Effective Tolman temperature induced by trace anomaly
Eur. Phys. J. C
Effective Tolman temperature induced by trace anomaly
Myungseok Eune 2
Yongwan Gim 0 1
Wontae Kim 1
0 Research Institute for Basic Science, Sogang University , Seoul, 04107 , Republic of Korea
1 Department of Physics, Sogang University , Seoul 04107 , Republic of Korea
2 Department of Civil Engineering, Sangmyung University , Cheonan 31066 , Republic of Korea
Despite the finiteness of stress tensor for a scalar field on the fourdimensional Schwarzschild black hole in the IsraelHartleHawking vacuum, the Tolman temperature in thermal equilibrium is certainly divergent on the horizon due to the infinite blueshift of the Hawking temperature. The origin of this conflict is due to the fact that the conventional Tolman temperature was based on the assumption of a traceless stress tensor, which is, however, incompatible with the presence of the trace anomaly responsible for the Hawking radiation. Here, we present an effective Tolman temperature which is compatible with the presence of the trace anomaly by using the modified StefanBoltzmann law. Eventually, the effective Tolman temperature turns out to be finite everywhere outside the horizon, and so an infinite blueshift of the Hawking temperature at the event horizon does not appear any more. In particular, it is vanishing on the horizon, so that the equivalence principle is exactly recovered at the horizon. A quantum black hole [1,2] in the IsraelHartleHawking vacuum [3,4] could be characterized by the Hawking temperature TH which is given by the surface gravity. The local temperature in a proper frame as the Tolman temperature can be defined in the form of the blueshifted Hawking temperature as [5,6]

Tloc = √
−g00(r )
which is infinite at the horizon due to the infinite blueshift of
the Hawking temperature, though it reduces to the Hawking
temperature at infinity.
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On the other hand, the renormalized stress tensor for a
conformal scalar field could be finite on the background
of the Schwarzschild black hole [7]. At infinity, the proper
energy density ρ is positive finite, which is consistent with
the Stefan–Boltzmann law as ρ = σ TH4, where σ = π 2/30.
If one considered a motion of an inertial observer [7–10],
the negative proper energy density could be found near the
horizon in various vacua and its role was also discussed in
connection with the information loss paradox [10]. However,
it might be interesting to note that the local temperature (1)
is infinite at the horizon, although the proper energy density
at the horizon rH is negative finite as ρ(rH) = −12σ TH4 as
seen in Ref. [7].
Now, it appears to be puzzling in that the Tolman
temperature at the horizon is positively divergent despite the negative
finite energy density there. More worse, the energy density
happens to vanish at a certain point outside the horizon [7],
but the local temperature (1) is positive finite at that point. In
these regards, the Tolman temperature runs contrary to the
finite renormalized stress tensor, which certainly requires that
the Stefan–Boltzmann law to relate the stress tensor to the
proper temperature should be appropriately modified in such
a way that they are compatible each other.
To resolve the above conflict between the finiteness of
the renormalized stress tensor and the divergent behavior of
the proper temperature, it is worth noting that the usual
Tolman temperature rests upon the traceless stress tensor;
however, the trace of the renormalized stress tensor is actually
not traceless because of the trace anomaly. So we should
find a modified Stefan–Boltzmann law in order to get the
proper temperature commensurate with the finite
renormalized stress tensor. In fact, this was successfully realized in
the twodimensional case where the stress tensor was
perfect fluid [11]. In this work, we would like to extend the
above issue to the case of the fourdimensional more
realistic Schwarzschild black hole, where the renormalized stress
tensor is no more isotropic.
Using the exact thermal stress tensor calculated in Ref. [7],
we solve the covariant conservation law and the equation for
the trace anomaly, and then obtain the proper quantities such
as the proper energy density and pressures written explicitly
in terms of the trace anomaly in Sect. 2. In Sect. 3, we derive
the effective Tolman temperature from the modified Stefan–
Boltzmann law based on thermodynamic analysis. It shows
that the effective Tolman temperature exactly reproduces the
Hawking temperature at infinity, but it has a maximum at a
finite distance outside the horizon and eventually it is
vanishing rather than divergent on the horizon. Finally, a conclusion
and a discussion are given in Sect. 4.
