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Einstein's action and the harmonic gauge in terms of Newtonian fields

Barak Kol, Michael Smolkin

TL;DR

This work recasts General Relativity in terms of non-relativistic gravitational (NRG) fields obtained from a temporal Kaluza-Klein reduction, deriving the full non-linear Einstein–Hilbert action and its harmonic gauge fixing in these variables. It provides a compact, time-dependent action for the NR fields, with explicit results in four dimensions and a clear pathway to read off post-Newtonian vertices for EFT-based two-body analyses. A Massive Kaluza-Klein extension is also presented, connecting to known 4+1 formulations and illustrating non-linear interactions with KK modes. The gauge-analysis shows the harmonic gauge is near-optimal for PN computations up to 2PN, offering guidance on gauge choices for efficient higher-order PN calculations and highlighting the role of bulk action symmetry in determining the best gauge.

Abstract

The "Newtonian" or non-relativistic decomposition of Einstein's gravitational field is useful in the post-Newtonian approximation. We obtain the full non-quadratic Einstein-Hilbert action in terms of these fields as well as the harmonic gauge fixing term and find fairly simple expressions. We discuss alternatives to the harmonic gauge.

Einstein's action and the harmonic gauge in terms of Newtonian fields

TL;DR

This work recasts General Relativity in terms of non-relativistic gravitational (NRG) fields obtained from a temporal Kaluza-Klein reduction, deriving the full non-linear Einstein–Hilbert action and its harmonic gauge fixing in these variables. It provides a compact, time-dependent action for the NR fields, with explicit results in four dimensions and a clear pathway to read off post-Newtonian vertices for EFT-based two-body analyses. A Massive Kaluza-Klein extension is also presented, connecting to known 4+1 formulations and illustrating non-linear interactions with KK modes. The gauge-analysis shows the harmonic gauge is near-optimal for PN computations up to 2PN, offering guidance on gauge choices for efficient higher-order PN calculations and highlighting the role of bulk action symmetry in determining the best gauge.

Abstract

The "Newtonian" or non-relativistic decomposition of Einstein's gravitational field is useful in the post-Newtonian approximation. We obtain the full non-quadratic Einstein-Hilbert action in terms of these fields as well as the harmonic gauge fixing term and find fairly simple expressions. We discuss alternatives to the harmonic gauge.

Paper Structure

This paper contains 9 sections, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. NRG fields
  3. NRG action
  4. Massive Kaluza-Klein action
  5. The gauge
  6. Harmonic gauge fixing term
  7. Comparing gauges for the PN limit
  8. Deriving the NRG action
  9. NRG action expanded

Figures (2)

  • Figure 1: Diagrammatic representation of the new static interaction obtained after gauging away $A^i\nabla_i\phi\dot\phi$ from the Einstein-Hilbert action.
  • Figure 2: New static Feynman diagrams obtained after gauging away $A^i\nabla_i\phi\dot\phi$ from the Einstein-Hilbert action. (a) and (b) replace figures 5(l) and 5(k) of Gilmore-Ross respectively.