软重定理中对数项的观测标志
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Observational signature of the logarithmic terms
in the soft-graviton theorem
Alok Laddha
1,*
and Ashoke Sen
2,†
1
Chennai Mathematical Institute, Siruseri, Chennai 603103, India
2
Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhusi, Allahabad 211019, India
(Received 8 November 2018; published 8 July 2019)
We show that the recently discovered logarithmic terms in the soft-graviton theorem induce a late time
component in the gravitational waveform that falls off as inverse power of time, producing a tail term
to the linear memory effect.
DOI: 10.1103/PhysRevD.100.024009
One of the reasons for the recent interest in the soft-
graviton theorem is its connection to the memory effect
[1–4]—the fact that a passing gravitational wave causes a
permanent change in the distance between two detectors
placed in its path [5–8]. This connection usually proceeds
via asymptotic symmetries [9–11] and has led to the
prediction of a new kind of memory effect associated with
the super-rotation symmetry [2]. In contrast, in Ref. [12],we
established a direct connection between the soft factors that
arise in soft theorems and the low frequency classical
radiation in a classical process by taking the classical limit
of the quantum scattering process. This has the advantage of
being valid in all space-time dimensions, irrespective of
whether or not the soft theorem can be related to an
asymptotic symmetry. However, while applying this for-
mula to four dimensions, we encounter a new phenomenon:
due to the long range force on the particles involved in the
scattering, the soft factor at the subleading order gets a
contribution proportional to the logarithm of the energy of
the soft radiation [13]. Our goal in this paper is to describe
the observational signature of this logarithmic term in the
soft-graviton theorem. We shall use ℏ ¼ c ¼ 8πG ¼ 1
units, although since we are analyzing classical radiation,
ℏ never enters any formula.
The result of this paper may be summarized as
follows. In any process that involves the breakup of a
massive object of mass M into another object of mass
M
0
≃ M and a set of light particles, the gravitational
waveform e
TT
ij
near future null infinity is given, at late
retarded time u,byEq.(7). A
ij
and B
ij
in this equation
are determined in terms of the final state mass and
velocity distribution of the light particles via Eqs. (11)
and (12). The coefficient A
ij
describes the standard
memory effect, while the co effici ent B
ij
describes a tail
term that falls off as an inverse power of u.
The s etup we shall investigate is a process in which a
system of mass M, describing the initial system, makes a
transition into a system of mass M
0
<M and some
matter/radiation that escapes the system. We shall work
in the rest f rame of the original system and assume for
simplicity that the total energy carried by the escaping
matter/radiation is small compared to the mass of the
original system so that M
0
≃ M and the recoil velocity of
the final system is small [14] .Anexampleofthiswould
be the merger of neutron stars where a large amount of
matter is ejected from the parent system but the total
amount of energy lost is still small compared to the mass
of the system that remains b ehind. Another example
would be th e merger of two black holes where the energy
is radiated away gravitationally, but we shall see that the
effects we shall describe vanish in the case where only
massless particles carry away the energy. Our focus will
be on the low frequency radiative com ponent of the
metric field h
μν
≡ ðg
μν
− η
μν
Þ=2.
In Ref. [13], a formula for the soft radiation was found in
a situation where a light particle of mass m scatters from a
heavy particle of mass M
0
. Here, a light particle refers to a
particle carrying energy ≪ M
0
. However, since the soft-
graviton theorem expresses the result as independent sums
over initial and final states, the result can be easily
generalized to the case where there are no light particles
in the initial state and multiple light particles in the final
state. We shall now state this result. For more details, we
refer the reader to Ref. [13].
Let t
0
≃ j
xjþM
0
ln j
xj=4π be the time at which the peak
of the gravitational radiation reaches the observer at
x.For
the radiative part of the trace reversed metric
*
aladdha@cmi.ac.in
†
sen@hri.res.in
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D 100, 024009 (2019)
2470-0010=2019=100(2)=024009(5) 024009-1 Published by the American Physical Society
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