal
I}^{C_{4}}_{lm}\,\delta_{nm}\delta_{k-4,l}\equiv{\cal I}^{C}_{lm,\,kn},$ (78c)
$\displaystyle\bullet\quad$ $\displaystyle\int d\Omega\cos
2\theta\cos^{2}\theta\,(Y_{k}^{n})^{*}Y_{l}^{m}$ $\displaystyle=$
$\displaystyle\,\,\frac{(-1)^{2m}(2(-8l(l+1)m^{2}+l(l+1)(2l(l+1)-7)+6m^{4})+30m^{2}+3)}{(2l-3)(2l-1)(2l+3)(2l+5)}\delta_{kl}\delta_{nm}$
$\displaystyle+(-1)^{2m-2l}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})}{4l^{2}+12l+5}}$
$\displaystyle\times\frac{8(-1)^{2m}(l(l+3)-7m^{2})+5(-1)^{2l}(2l-1)(2l+7)}{7(2l-1)(2l+3)(2l+7)}\delta_{nm}\delta_{k-2,l}$
$\displaystyle+2(-1)^{2m}\sqrt{\frac{((l+1)^{2}-m^{2})((l+2)^{2}-m^{2})((l+3)^{2}-m^{2})((l+4)^{2}-m^{2})}{(2l+1)(2l+3)^{2}(2l+5)^{2}(2l+7)^{2}(2l+9)}}\delta_{nm}\delta_{k-4,l}$
$\displaystyle\equiv$ $\displaystyle\,\,{\cal
I}^{D_{0}}_{lm}\,\delta_{kl}\delta_{nm}+{\cal
I}^{D_{2}}_{lm}\,\delta_{nm}\delta_{k-2,l}+{\cal
I}^{D_{4}}_{lm}\,\delta_{nm}\delta_{k-4,l}\equiv{\cal I}^{D}_{lm,\,kn}.$ (78d)

Next problem is it is not diagonalized with regard to $k$ and $l$. This
reflects the shape of the horizon of our 4D black hole is not a sphere.
Actually, it depends on ($\theta$, $\phi$) as in (52) (for zeroth law of black
hole thermodynamics, see Sec.B.2). In the next subsection, we diagonalize
these by redefining fields, which corresponds to rearrange appropriate bases.

###  B.5 2D effective near-horizon metrices

Using (78)-(78), we can write (76) as

$\displaystyle\\!\\!\\!\sum_{kn}\sum_{lm}\int dtdr\,(2m)^{2}\Big{\\{}\hskip
22.76219pt\phi_{kn}^{*}\Big{(}-\frac{2m}{r-2m}\Lambda_{lm,\,kn}-\frac{15m^{2}\varepsilon^{2}}{4\pi(r-2m)^{2}}{\cal
I}^{C}_{kn,\,lm}\Big{)}\partial_{t}\partial_{t}\phi_{lm}$ (79)
$\displaystyle\\!\\!\\!\hskip
108.12047pt+\,\phi_{kn}^{*}\partial_{r}\Big{(}\frac{r-2m}{2m}\Lambda_{lm,\,kn}-\frac{15m^{2}\varepsilon^{2}}{16\pi}{\cal
I}^{C}_{kn,\,lm}\Big{)}\partial_{r}\phi_{lm}\Big{\\}}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{kn}\sum_{lm}\int
dtdr\,(2m)^{2}\Lambda_{kn,\,lm}((g_{\rm
eff})^{tt}_{kn,\,lm}\partial_{t}\phi^{*}_{kn}\partial_{t}\phi_{lm}+(g_{\rm
eff})^{rr}_{kn,\,lm}\partial_{r}\phi^{*}_{kn}\partial_{r}\phi_{lm}),$

where $(g_{\rm eff})^{tt}_{kn,\,lm}$ and $(g_{\rm eff})^{rr}_{kn,\,lm}$ are
the 2D effective near-horizon metrices given as

$\displaystyle\Lambda_{kn,\,lm}=$
$\displaystyle\,\,\Big{(}1+\frac{3}{2}\sqrt{\frac{5}{\pi}}\left(1+3{\cal
I}^{A_{0}}_{lm}\right)\varepsilon+\frac{45}{2\pi}\left(-{\cal
I}^{B_{0}}_{lm}+3{\cal
I}^{D_{0}}_{lm}\right)\varepsilon^{2}\Big{)}\delta_{kl}\delta_{mn}$
$\displaystyle+\frac{9}{2}\Big{(}\sqrt{\frac{5}{\pi}}{\cal
I}^{A_{2}}_{lm}\varepsilon+\frac{5}{\pi}\left(-{\cal I}^{B_{2}}_{lm}+3{\cal
I}^{D_{2}}_{lm}\right)\varepsilon^{2}\Big{)}\delta_{k-2,l}\delta_{mn}+\frac{135}{2\pi}{\cal
I}^{D_{4}}_{lm}\varepsilon^{2}\delta_{k-4,l}\delta_{mn}$ $\displaystyle\equiv$
$\displaystyle\,\Lambda_{lm}^{(0)}\delta_{kl}\delta_{mn}+\Lambda_{lm}^{(2)}\delta_{k-2,l}\delta_{mn}+\Lambda_{lm}^{(4)}\delta_{k-4,l}\delta_{mn},$
(80a) $\displaystyle(g_{\rm eff})^{tt}_{kn,lm}\equiv$
$\displaystyle\,-\frac{2m}{r-2m-\frac{{\cal
I}^{C}_{kn,\,lm}}{\Lambda_{kn,\,lm}}\frac{15m\varepsilon^{2}}{8\pi
r}}=-\frac{2(m_{\rm eff})_{kn,\,lm}}{r-2(m_{\rm
eff})_{kn,\,lm}}+O(\varepsilon^{3}),$ (80b) $\displaystyle(g_{\rm
eff})^{rr}_{kn,lm}\equiv$ $\displaystyle\,-((g_{\rm eff})^{tt}_{kn,lm})^{-1},$
(80c) $\displaystyle(m_{\rm eff})_{kn,\,lm}\equiv$
$\displaystyle\,m+\frac{15m}{8\pi r}{\cal
I}^{C}_{kn,\,lm}\varepsilon^{2}+O(\varepsilon^{3})\equiv
m+\frac{\bar{\Delta}_{kn,\,lm}}{r}\varepsilon^{2},$ (80d)

where 1) $\frac{\varepsilon^{2}}{\Lambda_{kn,\,lm}}=1+O(\varepsilon^{3})$ in
(80d), 2) we defined $\Lambda_{lm}^{(0,2,4)}$, $\bar{\Delta}_{kn,\,lm}$ and
$(m_{\rm eff})_{kn,\,lm}$, and 3) $(m_{\rm eff})_{kn,\,lm}$ get depended on
$r$, which may be concerned. However metrices before the near-horizon limit in
SecA.5 satisfy Einstein equation, and

$(g_{\rm eff})^{tt}_{kn,lm}=-1+\frac{r}{2m}-\frac{15\,{\cal
I}^{C}_{kn,\,lm}{}}{16\,m\pi}\varepsilon^{2}+\frac{225\,({\cal
I}^{C}_{kn,\,lm}{})^{2}}{128\,m\pi^{2}r}\varepsilon^{4}+O\left(\varepsilon^{5}\right).$
(81)

Therefore, $r$-dependence is out of the analysis’s order. (Hawking temperature
and flux are obtained without any problems later.)

We perform the summation with regard to $k$ and $n$. Then, the indices $k$ and
$n$ in all the $(g_{\rm eff})^{tt}_{kn,\,lm}$, $(m_{\rm eff})_{kn,\,lm}$ and
$\bar{\Delta}_{kn,\,lm}$ become $l$ and $m$ for the delta-functions in (80a).
Therefore, to shorten the expressions of equations, we in what follows denote
these as

$\displaystyle(g_{\rm eff})^{tt}_{lm,\,lm}\rightarrow(g_{\rm
eff})^{tt}_{lm},\quad(m_{\rm eff})_{lm,\,lm}\rightarrow(m_{\rm
eff})_{lm},\quad\bar{\Delta}_{lm,\,lm}\rightarrow\bar{\Delta}_{lm}.$ (82)

In what follows, $tt$\- and $rr$-parts are basically same. We check $rr$-part
only at the checkpoint.

