[Paper Review] Spin Dependent Gravitational Tail Memory in $D=4$

Debodirna Ghosh, Biswajit Sahoo|arXiv (Cornell University)|Jun 20, 2021

Cosmology and Gravitation Theories40 references4 citations

TL;DR

This paper derives the leading spin-dependent gravitational tail memory at second post-Minkowskian (2 PM) order in four-dimensional spacetime, showing it behaves as $u^{-2}$ at large retarded time $u$. Using both a conjecture based on the classical limit of the quantum soft graviton theorem and a direct classical derivation from the Einstein equations, the authors confirm that the spin-dependent waveform at order $\omega \ln \omega$ in the frequency domain corresponds precisely to this $u^{-2}$ tail memory, resolving a long-standing question in classical gravitational memory effects.

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 $ω\lnω$ as a low-frequency expansion of gravitational waveform with frequency $ω$. 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.

Motivation & Objective

• To derive the spin-dependent gravitational tail memory at second post-Minkowskian (2 PM) order in four-dimensional spacetime.
• To establish the existence and form of the $u^{-2}$ tail memory effect arising from spin in gravitational waveforms.
• To validate the classical limit of the quantum soft graviton theorem up to sub-subleading order in the spin-dependent sector.
• To provide a direct classical derivation of the $\omega \ln \omega$-type waveform without relying on quantum soft theorems.

Proposed method

• Derives the gravitational waveform using an iterative solution of the Einstein equations with $G$ as a perturbative parameter.
• Applies the post-Minkowskian expansion to compute the metric perturbation $h_{\mu\nu}$ up to $O(G^2)$, including spin effects via the matter and gravitational energy-momentum tensors.
• Uses the time-Fourier transform to analyze the frequency-domain waveform $\tilde{e}_{\mu\nu}(\omega, \vec{x})$ in the low-frequency limit $\omega \to 0$, focusing on non-analytic terms like $\omega \ln \omega$.
• Performs detailed analysis of the gravitational energy-momentum tensor and its contribution to the waveform, particularly identifying terms with $\ln \omega$ and $\omega \ln \omega$ dependence.
• Compares the direct classical result with the conjecture from the classical limit of the quantum soft graviton theorem, confirming agreement at sub-subleading order.
• Applies causal and analytic techniques, including retarded propagators and Huygens' principle, to isolate the tail memory contribution from the full waveform.

Experimental results

Research questions

• RQ1 What is the form of the spin-dependent gravitational tail memory at second post-Minkowskian order in four spacetime dimensions?
• RQ2 Does the classical limit of the quantum soft graviton theorem correctly predict the spin-dependent $\omega \ln \omega$ term in the gravitational waveform?
• RQ3 Can the $u^{-2}$ tail memory effect be derived independently from the soft theorems using only classical general relativity?
• RQ4 How do spin degrees of freedom modify the structure of nonlinear gravitational memory effects in the tail regime?
• RQ5 What is the role of the gravitational energy-momentum tensor in generating non-analytic terms like $\omega \ln \omega$ in the frequency-domain waveform?

Key findings

• The leading spin-dependent gravitational tail memory appears at second post-Minkowskian (2 PM) order and scales as $u^{-2}$ at large retarded time $u$.
• The frequency-domain waveform contains a non-analytic term proportional to $\omega \ln \omega$, which corresponds to the $u^{-2}$ tail memory in the time domain.
• The direct classical derivation of the waveform from the Einstein equations reproduces exactly the same $\omega \ln \omega$ term predicted by the classical limit of the quantum soft graviton theorem.
• The spin-dependent contribution arises from the interplay between matter spin currents and the gravitational energy-momentum tensor at $O(G^2)$, with the $\omega \ln \omega$ term originating from specific logarithmic divergences in the retarded propagator integrals.
• The result confirms that spin effects generate a distinct, non-oscillatory memory effect beyond linear and nonlinear memory, contributing to the tail memory at order $u^{-2}$.
• The analysis shows that terms contributing to $\omega \ln \omega$ are robust and survive under careful regularization, while higher-order divergences vanish due to momentum conservation and spin current constraints.