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Carroll limit of four-dimensional gravity theories in the first order formalism

Amanda Guerrieri, Rodrigo F. Sobreiro

Abstract

We explore the ultra-relativistic limit of a class of four dimensional gravity theories, known as Lovelock-Cartan gravities, in the first order formalism. First, we review the well known limit of the Einstein-Hilbert action. A very useful scale symmetry involving the vierbeins and the boost connection is presented. Moreover, we explore the field equations in order to find formal solutions. Some remarkable results are obtained: Riemann and Weitzenböck like manifolds are discussed; Birkhoff's theorem is verified for the torsionless case; an explicit solution with non-trivial geometry is discussed; A quite general solution in the presence of matter is obtained. Latter, we consider the ultra-relativistic limit of the more general Lovelock-Cartan gravity. The previously scale symmetry is also discussed. The field equations are studied in vacuum and in the presence of matter. In comparison with the Einstein-Hilbert case, a few relevant results are found: Birkhoff's theorem is also verified for the torsionless case; A quite general solution in the presence of matter is obtained. This solution generalizes the previous case; Riemann and Weitzenböck like manifolds are derived in the same lines of the Einstein-Hilbert case.

Carroll limit of four-dimensional gravity theories in the first order formalism

Abstract

We explore the ultra-relativistic limit of a class of four dimensional gravity theories, known as Lovelock-Cartan gravities, in the first order formalism. First, we review the well known limit of the Einstein-Hilbert action. A very useful scale symmetry involving the vierbeins and the boost connection is presented. Moreover, we explore the field equations in order to find formal solutions. Some remarkable results are obtained: Riemann and Weitzenböck like manifolds are discussed; Birkhoff's theorem is verified for the torsionless case; an explicit solution with non-trivial geometry is discussed; A quite general solution in the presence of matter is obtained. Latter, we consider the ultra-relativistic limit of the more general Lovelock-Cartan gravity. The previously scale symmetry is also discussed. The field equations are studied in vacuum and in the presence of matter. In comparison with the Einstein-Hilbert case, a few relevant results are found: Birkhoff's theorem is also verified for the torsionless case; A quite general solution in the presence of matter is obtained. This solution generalizes the previous case; Riemann and Weitzenböck like manifolds are derived in the same lines of the Einstein-Hilbert case.

Paper Structure

This paper contains 16 sections, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lovelock-Cartan gravity
  3. Poincaré and Carroll algebras
  4. Carroll gravity
  5. Action and field equations
  6. Carroll-Riemann and Carroll- Weitzenböck manifolds
  7. Carroll-Riemann manifolds
  8. Carroll-Weitzenböck manifolds
  9. A non-trivial example
  10. Spherically symmetric solution
  11. Carroll-Cartan gravity
  12. Action and field equations
  13. Solutions in the presence of matter
  14. Almost general solution
  15. Carroll-Riemann and Carroll-Weitzenböck manifolds
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Illustration of the UR time $T(t)$ (red dashed line) and the lapse function $N(t)$ (blue solid line).