In (79), we consider to change the front factor $(2m)^{2}$ to $(2(m_{\rm
eff})_{lm})^{2}$. For this we evaluate $\frac{(2m)^{2}}{(2(m_{\rm
eff})_{lm})^{2}}\Lambda_{kn,\,lm}$. With

$\displaystyle\frac{(2m)^{2}}{(2(m_{\rm
eff})_{lm})^{2}}=1-\frac{2\bar{\Delta}_{lm}}{m\,r}\varepsilon^{2}+O(\varepsilon^{3}),$
(83)

and $\Lambda_{kn,\,lm}$ given in (80a), we can calculate in
$\varepsilon^{2}$-order as

$\displaystyle\bullet\quad\frac{(2m)^{2}}{(2(m_{\rm
eff})_{lm})^{2}}\Lambda_{lm}^{(0)}=\Lambda_{lm}^{(0)}-\frac{2\bar{\Delta}_{lm}}{m\,r}\varepsilon^{2}+O(\varepsilon^{3})\equiv\Theta_{lm}^{(0)},$
(84a) $\displaystyle\bullet\quad\frac{(2m)^{2}}{(2(m_{\rm
eff})_{lm})^{2}}\Lambda_{lm}^{(2)}=\Lambda_{lm}^{(2)}+O(\varepsilon^{3}),$
(84b) $\displaystyle\bullet\quad\frac{(2m)^{2}}{(2(m_{\rm
eff})_{lm})^{2}}\Lambda_{lm}^{(4)}=\Lambda_{lm}^{(4)}+O(\varepsilon^{3}).$
(84c)

Therefore, we can write the $tt$-part in (79) as

$\displaystyle\textrm{(\ref{cagttpteg})}\,\,=$ $\displaystyle\hskip
17.07164pt\sum_{l=0}^{l_{max}-4}\sum_{m=-l}^{l}\int d^{2}x\,(2(m_{\rm
eff})_{lm})^{2}({\cal L}^{(0)}_{lm}+{\cal L}^{(2)}_{lm}+{\cal L}^{(4)}_{lm})$
(85a) $\displaystyle+\sum_{l=l_{max}-3}^{l_{max}-2}\sum_{m=-l}^{l}\int
d^{2}x\,(2(m_{\rm eff})_{lm})^{2}({\cal L}^{(0)}_{lm}+{\cal L}^{(2)}_{lm})$
(85b) $\displaystyle+\sum_{l=l_{max}-1}^{l_{max}}\sum_{m=-l}^{l}\int
d^{2}x\,(2(m_{\rm eff})_{lm})^{2}{\cal L}^{(0)}_{lm},$ (85c) where
$\displaystyle\quad{\cal L}^{(K)}_{lm}\equiv\Lambda_{lm}^{(K)}(g_{\rm
eff})^{tt}_{l+K\,m}\partial_{t}\phi^{*}_{l+K\,m}\partial_{t}\phi_{lm}\quad\textrm{for
$K=0,2,4$},$ (85d)

$l_{max}$ is finally taken to $\infty$. The $\phi_{lm}$ with $l$ larger than
$l_{max}$ are zero, since no such $\phi_{lm}$ exist by definition. Calculation
from (79) to (85) proceeds irrelevantly of either $tt$\- or $rr$-part.

Focusing on (85a), we write its integrand as

$\begin{split}&\Omega_{lm}\Theta_{lm}^{(0)}\bigg{(}\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}+\bigg{(}\frac{(g_{\rm
eff})^{tt}_{l+2m}}{(g_{\rm
eff})^{tt}_{lm}}\frac{\Lambda_{lm}^{(2)}}{\Theta_{lm}^{(0)}}\partial_{t}\phi^{*}_{l+2m}+\frac{(g_{\rm
eff})^{tt}_{l+4m}}{(g_{\rm
eff})^{tt}_{lm}}\frac{\Lambda_{lm}^{(4)}}{\Theta_{lm}^{(0)}}\partial_{t}\phi^{*}_{l+4m}\bigg{)}\partial_{t}\phi_{lm}\bigg{)},\\\
&\Omega_{lm}\equiv(2(m_{\rm eff})_{lm})^{2}(g_{\rm
eff})^{tt}_{lm}.\end{split}$ (86)

$\Theta_{lm}^{(0)}$ are in (84a), and we defined $\Omega_{lm}$ to shorten the
expression. Rescaling as

$\displaystyle\phi_{lm}\to\frac{\phi_{lm}}{\big{(}\Theta_{lm}^{(0)}\big{)}^{1/2}}\quad\textrm{for
all $l,\,m$},$ (87)

we can rewrite (86) as

(86) $\displaystyle=$
$\displaystyle\\!\\!\\!\Omega_{lm}\bigg{(}\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}+\bigg{(}\frac{(g_{\rm
eff})^{tt}_{l+2\,m}}{(g_{\rm
eff})^{tt}_{lm}}\frac{\Lambda_{lm}^{(2)}}{\big{(}\Theta_{lm}^{(0)}\Theta_{l+2\,m}^{(0)}\big{)}^{1/2}}\partial_{t}\phi^{*}_{l+2\,m}$
(88) $\displaystyle\hskip 25.60747pt+\frac{(g_{\rm
eff})^{tt}_{l+4\,m}}{(g_{\rm
eff})^{tt}_{lm}}\frac{\Lambda_{lm}^{(4)}}{\big{(}\Theta_{lm}^{(0)}\Theta_{l+4\,m}^{(0)}\big{)}^{1/2}}\partial_{t}\phi^{*}_{l+4\,m}\bigg{)}\partial_{t}\phi_{lm}\bigg{)}.$

Rescaling above can be understood as follows. Writing $\phi_{lm}$ as
$(\Theta_{lm}^{(0)})^{-1/2}(\Theta_{lm}^{(0)})^{1/2}\phi_{lm}$, treat as
$\phi_{lm}^{(new)}\equiv(\Theta^{(0)}_{lm})^{1/2}\phi_{lm}$. At this time,
$S[\phi]=S[(\Theta^{(0)})^{-1/2}\phi^{(new)}]$ but of course ${\cal
D}\phi\not={\cal D}\phi^{(new)}={\cal J}{\cal D}\phi$ formally writing,
therefore,

$\displaystyle\int{\cal D}\phi\exp(-iS[\phi])\not=\int{\cal
D}\phi^{(new)}\exp(-iS[(\Theta^{(0)})^{-1/2}\phi^{(new)}]),$ (89)

and ${\cal J}$ involves $r$ somehow as can be seen from (91). However, ${\cal
J}$ would be finally just a finite numerical value, although we could not
evaluate its numerical value specifically. This is because our analysis is the
one with the $\Delta r~{}(=r-r_{h,4D})$ assumed small, therefore the values of
$r$ in our analysis are always some values slightly larger than $r_{h,4D}$.
Therefore, in (89), ${\cal J}$ is just some finite number and its r.h.s. can
be finally written as ${\cal J}Z_{\phi}~{}(Z_{\phi}=\textrm{l.h.s. of
(\ref{lanc})})$, and the effect by the rescaling of (87) is irrelevant in the
following analysis666 To understand this, let us consider the following path-
integral for a limited section, $r_{0}\leq r\leq r_{1}$,
$\displaystyle\mathcal{Z}_{\phi}\sim\int{\cal
D}\phi\exp(-i\int_{r_{0}}^{r_{1}}dr\,\phi(r)\,f(r)\,\phi(r)\,r^{-2}),$
$\displaystyle{\cal D}\phi\propto\prod_{\textrm{i=0 to
1}}d\phi(r_{i}),\quad\int_{r_{0}}^{r_{1}}dr\,\phi(r)\,f(r)\,\phi(r)\,r^{-2}\sim\sum_{\textrm{i=0
to 1}}\phi(r_{i})\,f(r_{i})\,\phi(r_{i})\,r_{i}^{-2}\,\Delta r,$ where $f(r)$
is some function with derivatives $\partial_{r}$. $r_{i}$ is the discretized
coordinate, which represents some points on $r_{0}\leq r\leq r_{1}$, and
$\Delta r$ corresponds to $dr$. “$\sim$” in the first line means, roughly
saying, an expression having been reached performing path-integral for
canonical momenta. Now, consider to change the variable of the path-integral
as $\phi\to\phi^{(new)}\equiv\phi\,r^{-1}$. At this time, the expression of
$\mathcal{Z}_{\phi}$ is changed as
$\displaystyle\mathcal{Z}_{\phi}\to\prod_{\textrm{i=0 to
1}}r_{i}\cdot\int{\cal
D}\phi^{(new)}\exp(-i\int_{r_{0}}^{r_{1}}dr\,\phi^{(new)}(r)\,f(r)\,\phi^{(new)}(r)),$
where $\displaystyle\prod_{\textrm{i=0 to 1}}d\phi(r_{i})\to\prod_{\textrm{i=0
to 1}}\,r_{i}\,d(\phi(r_{i}){r_{i}}^{-1})=\prod_{\textrm{i=0 to
1}}\,r_{i}\cdot\prod_{\textrm{i=0 to 1}}\,d(\phi^{(new)}(r_{i})).$
$\prod_{\textrm{i=0 to 1}}\,r_{i}$ corresponds to ${\cal J}$, which is some
finite number. The rescaling of (87) is likewise.. Even if ${\cal J}$ is a
divergent quantity it is still a number, therefore the conclusion is not
changed777 Let us just check the difference with the case of quantum
anomalies. For any gauge transformation, $Z_{\phi}=Z_{\phi^{\prime}}$.
However, for some kinds of gauge transformation, the path-integral measure is
not invariant, which we formally represent as ${\cal D}\phi^{\prime}={\cal
J}{\cal D}\phi$. Action is also not invariant in some cases, which we
represent as $S[\phi^{\prime}]=S[\phi]+\delta S[\phi]$. Therefore,
$Z_{\phi^{\prime}}=\int{\cal J}{\cal D}\phi\exp(-i(S[\phi]+\delta
S[\phi]))=\int{\cal D}\phi\exp(\log{\cal J}-i(S[\phi]+\delta S[\phi]))$, from
which $\log{\cal J}-i\delta S[\phi]=0$ is obtained. In evaluating $\log{\cal
J}$, some regularization is needed. As a result, $\log{\cal J}$ is given as
some function of field strengths (e.g. around Eq.(5.20) in [48]) and the
quantum anomaly is given, which is the different point from the case of (87).
. In any case, following a general formula: $\int{\cal D}\phi\exp(-i\int
dx^{4}\int dy^{4}\,\phi(x){\cal M}(x,y)\phi(y))\propto{\rm Det}\,{\cal M}$,
there is no difference finally in the results of the path-integral obtained
with/without (87).

