Comment on recent papers regarding next-to-leading order spin-spin effects in gravitational interaction

TL;DR

This paper critiques recent work on next-to-leading order spin-spin effects in gravitational interactions, clarifying the source of discrepancies and correcting a sign typo in prior calculations.

Contribution

It clarifies the correct form of the tetrad used in spin-spin calculations and identifies the true source of previous discrepancies as a sign typo.

Findings

The tetrad form used in Porto and Rothstein's paper is incorrect.

Discrepancies are due to a sign typo, not tetrad choice.

Next-to-leading order spin-squared dynamics are unaffected by the tetrad issue.

Abstract

It is argued that the tetrad in a recent paper by Porto and Rothstein on gravitational spin-spin coupling should not have the given form. The fixation of that tetrad was suggested by Steinhoff, Hergt, and Schaefer as a possible source for the disagreement found in the spin-squared dynamics. However, this inconsistency will only show up in the next-to-leading order spin-orbit dynamics and not in the spin-squared dynamics. Instead, the disagreement found at the next-to-leading order spin-squared level is due to a sign typo in the spin-squared paper by Porto and Rothstein.