Abstract
We derive the leading spin-dependent gravitational tail memory, which appears at the second post-Minkowskian (2 PM) order and behaves as $u^{-2}$ for large retarded time $u$. This result follows from classical soft graviton theorem at order $\omega\ln\omega$ as a low-frequency expansion of gravitational waveform with frequency $\omega$. First, we conjecture the result from the classical limit of quantum soft graviton theorem up to sub-subleading order in soft expansion and then we derive it for a classical scattering process without any reference to the soft graviton theorem. The final result of the gravitational waveform in the direct derivation completely agrees with the conjectured result.
Highlights
The observation of the permanent displacement between the mirrors of the gravitational wave detector relative to their initial distance after the passage of full gravitational radiation produced from an astrophysical scattering event is known as gravitational memory
We conjecture the gravitational waveform from the classical limit of the quantum soft graviton theorem up to sub-subleading order in a soft expansion, and we derive it for a classical scattering process without any reference to the soft graviton theorem
There has been a proposal of another kind of gravitational memory known as “tail memory,” which describes how the mirrors of a gravitational wave detector behave at a large retarded time before reaching their permanent displaced positions [11,12,13,14,15]
Summary
The observation of the permanent displacement between the mirrors of the gravitational wave detector relative to their initial distance after the passage of full gravitational radiation produced from an astrophysical scattering event is known as gravitational memory. From the classical limit of the quantum soft graviton theorem up to sub-subleading order in D 1⁄4 4, we get the following radiative mode of the gravitational waveform with frequency ω at distance R 1⁄4 jx⃗ j from the scattering center [12,16,17]: eμν ðω; x⃗ Þ ð−iÞ. Following the prescription of [12,17], if we replace ln jσj by − lnðω þ iεηaÞ in the classical angular momentum of Eq (2.3), it predicts the correct gravitational waveform, which has been independently verified in [13,14] up to sub-subleading order nonspinning particle scattering. For the scattering of nonspinning objects, we can read off the order ω ln ω gravitational waveform after setting Σa 1⁄4 0 and Σ0a 1⁄4 0 in the above relation
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