It is important to give attention to the consideration above that
$r$-direction is finite as it is related with the problem of boundary
condition (this is the problem arising even in the classical level) and a
fundamental supposition in the quantum field theory that spacetime spreads
infinitely (breaking of translational symmetry at the boundary in
$r$-direction may be needed to be cared, if to be exact).

What is being done in this section is just to obtain the expression of the
action, not obtaining the solution, therefore these problems are irrelevant in
this section. However, since we treat the quantum effect in the next section,
we give attention to these problems.

In the next section, we use the formulas of quantum anomaly. Although author
has not checked the derivation process of these one by one entirely, these are
obtained by once obtaining these in a flat spacetime, then by replacing the
derivatives in these with covariant derivatives [49]. In these derivation
process, the supposition that the spacetime spreads infinitely would be used.

Therefore, as what we do in fact, our study focuses on the vicinity of the
horizon before taking the quantum effect, and quantum effect is taken in after
focusing on the vicinity of the horizon, however, as the problem of how to
consider, it would be possible to consider by the following way: 1) performing
the analysis for the quantum anomaly in an infinitely spreading flat
spacetime, 2) then replacing that flat spacetime with the one in this study
with $r$-direction not limited, 3) then just focusing on the vicinity of the
horizon (not limiting $r$-direction), 4) we have used the formulas of quantum
anomaly at there. By considering like this, problems mentioned above are
considered not to arise in the analysis in this study.

We here would like to look at the calculation from (86) to (88) via (87) in
the $rr$-part, since $\Theta_{lm}^{(0)}$ depend on $r$ as can be seen in
(84a), and at (88) in the calculation of the $rr$-part, the following equation
appears, and which can be calculated as

$\displaystyle\frac{(g_{\rm eff})^{rr}_{l+K\,m}}{(g_{\rm
eff})^{rr}_{lm}}\frac{\Lambda_{lm}^{(K)}}{\Theta_{lm}^{(0)}}\partial_{r}\Big{(}\frac{\phi_{l+K\,m}}{\big{(}\Theta_{l+K\,m}^{(0)}\big{)}^{1/2}}\Big{)}\partial_{r}\Big{(}\frac{\phi_{lm}}{\big{(}\Theta_{lm}^{(0)}\big{)}^{1/2}}\Big{)}$
$\displaystyle=\frac{(g_{\rm eff})^{rr}_{l+K\,m}}{(g_{\rm
eff})^{rr}_{lm}}\frac{\Lambda_{lm}^{(K)}}{\Theta_{lm}^{(0)}}\Big{(}\\!\frac{\partial_{r}\Theta_{l+K\,m}^{(0)}\phi_{l+K\,m}}{2\big{(}\Theta_{l+K\,m}^{(0)}\big{)}^{3/2}}-\frac{\partial_{r}\phi_{l+K\,m}}{\big{(}\Theta_{l+K\,m}^{(0)}\big{)}^{1/2}}\Big{)}\Big{(}\\!\frac{\partial_{r}\Theta_{lm}^{(0)}\phi_{lm}}{2\big{(}\Theta_{lm}^{(0)}\big{)}^{3/2}}-\frac{\partial_{r}\phi_{lm}}{\big{(}\Theta_{lm}^{(0)}\big{)}^{1/2}}\Big{)}$
$\displaystyle=\frac{(g_{\rm eff})^{rr}_{l+K\,m}}{(g_{\rm
eff})^{rr}_{lm}}\frac{\Lambda_{lm}^{(K)}}{\Theta_{lm}^{(0)}}\frac{\partial_{r}\phi_{l+K\,m}}{\big{(}\Theta_{l+K\,m}^{(0)}\big{)}^{1/2}}\frac{\partial_{r}\phi_{lm}}{\big{(}\Theta_{lm}^{(0)}\big{)}^{1/2}}+O(\varepsilon^{3})\,,$
(90)

where $K=2,4$ and

$\displaystyle\frac{(g_{\rm eff})^{rr}_{l+K\,m}}{(g_{\rm eff})^{rr}_{lm}}\sim
1+\varepsilon^{2},\quad\Lambda_{lm}^{(K)}\sim\varepsilon^{K/2},\quad\Theta_{lm}^{(0)}\sim
1+\varepsilon+\left(1+r^{-1}\right)\varepsilon^{2},\quad\frac{\Lambda_{lm}^{(K)}}{\Theta_{lm}^{(0)}}\sim\varepsilon^{K/2},$
(91)

from the definitions of (80b), (80a) and (84a). Therefore, the extra terms
drop and the $rr$-part at (88) can be obtained in the same way with (88)
except $\partial_{t}$ and $(g_{\rm eff})^{tt}_{lm}$.

Then, for the parts in (88), the following calculation can be held in
$\varepsilon^{2}$-order:

$\displaystyle\frac{(g_{\rm eff})^{tt}_{l+K\,m}}{(g_{\rm
eff})^{tt}_{lm}}\frac{\Lambda_{lm}^{(K)}}{\big{(}\Theta_{lm}^{(0)}\Theta_{l+K\,m}^{(0)}\big{)}^{1/2}}=\frac{\Lambda_{lm}^{(K)}}{\big{(}\Theta_{lm}^{(0)}\Theta_{l+K\,m}^{(0)}\big{)}^{1/2}}+O(\varepsilon^{3})\equiv
2\overline{\Lambda}_{lm}^{(K)}$ (92) for all $l,\,m$, where $K=2,4$.

The one above can be actually checked with the $(g_{\rm eff})^{tt}_{lm}$,
$\Lambda_{lm}^{(K)}$ and $\Theta_{lm}^{(0)}$ given in (80b), (80a) and (84a)
respectively, and can hold in the case of $rr$, namely if $(g_{\rm
eff})^{tt}_{l+K\,m}$ and $(g_{\rm eff})^{tt}_{l\,m}$ are $(g_{\rm
eff})^{rr}_{l+K\,m}$ and $(g_{\rm eff})^{rr}_{l\,m}$. Using (92), we can write
(88) as

$\displaystyle(\ref{fcrsc4})\,=\,\Omega_{lm}\big{(}\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}+2\big{(}\overline{\Lambda}_{lm}^{(2)}\partial_{t}\phi^{*}_{l+2\,m}+\overline{\Lambda}_{lm}^{(4)}\partial_{t}\phi^{*}_{l+4\,m}\big{)}\partial_{t}\phi_{lm}\big{)}.$
(93)

We here define

$\displaystyle\Gamma_{lm}^{(2)}\equiv\overline{\Lambda}_{lm}^{(2)}\partial_{t}\phi_{l+2\,m},\quad\Gamma_{lm}^{(4)}\equiv\overline{\Lambda}_{lm}^{(2)}\partial_{t}\phi_{l+2\,m}+\overline{\Lambda}_{lm}^{(4)}\partial_{t}\phi_{l+4\,m},$
(94)

to shorten the expression of the equations ($\Gamma_{lm}^{(2)}$ is not used
immediately). Then,

$\displaystyle(\ref{fcsgf4})\,=\,\Omega_{lm}\big{(}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\big{|}^{2}-\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2m}\partial_{t}\phi_{l+2m}\big{)},$
(95)

where $\overline{\Lambda}_{lm}^{(K)}\sim\varepsilon^{K/2}$ from (92) and
$\Gamma_{lm}^{(4)}{}^{*}\Gamma_{lm}^{(4)}\,=\,\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2\,m}\partial_{t}\phi_{l+2\,m}+O(\varepsilon^{3})$.

