Proof of the classical soft graviton theorem in D = 4
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Springer
Received: March 3, 2020 Accepted: June 1, 2020 Published: June 25, 2020
Arnab Priya Saha, Biswajit Sahoo and Ashoke Sen Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhusi, Allahabad 211019, India
E-mail: [email protected], [email protected], [email protected] Abstract: Classical subleading soft graviton theorem in four space-time dimensions determines the gravitational wave-form at late and early retarded time, generated during a scattering or explosion, in terms of the four momenta of the ingoing and outgoing objects. This result was ‘derived’ earlier by taking the classical limit of the quantum soft graviton theorem, and making some assumptions about how to deal with the infrared divergences of the soft factor. In this paper we give a direct proof of this result by analyzing the classical equations of motion of gravity coupled to matter. We also extend the result to the electromagnetic wave-form generated during scattering of charged particles, and present a new conjecture on subsubleading corrections to the gravitational wave-form at early and late retarded time. Keywords: Scattering Amplitudes, Classical Theories of Gravity, Gauge Symmetry, Space-Time Symmetries ArXiv ePrint: 1912.06413
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)153
JHEP06(2020)153
Proof of the classical soft graviton theorem in D = 4
Contents 1 Introduction and summary
1
2 Some useful results
4 5 6 9 9 11 15 20
4 Generalizations 4.1 Soft photon theorem with electromagnetic interactions 4.2 Gravitational contribution to the soft photon theorem 4.3 Electromagnetic wave-form at early and late time 4.4 Electromagnetic contribution to the soft graviton theorem
21 21 23 25 25
5 New conjectures at the subsubleading order
27
6 Numerical estimate
30
A Derivation of some useful mathematical results A.1 Radiative field at large distance A.2 Late and early time behaviour from Fourier transformation
32 32 33
B Evaluation of some integrals
36
C Contribution from real gravitons
38
D Position space analysis of TbXµν
41
1
Introduction and summary
In a quantum theory of gravity, soft graviton theorem gives an amplitude with a set of finite energy external particles and one or more low energy external gravitons, in terms of the amplitude without the low energy gravitons [1–14]. However when we take the classical limit, there is a different manifestation of the same theorem — it determines the low frequency component of the gravitational wave-form produced during a scattering
–1–
JHEP06(2020)153
3 Proof of classical soft graviton theorem 3.1 General set-up 3.2 Leading order contribution 3.3 First order correction to the gravitational field 3.4 Subleading contribution to the matter stress tensor 3.5 Subleading contribution from the gravitational stress tensor 3.6 Gravitational wave-form at early and late time
R = |~x|,
n ˆ=
~x , R
n = (1, n ˆ) .
(1.1)
We shall consider the limit of large R and analyze only the terms of order 1/R in the gravitation
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