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The meaning of torsion in teleparallel theories

J. B. Formiga

Abstract

The ambiguity of the Weitzenböck connection and the meaning of torsion in teleparallel theories are investigated. A new postulate is added to teleparallel theories in order to remove the ambiguity and the inconsistencies in the calculation of the gravitational energy-momentum tensor and the like. In addition to the known restrictions on the spatial triad, it is shown that some restrictions on the congruence used to build the frame must also be imposed. Nevertheless, no restriction is imposed on a particular observer's worldline. The postulate and the restriction presented in this article are used to define what will be called here an ideal frame. This definition is applied to the Schwarzschild and the pp-wave spacetimes in the context of the Teleparallel Equivalent of General Relativity. In both cases, the results are very appealing and consistent; this includes the impossibility of making the gravitational energy density vanish along the accelerated observers' worldlines used here. Two promising interpretations for the Weitzenböck torsion are presented and discussed in detail. The possibility of having a well defined concept of an absolute vacuum in teleparallel theories is also discussed. Finally, some possible solutions to well-known problems of the f(T) theories are proposed.

The meaning of torsion in teleparallel theories

Abstract

The ambiguity of the Weitzenböck connection and the meaning of torsion in teleparallel theories are investigated. A new postulate is added to teleparallel theories in order to remove the ambiguity and the inconsistencies in the calculation of the gravitational energy-momentum tensor and the like. In addition to the known restrictions on the spatial triad, it is shown that some restrictions on the congruence used to build the frame must also be imposed. Nevertheless, no restriction is imposed on a particular observer's worldline. The postulate and the restriction presented in this article are used to define what will be called here an ideal frame. This definition is applied to the Schwarzschild and the pp-wave spacetimes in the context of the Teleparallel Equivalent of General Relativity. In both cases, the results are very appealing and consistent; this includes the impossibility of making the gravitational energy density vanish along the accelerated observers' worldlines used here. Two promising interpretations for the Weitzenböck torsion are presented and discussed in detail. The possibility of having a well defined concept of an absolute vacuum in teleparallel theories is also discussed. Finally, some possible solutions to well-known problems of the f(T) theories are proposed.

Paper Structure

This paper contains 57 sections, 4 theorems, 187 equations, 1 figure.

Table of Contents

  1. Revisiting the gravitational energy of the Schwarzschild spacetime with a new approach to the calculations
  2. Introduction
  3. Teleparallelism
  4. Hybrid machinery
  5. Mixing Cartesian unit vectors with spherical ones
  6. Changing the signature
  7. Mixing Cartesian and cylindrical unit vectors with spherical coordinates
  8. Accelerated observers in Schwarzschild spacetime
  9. Gravitational angular momentum density
  10. Gravitational energy
  11. Acceleration
  12. Energy density
  13. Conclusions
  14. On the teleparallel frame problem
  15. Introduction
  16. ...and 42 more sections

Key Result

Theorem \oldthetheorem

Let $x^\mu(\tau)$ represent the worldline of a general accelerated observer with proper time $\tau$. Assume that $\tensor{{e}}{_{(0)}^\mu}(\tau)=dx^\mu/d\tau$ is the observer's $4$-velocity and $\tensor{{e}}{_{(i)}}(\tau)$ is its spatial triad. Then, parallel transport (via Levi-Civita) its frame $\

Figures (1)

  • Figure 1: In this figure we show how to connect the observers in the region $z>0$ with those in $z<0$ so as to have $f(t',-z)=f(t',z)$ and $g(t',-z)=g(t',z)$ at a instant $t'$. While the curves on the right are $f(t,z)=|z|/\sqrt{z^2-t^2}$ and $g(t,z)=t/\sqrt{z^2-t^2}$, those on the left are given by $f(t,z)=|z|/\sqrt{z^2-(2t'-t)^2}$ and $g(t,z)=(2t'-t)/\sqrt{z^2-(2t'-t)^2}$. This ensures that all the observers are moving towards the right and in a symmetric way.

Theorems & Definitions (16)

  • Theorem \oldthetheorem
  • Definition 2.3.1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition 2.3.2
  • Definition 2.3.3
  • Definition 3.3.1
  • Definition 3.4.1
  • Definition 3.4.2
  • Definition 3.4.3
  • ...and 6 more