Performing the calculation regarding (85b) likewise, we can write (85) as

(88) $\displaystyle=$ $\displaystyle\hskip
8.53581pt\sum_{l=0}^{l_{max}-4}\,\sum_{|m|=0}^{l}\int
d^{2}x\,\Omega_{lm}\big{(}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\big{|}^{2}-\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2m}\partial_{t}\phi_{l+2m}\big{)}$
(96)
$\displaystyle\\!\\!\\!\\!+\sum_{l=l_{max}-3}^{l_{max}-2}\,\sum_{|m|=0}^{l}\int
d^{2}x\,\Omega_{lm}\big{(}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(2)}\big{|}^{2}-\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2m}\partial_{t}\phi_{l+2m}\big{)}$
$\displaystyle\\!\\!\\!\\!+\sum_{l=l_{max}-1}^{l_{max}}\,\sum_{|m|=0}^{l}\int
d^{2}x\,\Omega_{lm}\partial_{t}\phi_{lm}^{*}\partial_{t}\phi_{lm}+O(\varepsilon^{3}).$

Calculation for the $rr$-part from (88) to (96) can be proceeded without
problems, and the $rr$-part at (96) is also obtained basically same with (96).

Now we consider to do uniformly slide each
“$\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2m}\partial_{t}\phi_{l+2m}$”
appearing in the line of $l$ to the line of $l+2$ in (96). For this, let us
check $\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}$:

$\displaystyle\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}=\frac{405m^{3}({\cal
I}^{A_{2}}_{lm})^{2}}{2\pi(r-2m)}{\varepsilon^{2}}+O(\varepsilon^{3}),$ (97)

where ${\cal I}^{A_{2}}_{lm}$ are numbers in (78). Thus we can write
$\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}$ changing its $l$
to $l+2$ as

$\displaystyle\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}=\frac{({\cal
I}^{A_{2}}_{lm})^{2}}{({\cal
I}^{A_{2}}_{l+2\,m})^{2}}\Omega_{l+2\,m}\big{(}\overline{\Lambda}_{l+2\,m}^{(2)}\big{)}^{2}+O(\varepsilon^{3})\equiv\Omega_{l+2\,m}\Xi_{l+2\,m},$
(98)

where

$\displaystyle\frac{({\cal I}^{A_{2}}_{lm})^{2}}{({\cal
I}^{A_{2}}_{l+2\,m})^{2}}$
$\displaystyle=\frac{(2l+9)(2l+7)^{2}\left((l+1)^{2}-m^{2}\right)\left((l+2)^{2}-m^{2}\right)}{(2l+1)(2l+3)^{2}\left((l+3)^{2}-m^{2}\right)\left((l+4)^{2}-m^{2}\right)},$
(99a) $\displaystyle\Xi_{l+2\,m}$ $\displaystyle\equiv\frac{({\cal
I}^{A_{2}}_{lm})^{2}}{({\cal
I}^{A_{2}}_{l+2\,m})^{2}}\big{(}\overline{\Lambda}_{l+2\,m}^{(2)}\big{)}^{2}.$
(99b)

With (98), we can replace as

$\displaystyle\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2\,m}\partial_{t}\phi_{l+2\,m}\rightarrow\Omega_{l+2\,m}\Xi_{l+2\,m}\partial_{t}\phi^{*}_{l+2\,m}\partial_{t}\phi_{l+2\,m}\quad\textrm{for
all $l$, $m$}.$ (100)

Therefore, uniformly sliding each
“$\Omega_{lm}\big{(}\overline{\Lambda}_{lm}^{(2)}\big{)}^{2}\partial_{t}\phi^{*}_{l+2m}\partial_{t}\phi_{l+2m}$”
by $2$ regarding $l$ in (96),

(96) $\displaystyle=$ $\displaystyle\hskip
17.07164pt\sum_{l=0}^{1}\sum_{|m|=0}^{l}\int
d^{2}x\,\Omega_{lm}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\big{|}^{2}$
(101) $\displaystyle+\,\sum_{l=2}^{l_{max}-4}\sum_{|m|=0}^{l-2}\int
d^{2}x\,\Omega_{lm}\big{(}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\big{|}^{2}-\Xi_{lm}\,\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}\big{)}$
$\displaystyle+\,\sum_{l=2}^{l_{max}-4}\sum_{|m|=l-1}^{l}\int
d^{2}x\,\Omega_{lm}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\big{|}^{2}$
$\displaystyle+\,\sum_{l=l_{max}-3}^{l_{max}-2}\sum_{|m|=0}^{l-2}\int
d^{2}x\,\Omega_{lm}\big{(}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(2)}\big{|}^{2}-\Xi_{lm}\,\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}\big{)}$
$\displaystyle+\,\sum_{l=l_{max}-3}^{l_{max}-2}\sum_{|m|=l-1}^{l}\int
d^{2}x\,\Omega_{lm}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(2)}\big{|}^{2}$
$\displaystyle+\,\sum_{l=l_{max}-1}^{l_{max}}\sum_{|m|=0}^{l-2}\int
d^{2}x\,\Omega_{lm}\big{(}\partial_{t}\phi_{lm}^{*}\partial_{t}\phi_{lm}-\Xi_{lm}\,\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}\big{)}$
$\displaystyle+\,\sum_{l=l_{max}-1}^{l_{max}}\sum_{|m|=l-1}^{l}\int
d^{2}x\,\Omega_{lm}\partial_{t}\phi_{lm}^{*}\partial_{t}\phi_{lm}+O(\varepsilon^{3}).$

We once again perform the rescaling of the fields as

$\displaystyle\phi_{l\,m}\rightarrow\frac{\phi_{l\,m}}{(1-\Xi_{lm})^{1/2}}\quad\textrm{for
$l=2,3,\cdots,l_{max}$ ($l=0,1$ are not included)}$ (102) and
$|m|=0,1,\cdots,l-2$ for each $l$.

This rescaling is possible by the same reason written around (89). At this
time, $\overline{\Lambda}_{lm}^{(K)}\sim\varepsilon^{K/2}$ (see under (95))
and $\Gamma_{lm}^{(K)}$ and $\partial_{t}\phi\,\Gamma_{lm}^{(K)*}$ can stay
same in $\varepsilon^{2}$-order as

$\displaystyle\Gamma_{lm}^{(K)}\rightarrow\Gamma_{lm}^{(K)}+O(\varepsilon^{3}),\quad\partial_{t}\phi\,\Gamma_{lm}^{(K)*}\rightarrow\partial_{t}\phi\,\Gamma_{lm}^{(K)*}+O(\varepsilon^{3})$
(103) for all $l$ and $m$, where $K=2,4$.

Therefore, we can exchange the lines in (101) with the squared form as

$\displaystyle\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(K)}\big{|}^{2}-\Xi_{lm}\,\partial_{t}\phi^{*}_{lm}\partial_{t}\phi_{lm}\rightarrow\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(K)}\big{|}^{2},$
(104)

where “$\rightarrow$” means the rescaling (102). Therefore, we can write (101)
as

(101) $\displaystyle=$ $\displaystyle\hskip
17.07164pt\sum_{l=0}^{l_{max}-4}\sum_{m=-l}^{l}\int
d^{2}x\,\Omega_{lm}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\big{|}^{2}$
(105) $\displaystyle+\,\sum_{l=l_{max}-3}^{l_{max}-2}\sum_{m=-l}^{l}\int
d^{2}x\,\Omega_{lm}\big{|}\partial_{t}\phi_{lm}+\Gamma_{lm}^{(2)}\big{|}^{2}$
$\displaystyle+\,\sum_{l=l_{max}-1}^{l_{max}}\sum_{m=-l}^{l}\int
d^{2}x\,\Omega_{lm}\partial_{t}\phi_{lm}^{*}\partial_{t}\phi_{lm}.$

We here would like to give attention to the $rr$-part. The point to be checked
between (96) and (105) is the manipulation (102): whether
$\partial_{r}\big{(}\frac{\phi_{l\,m}}{(1-\Xi_{lm})^{1/2}}\big{)}=\frac{\partial_{r}\phi_{l\,m}}{(1-\Xi_{lm})^{1/2}}+O\big{(}\varepsilon^{3}\big{)}$
can be held or not at (102) in the calculation of the $rr$-part. For this, let
us check the $r$-dependence of $\Xi_{lm}$: $\Xi_{lm}=\frac{({\cal
I}^{A_{2}}_{lm})^{2}}{({\cal
I}^{A_{2}}_{l+2\,m})^{2}}\Big{(}\frac{\Lambda_{lm}^{(2)}}{2\big{(}\Theta_{lm}^{(0)}\Theta_{l+2\,m}^{(0)}\big{)}^{1/2}}\Big{)}^{2}$,
where $\frac{({\cal I}^{A_{2}}_{lm})^{2}}{({\cal I}^{A_{2}}_{l+2\,m})^{2}}$
are numbers, $\Lambda_{lm}^{(2)}\sim\varepsilon$ and $\Theta_{lm}^{(0)}\sim
1+\varepsilon+(1+r^{-1})\varepsilon^{2}$. Therefore, $\Xi_{lm}$ is independent
of $r$ in $\varepsilon^{2}$-order. If so, the equation above can be held, and
the $rr$-part at (105) can be obtained as (105) as well as the $tt$-part
without the difference of $\partial_{r}$ and $(g_{\rm eff})^{rr}_{lm}$.

