Physics Letters B 798 (2019) 135020
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Test of quantum atmosphere in the dimensionally reduced
Schwarzschild black hole
Myungseok Eune
a
, Wontae Kim
b,∗
a
Gyedang College of General Education, Sangmyung University, Cheonan, 31066, Republic of Korea
b
Department of Physics, Sogang University, Seoul, 04107, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received
21 August 2019
Received
in revised form 2 October 2019
Accepted
8 October 2019
Available
online 11 October 2019
Editor:
M. Cveti
ˇ
c
Keywords:
Hawking
temperature
Stefan-Boltzmann
law
Tolman
temperature
Hartle-Hawking
vacuum
Unruh
vacuum
It has been suggested by Giddings that the origin of Hawking radiation in black holes is a quantum
atmosphere of near-horizon quantum region by investigating both the total emission rate and the
stress tensor of Hawking radiation. Revisiting this issue in the exactly soluble model of a dimensionally
reduced Schwarzschild black hole, we shall confirm that the dominant Hawking radiation in the Unruh
vacuum indeed occurs at the quantum atmosphere, not just at the horizon by exactly calculating the
out-temperature responsible for outgoing Hawking particle excitations. Consequently we show that the
out-temperature vanishes at the horizon and has a peak at a scale whose radial extent is set by the
horizon radius, and then decreases to the Hawking temperature at infinity. We also discuss bounds of
location of the peak for the out-temperature in our model.
© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Hawking radiation as information carrier [1]leads to black hole
complementarity in such a way that there would be no contra-
dictory
physical observations between static and freely falling ob-
servers
[2]. In connection with black hole complementarity, one
of the solutions to the firewall paradox [3]is that the infalling
observer crossing the horizon could find the firewall of high fre-
quency
quanta after the Page time [4]. Thus, the Hawking radiation
at the horizon should be highly excited beyond the Planckian scale.
The existence of the firewall might also be explained by the Tol-
man
temperature [5,6]since the Hawking radiation at infinity is
ascribed to the infinitely blue-shifted radiation at the horizon.
On
the other hand, Unruh showed numerically that the pro-
cess
of thermal particle creation is low-energy behavior so that
the highest frequency mode does not matter for the thermal emis-
sion
[7]. It was also claimed that the Hawking radiation can be
retrieved by an alternative Boulware accretion scenario without
recourse to a pair creation scenario at the horizon [8]. Recently,
Giddings raised a refined question regarding the origin of the
Hawking radiation in the Unruh vacuum [9]. He investigated both
the total emission rate and the stress tensor of Hawking radiation
*
Corresponding author.
E-mail
addresses: eunems@smu.ac.kr (M. Eune), wtkim@sogang.ac.kr (W. Kim).
and then concluded that the origin of Hawking radiation is the
near-horizon quantum region of the quantum atmosphere whose
radial extent is set by the horizon radius scale. Subsequently, there
have been some related works to the quantum atmosphere; anal-
yses
of the stress tensor and the effective temperature [10–12],
and calculations of emission rate of Hawking radiation in arbitrary
dimensions [13].
In
particular, from the analysis of a stress tensor, it was claimed
that in the Unruh vacuum the Tolman temperature near the hori-
zon
does not originate from the out-going particles but the in-
going
particles from the fact that the negative influx would transi-
tion
to the positive outward flux over the quantum region outside
the horizon [9]. It means that the out-going Hawking radiation
originates from the quantum atmosphere. This claim was also dis-
cussed
by employing the local temperature responsible for the out-
going
particles [10]; thus the local temperature related to the out-
going
particles must be finite over the whole region, in particular,
it has a peak at a macroscopic distance outside the horizon. The
crucial difference from conventional results comes from a modi-
fication
of the Stefan-Boltzmann law for the out-going particles.
In Ref. [11], the authors also advocated the quantum atmosphere
with two different arguments. Heuristically, the first was based on
the gravitational Schwinger effect for particle production by the
tidal force outside a black hole horizon. Next, the second argu-
ment
of our concern made use of a calculation of the stress tensor
to derive the energy density for an observer at a constant Kruskal
https://doi.org/10.1016/j.physletb.2019.135020
0370-2693/
© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.