Schwarzschild黑洞的量子校正
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Eur. Phys. J. C (2017) 77:243
DOI 10.1140/epjc/s10052-017-4802-0
Regular Article - Theoretical Physics
Quantum corrections to Schwarzschild black hole
Xavier Calmet
a
, Basem Kamal El-Menoufi
b
Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, UK
Received: 17 March 2017 / Accepted: 1 April 2017 / Published online: 17 April 2017
© The Author(s) 2017. This article is an open access publication
Abstract Using effective field theory techniques, we com-
pute quantum corrections to spherically symmetric solu-
tions of Einstein’s gravity and focus in particular on the
Schwarzschild black hole. Quantum modifications are covari-
antly encoded in a non-local effective action. We work to
quadratic order in curvatures simultaneously taking local and
non-local corrections into account. Looking for solutions per-
turbatively close to that of classical general relativity, we find
that an eternal Schwarzschild black hole remains a solution
and receives no quantum corrections up to this order in the
curvature expansion. In contrast, the field of a massive star
receives corrections which are fully determined by the effec-
tive field theory.
1 Introduction
With the recent discovery of gravitational waves, we have
entered a new era in astrophysics in which we will be prob-
ing black hole physics directly. Besides gravitational waves,
the Event Horizon Telescope will soon be in a situation to
probe the horizon of Sgr A
∗
, the black hole at the center of
our galaxy, by studying its shadow directly. Recent progress
in observational astronomy thus prompts us to try and under-
stand quantum effects on black holes, as we should soon be
in a situation able to confront theory with observations. Some
of the questions that we can hope to probe with such obser-
vations are whether the horizons of black holes are more
violent regions of space-time, in contrast to what is expected
in general relativity, and what physical effect resolves the
curvature singularity at r = 0. Both questions are crucial
if we want to understand the information paradox linked to
Hawking radiation [1].
Black holes are amongst the simplest and yet most mys-
terious objects in our universe. According to the no-hair the-
a
e-mail: x.calmet@sussex.ac.uk
b
e-mail: b.elmenoufi@sussex.ac.uk
orem, they are described by only a few parameters: their
masses, angular momenta and charges. Despite this apparent
simplicity, they are incredibly challenging as understanding
their physics requires merging quantum mechanics and gen-
eral relativity, which is one of the remaining holy grails of
theoretical physics (see e.g. [2] for a recent review).
While we are far away from having a consistent theory
of quantum gravity valid at all energy scales, effective field
theory (EFT) techniques applied to general relativity lead to
a consistent theory of quantum gravity valid up to an energy
scale close to the Planck scale [3,4]. Several universal fea-
tures of quantum gravity can be identified using these EFT
methods. The most intriguing of these features is the dynam-
ical non-locality of space-time induced at short distances by
quantum effects.
This non-locality has been shown to have interesting fea-
tures in cosmology and could indeed avoid the big crunch sin-
gularity in a collapsing universe [5]. Investigating the effects
of this non-locality in black hole physics is our main motiva-
tion to consider quantum corrections to spherically symmet-
ric solutions in general relativity. In particular, we revisit the
issue of quantum corrections to the Schwarzschild black hole
solution which have been studied in the past [6,7]. We iden-
tify a complication which has not been realized previously,
namely that of how to define a black hole.
A mathematically consistent way to define a black hole
is to define it as a static vacuum solution, i.e., an eternal
black hole. If this definition is adopted, we obtain a result
that differs from previous investigations. In particular, we
show that the classical black hole solution remains a solu-
tion in quantum gravity up to quartic order in the non-local
curvature expansion. This result is obtained by looking for
perturbative solutions around the Schwarzschild black hole
metric. Nevertheless, higher curvature operators give rise
to non-trivial corrections. The non-local operators that led
to singularity avoidances in [5] thus do not affect the sin-
gularity in the case of the Schwarzschild black hole solu-
tion.
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