Since it can be written as follows:

$\displaystyle\partial_{t}\phi_{lm}+\Gamma_{lm}^{(4)}\,=$
$\displaystyle\,\,\partial_{t}(\phi_{lm}+\overline{\Lambda}_{lm}^{(2)}\phi_{l+2\,m}+\overline{\Lambda}_{lm}^{(4)}\phi_{l+4\,m}),$
(106a) $\displaystyle\partial_{t}\phi_{lm}+\Gamma_{lm}^{(2)}\,=$
$\displaystyle\,\,\partial_{t}(\phi_{lm}+\overline{\Lambda}_{lm}^{(2)}\phi_{l+2\,m}),$
(106b)

let us perform the redefinition of the fields as

$\displaystyle\bullet\quad\varphi_{lm}\equiv\,\phi_{lm}+\overline{\Lambda}_{lm}^{(2)}\phi_{l+2\,m}+\overline{\Lambda}_{lm}^{(4)}\phi_{l+4\,m}\quad\textrm{for
$l=0,1,\cdots,l_{max}-4$,}$ (107a)
$\displaystyle\bullet\quad\varphi_{lm}\equiv\,\phi_{lm}+\overline{\Lambda}_{lm}^{(2)}\phi_{l+2\,m}\qquad\qquad\qquad\hskip
5.69054pt\,\textrm{for $l=l_{max}-3,l_{max}-2$,}$ (107b)
$\displaystyle\bullet\quad\varphi_{lm}\equiv\,\phi_{lm}\qquad\qquad\qquad\qquad\qquad\qquad\,\,\textrm{for
$l=l_{max}-1,l_{max}$,}$ (107c)

where $m$ above are $0,\pm 1,\cdots,\pm(l-2)$ for each $l$. The leadings of
$\overline{\Lambda}_{lm}^{(K)}$ is $\varepsilon^{K/2}$.

(87) and (102) are rescalings which can be absorbed as configurations of the
path-integral for $\phi_{lm}$ for the reason written around (89), however
(107) is recombinations. Therefore, the Jacobian for
$\phi_{lm}\rightarrow\varphi_{lm}$, should be checked. Forming a matrix
according to (107) We can check it gives unit.

With $\varphi_{lm}$ above, we can finally obtain the decoupled 2D effective
action which is equivalent with (70) as a action in the range of
$\varepsilon^{2}$ as

$\displaystyle(\ref{esqcml})\,=\,\sum_{l=0}^{l_{max}}\sum_{|m|=0}^{l}\int
d^{2}x\,\Phi_{lm}((g_{\rm
eff})^{tt}_{lm}\,\partial_{t}\varphi^{*}_{lm}\partial_{t}\varphi_{lm}+(g_{\rm
eff})^{rr}_{lm}\,\partial_{r}\varphi^{*}_{lm}\partial_{r}\varphi_{lm}),$ (108)

where 1) $\Phi_{lm}=\big{(}2(m_{\rm eff})_{lm}\big{)}^{2}$ and the 2D
effective metrices are given in (80). 2) Einstein equation can be satisfied
with these effective metrices. 3) Since the labels distinguishing the
effective metrices are irrelevant of the spins, the effective metrices would
not be changed if we considered fermions [50, 51, 52] and higher spin fields
[53].

Lastly, the behavior (91) is critical in the feasibility of the analysis in
this subsection.

##  Appendix C Hawking Temperature in the effective 2D

We have obtained the 2D effective metrices, which are labeled by spherical
harmonics modes. From these, we can naively expect 1) existence of various
Hawking temperatures for each effective metric, 2) correspondingly, breaking
of the zeroth law of the black hole thermodynamics. (Furthermore, 3) $(m_{\rm
eff})_{kn,\,lm}$ get depended on $r$ as in $(\ref{mdfeft})$, though this is
not problems in the analysis’s order.) Hence, let us check the Hawking
temperature.

We can obtain the position of the horizon in the 2D picture from $(g_{\rm
eff})_{{tt},\,{lm}}=0$ as

$\displaystyle(r_{h,2D})_{lm}=2m+\frac{15m\,{\cal
I}^{C_{0}}_{lm}}{8\pi}\,\varepsilon^{2}+O(\varepsilon^{3}).$ (109)

As this is labeled by spherical harmonics modes, we can expect the points
above. However, the Hawking temperature obtained from the 2D effective
metrices with the one above is

$\displaystyle T_{H}={1}/{8\pi m}+O(\varepsilon^{3}).$ (110)

This is just that in the 4D Schwarzschild and free from the concerns above.
The reason of this is the same with those written in Sec.B.2, where it is
considered replacing with (81).

The original 4D and effective 2D spacetimes are different each other. However
the Hawking temperature in the effective 2D spacetime is generally considered
to coincide with the one in the original 4D spacetime. Actually the one above
coincides with (66).

##  Appendix D Hawking flux by anomaly cancellation

We call the anomaly cancellation method as “anomaly cancellation”. Since our
U(1) gauge field does not arise chiral anomalies, we ignore it as mentioned
under (69b). Hence, we do not consider the Hawking flux of the electric
charged current.

###  D.1 Set up of the radial direction

The key point in the anomaly cancellation is the fact of no outgoing modes on
the horizon at the classical level. To treat this situation in the anomaly
cancellation, some interval from $(r_{h,2D})_{lm}$ in the radial direction are
sharply divided as follows:888 Radial direction is sharply divided with
$\epsilon$ in all the papers of the anomaly cancellation, which is unnatural.
This problem is commented in Chap.4 in [46] and treated in [54]. There is one
more artificial point in the anomaly cancellation, which is to use two
anomalies, gravitational and consistent anomalies. [55] cares this point.

$\displaystyle(r_{h,2D})_{lm}\leq$
$\displaystyle\,r\leq(r_{h,2D})_{lm}+\epsilon_{lm},$ (111a)
$\displaystyle(r_{h,2D})_{lm}+\epsilon_{lm}<$
$\displaystyle\,r\leq(r_{o})_{lm}.$ (111b)

  * •

$\epsilon_{lm}$ represent the divided points, which are finally taken to zero,

  * •

$(r_{o})_{lm}$ mean the positions put by hand reasonably supposing that it is
the maximum of the $r$ to where the description by the 2D effective action
(108) is possible.

  * •

(111a) is the region where supposed only ingoing modes exist at the classical
level,

  * •

(111b) is the region where both ingoing and outgoing modes exist at classical
level.

We refer to the two regions, (111a) and (111b), as the regions $\cal H$ and
$\cal O$, respectively. In what follows we suppose the following corresponding
in the 2D effective picture:

the outgoing modes $\to$ the right-hand modes, the ingoing modes $\to$ the
left-hand modes.

###  D.2 Hawking flux of the energy-momentum tensors

We consider the distribution function in the region $(r_{h,2D})_{lm}\leq
r\leq(r_{o})_{lm}$ as

$\displaystyle Z\left[(g_{\rm eff})_{lm}^{\mu\nu},\Phi_{lm}\right]=\int{\cal
D}\varphi_{lm}\,\exp iS_{\textrm{2D}}((g_{\rm
eff})_{lm}^{\mu\nu},\Phi_{lm},\varphi_{lm}),$ (112)

where $(g_{\rm eff})_{lm}^{\mu\nu}$, $\Phi_{lm}$ and $\varphi_{lm}$ are those
in (108). Then, consider degrees of gauge freedom of general coordinate
transformation in that region as

$\displaystyle x^{\mu}\mapsto
x^{\prime}{}^{\mu}=x^{\mu}-\eta^{\mu}\left(x^{\mu}\right).$ (113)

Variation toward these can be written as

$\displaystyle(\delta Z)_{lm}=\Big{(}\delta_{L}(g_{\rm
eff})_{lm}^{\mu\nu}\frac{\delta}{\delta_{L}(g_{\rm
eff})_{lm}^{\mu\nu}}+\delta_{L}A_{\mu\,lm}\frac{\delta}{\delta_{L}A_{\mu,\,lm}}+\delta_{L}\Phi_{lm}\frac{\delta}{\delta_{L}\Phi_{lm}}\Big{)}Z$
(114) $\displaystyle\textrm{with}\quad\delta_{L}(g_{\rm eff})_{lm}^{\mu\nu}=$
$\displaystyle\,-(\nabla^{\mu}_{lm}\eta^{\nu}+\nabla^{\nu}_{lm}\eta^{\mu}_{lm}),$
(115a) $\displaystyle\delta_{L}A_{\mu,\,lm}=$
$\displaystyle\,\nabla_{\mu,\,lm}\eta^{\nu},\,A_{\nu\,lm}+\eta^{\nu}\nabla_{\nu,\,lm}A_{\mu,\,lm},$
(115b) $\displaystyle(\delta_{L}A^{\mu}_{lm}=$
$\displaystyle\,-\nabla^{\mu}_{lm}\eta^{\nu},\,A_{\nu\,lm}+\eta^{\nu}\nabla_{\nu,\,lm}A^{\mu}_{lm})$
$\displaystyle\delta_{L}\Phi_{lm}=$
$\displaystyle\,\,\eta^{\mu}\partial_{\mu}\Phi_{lm},$ (115c)

where $\delta_{L}$ means Lie derivative. $l$ and $m$ are not summed. We keep
$A_{\mu,\,lm}$ just in case.