2 Proper quantities in terms of trace anomaly We start with a fourdimensional Schwarzschild black hole governed by the static line element as 1
ds2 = − f (r )dt 2 + f (r ) dr 2 + r 2(dθ 2 + sin2 θ dφ2),
where the metric function is f (r ) = 1 − 2G M/r . The
renormalized stress tensor for a conformal scalar field on the
Schwarzschild black hole was obtained in the Israel–Hartle–
Hawking vacuum [3,4] by using the Gaussian
approximation [7]
anomaly reduces to
Tμμ = 28810π 2 Rμνρσ Rμνρσ = 60Mπ 22r 6 ,
and then the trace for the stress tensor (3) is exactly in accord
with the conformal anomaly (5).
In contrast to the twodimensional case [11], the stress
tensor appears anisotropic in the spherically symmetric black
hole in four dimensions, and so the form of the stress
tensor (3) should be generically written as [29,30]
T μν = (ρ + pt )uμuν + pt gμν + ( pr − pt )n(μr)n(νr).
The proper velocity uμ is a timelike unit vector satisfying
μ
uμuμ = −1, n(r) is the unit spacelike vector in the radial
direction, and n(μθ) and n(μφ) are the unit normal vectors,
which are orthogonal to n(μr), satisfying gμν n(μi)n( j) = δi j
μ
and n(μi)uμ = 0 where i, j = r, θ , φ. Thus the spacelike unit
normal vectors are determined as
with the proper velocity
where it is finite everywhere.
On general grounds, the trace anomaly can be written in
the form of curvature invariants as
where F = Rμνρσ Rμνρσ − 2 Rμν Rμν + R2/3 and G =
Rμνρσ Rμνρσ − 4Rμν Rμν + R2 [12–16]. There have been a
lot of applications of trace anomalies to Hawking radiation
and black hole thermodynamics in wide variety of cases of
interest [17–28]. The coefficients α and β are related to the
number of conformal fields such as real scalar fields NS,
Dirac (fermion) fields NF, and vector fields NV, such that
they are fixed as α = (120(4π )2)−1(NS + 6NF + 12NV) and
β = −(360(4π )2)−1(NS + 11NF + 62NV). For the Ricci
flat spacetime with a single conformal scalar field, the trace
for the frame dropped from rest. Then, from Eqs. (3), (6),
(7), and (8), the proper energy density and pressures can be
explicitly calculated by using the following relations:
ρ = Tμν uμuν , pr = Tμν n(μr)n(νr),
pt = Tμν n(μθ)n(θ) = Tμν n(φ)n(νφ),
ν μ
where the proper flux along xi direction can also be obtained
by using the relation Fi = −Tμν uμn(νi) but it trivially
vanishes in thermal equilibrium [3,4].
Note that the energy density and pressures are not
independent, as seen from the trace relation,
From Eqs. (3), (5), and (9), we find the additional relation
which characterizes the anisotropy between the tangential
pressure and radial pressure.
Let us now express the proper energy density and
pressures formally in terms of the trace anomaly for our purpose.
From Eq. (6), the covariant conservation law for the
energymomentum tensor is rewritten as
Plugging Eqs. (10) and (11) into Eq. (12), one can obtain the
simplified form of
which can be solved as
where C0 is an integration constant. Additionally, from
Eqs. (10) and (11), the tangential pressure and energy density
can also be obtained:
1
pt = f 2
3
ρ = f 2
C0 − 4
C0 − 2
The above proper quantities defined in freely falling frames
were related to the trace anomaly conveniently, which will
be used in the next section.
3 Effective Tolman temperature
In this section, we derive the proper temperature for the
background of the fourdimensional Schwarzschild black hole
based on the modified Stefan–Boltzmann law. First of all,
we note that the volume of the system in the radial proper
frame can be changed only along the radial direction on the
spherically symmetric black hole, and thus obtain the
thermodynamic first law written as
dU = T dS − pr dV
without recourse to the tangential work. From Eq. (17), one
can immediately get
= T
− pr ,
and then, from the Maxwell relations such as (∂ S/∂ V )T =
(∂ pr /∂ T )V , we obtain
− pr .
Using the fact that the trace anomaly is independent of
temμ
perature as ∂T Tμ = 0 [31], from Eqs. (10) and (11), we also
obtain
and it can be compactly written in terms of the trace anomaly
as
Plugging Eqs. (20) and (21) into Eq. (19), we get
which is solved as
From Eqs. (10) and (11), the radial and tangential pressure
are also derived as
respectively. The integration constant γ is related to the
Stefan–Boltzmann constant σ as γ = σ/3 = π 2/90 for
a conformal scalar field [32]. For the traceless case, the
modified Stefan–Boltzmann law (23) simply reduces to the usual
one. The proper energy density in Eq. (23) is not necessarily
positive definite thanks to the trace anomaly, so that the
negative energy states are naturally permitted in this extended
setting.
