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Non-relativistic regime and topology: topological term in the Einstein equation

Quentin Vigneron

Abstract

We study the non-relativistic (NR) limit of relativistic spacetimes in relation with the topology of the Universe. We first show that the NR limit of the Einstein equation is only possible in Euclidean topologies, i.e. for which the covering space is $\mathbb{E}^3$. We interpret this result as an inconsistency of general relativity in non-Euclidean topologies and propose a modification of that theory which allows for the limit to be performed in any topology. For this, a second reference non-dynamical connection is introduced in addition to the physical spacetime connection. The choice of reference connection is related to the covering space of the spacetime topology. Instead of featuring only the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference Ricci tensors. This theory should be considered instead of general relativity if one wants to study a model universe with a non-Euclidean topology and admitting a non-relativistic limit.

Non-relativistic regime and topology: topological term in the Einstein equation

Abstract

We study the non-relativistic (NR) limit of relativistic spacetimes in relation with the topology of the Universe. We first show that the NR limit of the Einstein equation is only possible in Euclidean topologies, i.e. for which the covering space is . We interpret this result as an inconsistency of general relativity in non-Euclidean topologies and propose a modification of that theory which allows for the limit to be performed in any topology. For this, a second reference non-dynamical connection is introduced in addition to the physical spacetime connection. The choice of reference connection is related to the covering space of the spacetime topology. Instead of featuring only the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference Ricci tensors. This theory should be considered instead of general relativity if one wants to study a model universe with a non-Euclidean topology and admitting a non-relativistic limit.
Paper Structure (51 sections, 1 theorem, 120 equations, 6 figures)

This paper contains 51 sections, 1 theorem, 120 equations, 6 figures.

Table of Contents

  1. Introduction and summary
  2. Definitions
  3. What we mean by Euclidean and non-Euclidean topologies
  4. Notations
  5. The Non-relativistic limit of Lorentzian spacetimes
  6. What limit?
  7. Galilean structures
  8. Status of the 4-manifold within the limit
  9. The limit
  10. Local or global limit?
  11. Interpretation of $\lambda$
  12. Limit of $g$-timelike vectors
  13. Limit of the Lorentzian Riemann tensor
  14. Non-relativistic limit of the Einstein equation
  15. Limit of the energy-momentum tensor
  16. ...and 36 more sections

Key Result

Theorem 1

A solution of the Einstein equation on a 4-manifold $\mathcal{M} = \mathbb{R}\times\Sigma$ with $\Sigma$ closed, and which has a non-relativistic limit everywhere, as defined in Section sec::the_limit, requires the topology of the 3-manifold $\Sigma$ to be Euclidean. The limit is the cosmological Ne

Figures (6)

  • Figure 1: In this scheme, the second and third columns represent a Universe with, respectively, a Euclidean spatial topology and a non-Euclidean one. The second and third lines represent the relativistic and non-relativistic regimes. The former is defined from local Lorentzian invariance, and is mathematically described by Lorentzian structures solution of the Einstein equation, which is defined in any topology. The latter is defined from local Galilean invariance and is mathematically described via Galilean structures (Section \ref{['sec::Gal_struct']}) solution of the Newton-Cartan equation (i.e. Newton's theory), only in the Euclidean case. "(1) What non-relativistic equation should we consider in non-Euclidean topologies?", and "(2) Is the non-relativistic limit of the Einstein's equation possible in non-Euclidean topologies?" were the two questions that initiated the work of this paper.
  • Figure 2: The answer to the first question was found in 1976_Kunzle2022_Vigneron_b: the non-relativistic theory on non-Euclidean topologies is given by a modification of the Newton-Cartan equation, in which a spatial curvature term is added.
  • Figure 3: Two new results appear on this third scheme: i) the NR limit fundamentally corresponds to a limit of structures, from a Lorentzian structure to a Galilean structure (Sections \ref{['sec::gen_the_limit']} and \ref{['sec::the_limit']}), i.e. it is a limit between Lorentzian and Galilean invariance; ii) the NR limit of the Einstein equation is only possible in 4-manifolds with a Euclidean spatial topology (Section \ref{['sec::limit_Einstein']}).
  • Figure 4: If we require the NR limit to exist in any topologies, then the relativistic equation to consider should feature an additional term with respect to the Einstein equation. "(3) What is that term?" is the third question we are interested in in this paper. Several arguments for why this question is fundamental are drawn in Section \ref{['sec::Question']}.
  • Figure 5: By adding in the Einstein equation a non-dynamical, reference, spacetime curvature $\bar{R}_{\mu\nu}^{\mathbb R\times\tilde{\Sigma}}$ related to the covering space ${\mathbb R\times\tilde{\Sigma}}$ of the spacetime topology, we allow for the NR limit to be performed in any topologies. In the follow up paper of this study 2023_Vigneron_et_al_b, we show that this new relativistic equation implies that the expansion becomes blind to the spatial curvature (i.e. $\Omega_{\not= K} = 1, \ \forall \Omega_K$).
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1: Non-relativistic limit
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Conjecture 1
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 2 more