Each $(\delta Z)_{lm}$ should vanishes, from which we can obtain the
conservation laws for the energy-momentum tensors at the classical level from
$(\delta Z)_{lm}=0$. Aside from these, quantum anomalies exist as [56]

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu,\,lm}$
$\displaystyle=\pm\frac{1}{96\pi(-(g_{\rm
eff})_{lm})^{1/2}}\epsilon^{\beta\delta}\partial_{\delta}\partial_{\alpha}\Gamma^{\alpha}_{\nu\beta,\,lm}\equiv\mathscr{A}_{\nu,\,lm}^{\pm},$
(116a) ($+/-$ $\to$ left- / right-hand mode’s contributions)
$\displaystyle\nabla_{\mu}\widetilde{T}^{\mu}{}_{\nu,\,lm}$
$\displaystyle=\mp\frac{1}{96\pi(-(g_{\rm
eff})_{lm})^{1/2}}\epsilon_{\mu\nu}\partial^{\mu}R_{lm}\equiv\widetilde{\mathscr{A}}_{\nu,\,lm}^{\mp},$
(116b) ($-/+$ $\to$ left- / right-hand mode’s contributions)

where $\epsilon^{tr}=1$ and $\epsilon_{\mu\nu}=(g_{\rm
eff})_{\mu\alpha,\,lm}(g_{\rm eff})_{\nu\beta,\,lm}\epsilon^{\alpha\beta}$.
Top and bottom are the consistent and covariant anomalies.
$\widetilde{T}^{\mu}{}_{\nu,\,lm}$ follow the boundary condition as

$\displaystyle(\widetilde{T}_{H})_{\mu\nu,\,lm}\big{|}_{r=(r_{h,2D})_{lm}}=0.$
(117)

The conservation laws in the anomaly cancellation are given combining these as

$\displaystyle\nabla_{\mu}T^{\mu}{}_{\nu,\,lm}=$
$\displaystyle\,F_{\mu\nu,\,lm}J^{\mu}_{lm}+A_{\nu,\,lm}\nabla_{\mu}J^{\mu}_{lm}-\frac{\partial_{\nu}\Phi_{lm}}{(-(g_{\rm
eff})_{lm})^{1/2}}\frac{\delta
S_{\textrm{2D}}}{\delta_{L}\Phi_{lm}}+\textrm{both/either
$\mathscr{A}_{\nu,\,lm}^{\pm}$},$ (118a)
$\displaystyle\nabla_{\mu}\widetilde{T}^{\mu}{}_{\nu,\,lm}=$
$\displaystyle\,F_{\mu\nu,\,lm}J^{\mu}_{lm}+A_{\nu,\,lm}\nabla_{\mu}J^{\mu}_{lm}-\frac{\partial_{\nu}\Phi_{lm}}{(-(g_{\rm
eff})_{lm})^{1/2}}\frac{\delta
S_{\textrm{2D}}}{\delta_{L}\Phi_{lm}}+\textrm{both/either
$\widetilde{\mathscr{A}}_{\nu,\,lm}^{\mp}$},$ (118b)

where $J^{\mu}_{lm}\equiv\frac{1}{(-(g_{\rm eff})_{lm})^{1/2}}\frac{\delta
S_{\textrm{2D}}}{\delta_{L}A_{\mu,\,lm}}$ and
$T_{\mu\nu,\,lm}\equiv\frac{2}{(-(g_{\rm eff})_{lm})^{1/2}}\frac{\delta
S_{\textrm{2D}}}{\delta_{L}(g_{\rm eff})^{\mu\nu}_{lm}}$. “both” or “either”
is taken according to both left- and right-hand modes exist or not. Anomalies
vanish in “both” as the left- and right-hand modes cancel each other.

We show (118) in our case by calculating these for the case $\nu=t$ and $r$
respectively using (80b), (80c) and (80b) etc as

$\displaystyle\partial_{r}T^{r}{}_{t,\,lm}=\textrm{both/either
$\mathscr{A}_{t,\,lm}^{\pm}$}(=\pm\partial_{r}N^{r}{}_{t,\,lm}),\quad\partial_{r}T^{r}{}_{r,\,lm}=0,$
(119a)
$\displaystyle\partial_{r}\widetilde{T}^{\mu}{}_{t,\,lm}=\textrm{both/either
$\widetilde{\mathscr{A}}_{t,\,lm}^{\mp}$}(=\pm\partial_{r}\widetilde{N}^{r}{}_{t,\,lm}),\quad\partial_{r}\widetilde{T}^{r}{}_{r,\,lm}=0,$
(119b) $\displaystyle N^{r}{}_{t,\,lm}=(f^{\prime
2}+ff^{\prime\prime})/192\pi,\quad\widetilde{N}^{r}{}_{t,\,lm}=(ff^{\prime\prime}-(f^{\prime})^{2}/2)/96\pi,$

where $f$ means $-(g_{\rm eff})_{tt,\,lm}$ and ′ means $\partial_{r}$. We have
used the facts that our gauge fields are ignoble (see under (69b)) and our
dilaton is time-independent with our killing vector.

We give the expressions of the energy-momentum tensors we employ as

$\displaystyle T^{\mu}{}_{\nu,\,lm}=$
$\displaystyle\,(T_{o})^{\mu}{}_{\nu,\,lm}\Theta_{lm}+(T_{H})^{\mu}{}_{\nu,\,lm}H_{lm},$
(120a) $\displaystyle\widetilde{T}^{\mu}{}_{\nu,\,lm}=$
$\displaystyle\,(\widetilde{T}_{o})^{\mu}{}_{\nu,\,lm}\Theta_{lm}+(\widetilde{T}_{H})^{\mu}{}_{\nu,\,lm}H_{lm},$
(120b)

where $\Theta_{lm}$ mean the step function
$\theta\left(r-\left((r_{h,2D})_{lm}+\epsilon_{lm}\right)\right)$ and $H_{lm}$
is $1-\Theta_{lm}$. Therefore,

$\displaystyle\textrm{$(T_{H})^{\mu}{}_{\nu,\,lm}$ include only the right-hand
modes},$ (121a) $(T_{o})^{\mu}{}_{\nu,\,lm}$ include both hand modes, which
leads no anomalies. (121b)

$(\widetilde{T}_{H})^{\mu}{}_{\nu,\,lm}$ and
$(\widetilde{T}_{o})^{\mu}{}_{\nu,\,lm}$ are likewise. Sharp expressions of
(120) is rooted in setting (111).

From (119) and (120) with (121), we can obtain the identities we consider as

$\displaystyle\partial_{r}(T_{H})^{r}{}_{t,\,lm}=\,\mathscr{A}_{t,\,lm}^{+}=\partial_{r}N^{r}{}_{t,\,lm}\quad{\rm
and}\quad\partial_{r}(T_{o})^{r}{}_{t,\,lm}=\,0,$ (122a)
$\displaystyle\partial_{r}(\widetilde{T}_{H})^{r}{}_{t,\,lm}=\,\widetilde{\mathscr{A}}_{t,\,lm}^{-}-\partial_{r}\widetilde{N}^{r}{}_{t,\,lm}\quad{\rm
and}\quad\partial_{r}(\widetilde{T}_{o})^{r}{}_{t,\,lm}=\,0.$ (122b)

From these, we can get the expressions of $(T_{H,\,o})^{r}{}_{t,\,lm}$ and
$(\widetilde{T}_{H,\,o})^{r}{}_{t,\,lm}$ as