From Eqs. (23), (24), and (25), the proper temperature is
obtained as
T =
3 T μ
3 T μ
where we used Eqs. (14), (15), and (16). In the absence of
the trace anomaly, the proper temperature (27) reduces to the
usual Tolman temperature [5, 6]. Requiring that the proper
temperature (27) be coincident with the Hawking
temperature TH at infinity, we can fix the constant as C0 = γ 1/4TH.
Finally, plugging the trace anomaly (5) into Eq. (27), we
obtain
− 21
1 − 28
which can be neatly factorized as
1 −
− 21
It seems to be interesting to note that the blueshift factor
in the denominator related to the origin of the divergence at
the horizon can be canceled out, so that the effective Tolman
temperature is written as
1 −
n=1
n−1 1/4
Thus the redshift factor responsible for the infinite blueshift
of the Hawking temperature on the horizon does not appear
any more in the effective Tolman temperature. As seen from
Fig. 1, the behavior of the temperature (30) shows that it is
finite everywhere and approaches the Hawking temperature
at infinity. In particular, it is vanishing on the horizon, so that
the freely falling observer from rest does not see any excited
particles. On the contrary to the naively expected divergence
from the usual Tolman temperature at the horizon, the high
frequency quanta could not be found on the horizon, which
would be compatible with the result that the equivalence
principle could be recovered at the horizon [33].
The divergent dashed curve near the horizon in Fig. 1 could
be made finite by taking into account the quantum effect
via the trace anomaly, which is reminiscent of the vanishing
Fig. 1 The dashed curve shows the behavior of the usual Tolman
temperature of being divergent on the horizon. The solid curve is the
effective Tolman temperature, which is finite everywhere. In
particular, it vanishes at the horizon and has a maximum Tmax ∼ 1.51TH
at rc ∼ 1.31rH . All the curves approach the Hawking temperature at
infinity, whereas they are very different from each other near the horizon
where quantum effects are significant
Hawking temperature in the noncommutative Schwarzschild
black hole based on the different assumptions of quantization
rules [34]. The proper temperature based on the effective
temperature method is also compatible with the present result in
the sense that the proper temperature vanishes at the horizon
[35].
4 Conclusion and discussion
It has been widely believed that the Tolman temperature is
divergent at the horizon due to the infinite blueshift of the
Hawking radiation. However, the usual Stefan–Boltzmann
law assuming the traceless stress tensor should be
consistently modified in order to discuss the case where the stress
tensor is no longer traceless in the process of the
Hawking radiation. From the modified Stefan–Boltzmann law, we
obtained the effective Tolman temperature without the
redshift factor related to the origin of the divergence at the
horizon, so that it is finite everywhere outside the black hole
horizon.
The intriguing behavior of the effective Tolman
temperature on the horizon may be understood by the Unruh
effect [36]. The static metric (2) near the horizon can be
written by the Rindler metric for a large black hole whose
curvature scale is negligible. The Unruh temperature is divergent
due to the infinite acceleration of the frame where the fixed
detector is very close to the horizon. So the Unruh
temperature is equivalent to the locally fiducial temperature for the
Schwarzschild black hole [33]. Conversely speaking, based
on the equivalence principle, the Unruh temperature
measured by the geodesic detector should vanish on the horizon
since the proper acceleration of the geodesic detector
vanishes. In this regard, it appears natural to conclude that the
freely falling observer from rest does not see any excited
particles on the horizon in thermal equilibrium and thus the
effective Tolman temperature vanishes at the horizon.
On the other hand, AMPS argument is that the firewall on
the horizon should be defined in an evaporating black hole
rather than the black hole in thermal equilibrium [37]. The
firewall is certainly characterized by the divergent proper
temperature in that the average frequency ω of an excited
particle with a thermal bath can be identified with the proper
temperature as ω ∼ T . Using the advantage of the
effective Tolman temperature, we find the reason why the firewall
could not exist in thermal equilibrium: the fact that the
redshift factor responsible for the divergence at the horizon could
be canceled out.
Acknowledgements We would like to thank JeongHyuck Park and
Edwin J. Son for exciting discussions. W. Kim was supported by the
National Research Foundation of Korea (NRF) grant funded by the
Korea government (MSIP) (2017R1A2B2006159).
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