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Torsional Newton-Cartan gravity from the large $c$ expansion of General Relativity

Dieter Van den Bleeken

Abstract

We revisit the manifestly covariant large $c$ expansion of General Relativity, $c$ being the speed of light. Assuming the relativistic connection has no pole in $c^{-2}$, this expansion is known to reproduce Newton-Cartan gravity and a covariant version of Post-Newtonian corrections to it. We show that relaxing this assumption leads to the inclusion of twistless torsion in the effective non-relativistic theory. We argue that the resulting TTNC theory is an effective description of a non-relativistic regime of General Relativity that extends Newtonian physics by including strong gravitational time dilation.

Torsional Newton-Cartan gravity from the large $c$ expansion of General Relativity

Abstract

We revisit the manifestly covariant large expansion of General Relativity, being the speed of light. Assuming the relativistic connection has no pole in , this expansion is known to reproduce Newton-Cartan gravity and a covariant version of Post-Newtonian corrections to it. We show that relaxing this assumption leads to the inclusion of twistless torsion in the effective non-relativistic theory. We argue that the resulting TTNC theory is an effective description of a non-relativistic regime of General Relativity that extends Newtonian physics by including strong gravitational time dilation.

Paper Structure

This paper contains 26 sections, 70 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Expansion of the geometry
  3. Starting ansatz
  4. Diffeomorphisms
  5. Metric invertibility
  6. LO
  7. NLO
  8. NNLO
  9. Metric compatibility
  10. Expansion of the Einstein equations
  11. Setup
  12. LO: Twistless torsion
  13. NLO & NNLO: TTNC gravity
  14. LO
  15. NLO
  16. ...and 11 more sections