$\displaystyle(T_{H})^{r}{}_{t,\,lm}=(c_{H})^{r}{}_{t,\,lm}+\int_{(r_{h,2D})_{lm}}^{r}\\!\\!\\!\\!dr\,\partial_{r}N^{r}{}_{t,\,lm},\quad(T_{o})^{r}{}_{t,\,lm}=(c_{o})^{r}{}_{t,\,lm},$
(123a)
$\displaystyle(\widetilde{T}_{H})^{r}{}_{t,\,lm}=(\widetilde{c}_{H})^{r}{}_{t,\,lm}+\int_{(r_{h,2D})_{lm}}^{r}\\!\\!\\!\\!dr\,\partial_{r}\widetilde{N}^{r}{}_{t,\,lm},\quad(\widetilde{T}_{o})^{r}{}_{t,\,lm}=(\widetilde{c}_{o})^{r}{}_{t,\,lm},$
(123b)

where ($(c_{H})^{r}{}_{t,\,lm}$, $(\widetilde{c}_{H})^{r}{}_{t,\,lm}$) and
($(\widetilde{c}_{o})^{r}{}_{t,\,lm}$, $(\widetilde{c}_{o})^{r}{}_{t,\,lm}$)
are integral constants. The former two are the values of those at horizon, the
latter two are the values of those at $r=(r_{o})_{lm}$.
$(\widetilde{c}_{o})^{r}{}_{t,\,lm}$ is identified with the total amount of
the Hawking flux (e.g. [43, 57]999 $(\widetilde{c}_{o})^{r}{}_{t,\,lm}$ can be
identified with the value of the total amount of the black-body radiation
through the identification of $T^{r}{}_{t}$ with that ((28) in [43]), where
the fermion case is considered in [43, 57] to avoid the problem of
superradiance supposing that it would be the same with the bosonic case. Once
one has checked that the value of $T^{r}{}_{t}$ can agree to the black-body
radiations in the 3 kinds of the fundamental 4D black holes (Schwarzschild,
Kerr and charged), all the papers concerning the anomaly cancellation compute
the value of $T^{r}{}_{t}$ in various black holes, and consider that it always
represents the total amount of the black-body radiation. We in this study also
follow this way. ).

We consider an equation obtained from (123) as101010 There is one point. We
can see a quantity:
$(\widetilde{T}_{H})^{r}{}_{t,\,lm}-\left((\widetilde{c}_{o})^{r}{}_{t,\,lm}-(c_{o})^{r}{}_{t,\,lm}\right)$
appears when obtaining (124) from (123). We can see we should redefine it as a
new $(\widetilde{T}_{H})^{r}{}_{t,\,lm}$ so that new
$(\widetilde{T}_{H})^{r}{}_{t,\,lm}$ can vanish at the horizon as in (117) by
appropriately taking the integral constants,
$(\widetilde{c}_{o})^{r}{}_{t,\,lm}$ and $(c_{o})^{r}{}_{t,\,lm}$. This is
because $(\widetilde{T}_{H})^{r}{}_{t,\,lm}$ should vanish at the horizon to
get (124) (or (24) in [43] or (36) in [57]), however it does not if it is as
it is. We can see this as
$(\widetilde{T}_{H})^{r}{}_{t,\,lm}\big{|}_{r=(r_{h,2D})_{lm}}=-\widetilde{N}^{r}{}_{t,\,lm}\big{|}_{r=(r_{h,2D})_{lm}}\not=0$
with $f^{\prime}\big{|}_{r=(r_{h,2D})_{lm}}\not=0$. (This is not written in
any papers such as [43, 57]. Further, [56] is referred at (36) in [57], so
look it. Then its (6.21) corresponds to (124). There should be some integral
constants there when ${\cal P}_{\mu\nu}$ is obtained by performing integration
in (6.22), however no comment about this point there.)

$\displaystyle(\widetilde{T}_{H})^{r}{}_{t,\,lm}-(T_{H})^{r}{}_{t,\,lm}=(ff^{\prime\prime}-2(f^{\prime})^{2})/192\pi.$
(124)

We can obtain the value of $(c_{H})^{r}{}_{t,\,lm}$ from (124) with (117) as

$\displaystyle(c_{H})^{r}{}_{t,\,lm}=-\frac{1}{192\pi}(ff^{\prime\prime}-2(f^{\prime})^{2})\big{|}_{r=(r_{h,2D})_{lm}}=\frac{(f^{\prime})^{2}}{96\pi}\Big{|}_{r=(r_{h,2D})_{lm}}=\frac{\pi}{6}T_{H}^{2},$
(125)

where $f^{\prime}|_{r=(r_{h,2D})_{lm}}=4\pi T_{H}$, ($T_{H}$ is (110)).
Variation for (113) can be written as

$\displaystyle(\delta S_{\textrm{2D}})_{lm}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\int d^{2}x(-(g_{\rm
eff})_{lm})^{1/2}\,\eta^{\nu}\nabla_{\mu,\,lm}T^{\mu}{}_{\nu,\,lm}$ (126)
$\displaystyle=$ $\displaystyle\\!\\!\\!-\int dx^{2}\,\eta^{t}(\hskip
7.11317pt((T_{o})^{r}{}_{t,\,lm}-(T_{H})^{r}{}_{t,\,lm}+N^{r}{}_{t,\,lm})\delta\left(r-((r_{h,2D})_{lm}+\epsilon_{lm})\right)$
$\displaystyle\qquad\qquad\quad\\!+\partial_{r}(N^{r}{}_{t,\,lm}H)).$

$\epsilon_{lm}$ are taken to zero as the near-horizon limit. The last term
will vanish [42, 43, 58]111111
$\displaystyle\int_{(r_{h,2D})_{lm}}^{(r_{h,2D})_{lm}+\epsilon}dr\,\partial_{r}\,(\cdots)\to
0$ with $\epsilon\to 0$. .

$(\delta S_{\textrm{2D}})_{lm}$ should vanish, from which
$(c_{o})^{r}{}_{t,\,lm}$, the total amount of the Hawking flux, are determined
as

$\displaystyle(c_{o})^{r}{}_{t,\,lm}=(c_{H})^{r}{}_{t,\,lm}-N^{r}{}_{t,\,lm}\big{|}_{r=(r_{h,2D})_{lm}}=\pi
T_{H}^{2}/12.$ (127)

This result is the same with just the Schwarzschild [42, 43, 57]. The reason
of this is that the Hawking flux is determined from the $f$, $f^{\prime}$ and
$f^{\prime\prime}$ at $r=(r_{h,2D})_{lm}$ as in (124) and (125), however these
are not changed from just the Schwarzschild as can be seen from (81). This is
the same situation with the Hawking temperature in Sec.B.2 and C.

##  Appendix E Comment on result in terms of the information paradox

As mentioned under (110), the result (127) would be the one in the original 4D
black hole, and if the correction is to $\varepsilon^{2}$-order and
$\phi$-independent, we could conclude by the logic in Sec.6 the black-body
radiation of the supertranslated black holes would be always thermal.

Important problem for us is the information paradox. As an insight obtained
from this work, the Hawking temperature and flux could not be the solution as
we have found there is no breaking of the thermal flux in the range of the
analysis in this paper.

Supertranslated black hole spacetimes would be normal in reality and how to be
supertranslated is determined by the initial configuration [6]. Therefore, the
information of the initial configuration would be stored in the configuration
of the asymptotic region of the spacetime, again which would be the key of the
information paradox.

## References

  * [1] https://inspirehep.net/conferences/1772425
  * [2] F. L. Lin and S. Takeuchi, “Hawking Flux from a Black Hole with Nonlinear Supertranslation Hair,” Phys. Rev. D 102, no.4, 044004 (2020) [arXiv:2004.07474 [hep-th]].
  * [3] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A A269, 21-52 (1962)
  * [4] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A A270, 103-126 (1962)
  * [5] A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” [arXiv:1703.05448 [hep-th]].
  * [6] G. Compere and J. Long, “Classical static final state of collapse with supertranslation memory,” Class. Quant. Grav. 33, no. 19, 195001 (2016) [arXiv:1602.05197 [gr-qc]].
  * [7] C. S. Chu and Y. Koyama, “Soft Hair of Dynamical Black Hole and Hawking Radiation,” JHEP 04, 056 (2018) [arXiv:1801.03658 [hep-th]].
  * [8] R. Javadinezhad, U. Kol and M. Porrati, “Comments on Lorentz Transformations, Dressed Asymptotic States and Hawking Radiation,” JHEP 01, 089 (2019) [arXiv:1808.02987 [hep-th]].
  * [9] M. Maitra, D. Maity and B. R. Majhi, “Near horizon symmetries, emergence of Goldstone modes and thermality,” Eur. Phys. J. Plus 135, no.6, 483 (2020) [arXiv:1906.04489 [hep-th]].
  * [10] W. Y. Wen, “Dressed tunneling in soft hair,” Phys. Lett. B 820, 136578 (2021) [arXiv:2103.00516 [hep-th]].
  * [11] G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105, 111103 (2010) [arXiv:0909.2617 [gr-qc]].
  * [12] S. W. Hawking, M. J. Perry and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. 116, no.23, 231301 (2016) [arXiv:1601.00921 [hep-th]].
  * [13] M. S. Pshirkov, D. Baskaran and K. A. Postnov, “Observing gravitational wave bursts in pulsar timing measurements,” Mon. Not. Roy. Astron. Soc. 402, 417 (2010) [arXiv:0909.0742 [astro-ph.CO]].
  * [14] R. van Haasteren and Y. Levin, “Gravitational-wave memory and pulsar timing arrays,” Mon. Not. Roy. Astron. Soc. 401, 2372 (2010) [arXiv:0909.0954 [astro-ph.IM]].
  * [15] N. Seto, “Search for Memory and Inspiral Gravitational Waves from Super-Massive Binary Black Holes with Pulsar Timing Arrays,” Mon.Not.Roy.Astron.Soc. 400 (2009) L38 [arXiv:0909.1379 [astro-ph.GA]].
  * [16] J. B. Wang, G. Hobbs, W. Coles, R. M. Shannon, X. J. Zhu, D. R. Madison, M. Kerr, V. Ravi, M. J. Keith, R. N. Manchester, Y. Levin, M. Bailes, N. D. R. Bhat, S. Burke-Spolaor, S. Dai, S. Oslowski, W. van Straten, L. Toomey, N. Wang and L. Wen, “Searching for gravitational wave memory bursts with the Parkes Pulsar Timing Array,” Mon. Not. Roy. Astron. Soc. 446, 1657-1671 (2015) [arXiv:1410.3323 [astro-ph.GA]].
  * [17] Z. Arzoumanian et al. [NANOGrav], “NANOGrav Constraints on Gravitational Wave Bursts with Memory,” Astrophys. J. 810, no.2, 150 (2015) [arXiv:1501.05343 [astro-ph.GA]].
  * [18] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen, “Detecting gravitational-wave memory with LIGO: implications of GW150914,” Phys. Rev. Lett. 117, no.6, 061102 (2016) [arXiv:1605.01415 [astro-ph.HE]].
  * [19] S. Pasterski, A. Strominger and A. Zhiboedov, “New Gravitational Memories,” JHEP 12, 053 (2016) [arXiv:1502.06120 [hep-th]].
  * [20] M. Pate, A. M. Raclariu and A. Strominger, “Color Memory: A Yang-Mills Analog of Gravitational Wave Memory,” Phys. Rev. Lett. 119, no.26, 261602 (2017) [arXiv:1707.08016 [hep-th]].
  * [21] L. Bieri and D. Garfinkle, “An electromagnetic analogue of gravitational wave memory,” Class. Quant. Grav. 30, 195009 (2013) [arXiv:1307.5098 [gr-qc]].
  * [22] L. Susskind, “Electromagnetic Memory,” [arXiv:1507.02584 [hep-th]].
  * [23] S. Pasterski, “Asymptotic Symmetries and Electromagnetic Memory,” JHEP 09, 154 (2017) [arXiv:1505.00716 [hep-th]].
  * [24] A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 07, 151 (2014) [arXiv:1308.0589 [hep-th]].
  * [25] T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “New Symmetries of Massless QED,” JHEP 10, 112 (2014) [arXiv:1407.3789 [hep-th]].
  * [26] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07, 152 (2014) [arXiv:1312.2229 [hep-th]].
  * [27] T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05, 151 (2015) [arXiv:1401.7026 [hep-th]].
  * [28] F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” [arXiv:1404.4091 [hep-th]].
  * [29] D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix,” JHEP 08, 058 (2014) [arXiv:1406.3312 [hep-th]].
  * [30] D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “2D Stress Tensor for 4D Gravity,” Phys. Rev. Lett. 119, no.12, 121601 (2017) [arXiv:1609.00282 [hep-th]].
  * [31] T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP 08, 050 (2017) [arXiv:1701.00496 [hep-th]].
  * [32] T. He, P. Mitra and A. Strominger, “2D Kac-Moody Symmetry of 4D Yang-Mills Theory,” JHEP 10, 137 (2016) [arXiv:1503.02663 [hep-th]].
  * [33] A. Bagchi, R. Basu, A. Kakkar and A. Mehra, “Flat Holography: Aspects of the dual field theory,” JHEP 12, 147 (2016) [arXiv:1609.06203 [hep-th]].
  * [34] S. Pasterski, S. H. Shao and A. Strominger, “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,” Phys. Rev. D 96, no.6, 065026 (2017) doi:10.1103/PhysRevD.96.065026 [arXiv:1701.00049 [hep-th]].
  * [35] C. Cardona and Y. t. Huang, “S-matrix singularities and CFT correlation functions,” JHEP 08, 133 (2017) [arXiv:1702.03283 [hep-th]].
  * [36] S. Pasterski and S. H. Shao, “Conformal basis for flat space amplitudes,” Phys. Rev. D 96, no.6, 065022 (2017) [arXiv:1705.01027 [hep-th]].
  * [37] S. Pasterski, S. H. Shao and A. Strominger, “Gluon Amplitudes as 2d Conformal Correlators,” Phys. Rev. D 96, no.8, 085006 (2017) [arXiv:1706.03917 [hep-th]].
  * [38] A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 01, 086 (2016) [arXiv:1411.5745 [hep-th]].
  * [39] S. W. Hawking, M. J. Perry and A. Strominger, “Superrotation Charge and Supertranslation Hair on Black Holes,” JHEP 05, 161 (2017) [arXiv:1611.09175 [hep-th]].
  * [40] A. Strominger, “Black Hole Information Revisited,” [arXiv:1706.07143 [hep-th]].
  * [41] E. Berti, V. Cardoso and C. M. Will, “On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA,” Phys. Rev. D 73, 064030 (2006) [gr-qc/0512160].
  * [42] S. P. Robinson and F. Wilczek, “A Relationship between Hawking radiation and gravitational anomalies,” Phys. Rev. Lett. 95, 011303 (2005) [gr-qc/0502074].
  * [43] S. Iso, H. Umetsu and F. Wilczek, “Hawking radiation from charged black holes via gauge and gravitational anomalies,” Phys. Rev. Lett. 96, 151302 (2006) [arXiv:hep-th/0602146 [hep-th]].
  * [44] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” Phys. Rev. Lett. 85, 5042-5045 (2000) [arXiv:hep-th/9907001 [hep-th]].
  * [45] D. Lust and W. Vleeshouwers, “Black Hole Information and Thermodynamics,” [arXiv:1809.01403 [gr-qc]].
  * [46] K. Umetsu, “Recent Attempts in the Analysis of Black Hole Radiation,” arXiv:1003.5534 [hep-th].
  * [47] https://en.wikipedia.org/wiki/Clebsch-Gordan$\\_$coefficients
  * [48] K. Fujikawa and H. Suzuki, “Path integrals and quantum anomalies,”
  * [49] T. Kimura and T. Ohta “Classical and quantum gravitational theory,” McGraw-Hill Co., Inc., 1989 (Japanese),
  * [50] R. Li, S. Li and J. R. Ren, “Hawking Radiation of Fermionic Field and Anomaly in 2+1 Dimensional Black Holes,” Class. Quant. Grav. 27, 155011 (2010) [arXiv:1005.3615 [hep-th]].
  * [51] R. Becar and P. A. Gonzalez, “Hawking Radiation for Scalar and Dirac Fields in Five Dimensional Dilatonic Black Hole via Anomalies,” Int. J. Mod. Phys. D 21, 1250030 (2012) [arXiv:1104.0356 [gr-qc]].
  * [52] P. J. Mao, R. Li, L. Y. Jia and J. R. Ren, “Hawking radiation of Dirac particles from the Myers-Perry black hole,” Eur. Phys. J. C 71, 1527 (2011).
  * [53] S. Iso, T. Morita and H. Umetsu, “Hawking radiation via higher-spin gauge anomalies,” Phys. Rev. D 77, 045007 (2008) [arXiv:0710.0456 [hep-th]].
  * [54] K. Umetsu, “Ward Identities in the derivation of Hawking radiation from Anomalies,” Prog. Theor. Phys. 119, 849 (2008) [arXiv:0804.0963 [hep-th]].
  * [55] R. Banerjee and S. Kulkarni, “Hawking radiation, effective actions and covariant boundary conditions,” Phys. Lett. B 659, 827 (2008) [arXiv:0709.3916 [hep-th]].
  * [56] R. A. Bertlmann and E. Kohlprath, “Two-dimensional gravitational anomalies, Schwinger terms and dispersion relations,” Annals Phys. 288, 137 (2001) [hep-th/0011067].
  * [57] S. Iso, H. Umetsu and F. Wilczek, “Anomalies, Hawking radiations and regularity in rotating black holes,” Phys. Rev. D 74, 044017 (2006) [arXiv:hep-th/0606018 [hep-th]].
  * [58] K. Murata and J. Soda, “Hawking radiation from rotating black holes and gravitational anomalies,” Phys. Rev. D 74, 044018 (2006) [hep-th/0606069].