The teleparallel equivalent of general relativity (1303.3897v1)

Abstract: A review of the teleparallel equivalent of general relativity is presented. It is emphasized that general relativity may be formulated in terms of the tetrad fields and of the torsion tensor, and that this geometrical formulation leads to alternative insights into the theory. The equivalence with the standard formulation in terms of the metric and curvature tensors takes place at the level of field equations. The review starts with a brief account of the history of teleparallel theories of gravity. Then the ordinary interpretation of the tetrad fields as reference frames adapted to arbitrary observers in space-time is discussed, and the tensor of inertial accelerations on frames is obtained. It is shown that the Lagrangian and Hamiltonian field equations allow to define the energy, momentum and angular momentum of the gravitational field, as surface integrals of the field quantities. In the phase space of the theory, these quantities satisfy the algebra of the Poincar\'{e} group.

• The paper demonstrates that TEGR, using tetrad fields and torsion tensors, yields field equations equivalent to Einstein's general relativity.
• It details how gravitational energy, momentum, and angular momentum are rigorously derived through surface integrals in the Hamiltonian framework.
• The work paves the way for extensions like f(T) gravity, offering novel approaches to cosmological challenges and unification theories.

The Teleparallel Equivalent of General Relativity: An Overview

The paper "The Teleparallel Equivalent of General Relativity" by J. W. Maluf offers a comprehensive review of the theory known as the Teleparallel Equivalent of General Relativity (TEGR). TEGR presents an alternative formulation of Einstein's general relativity, utilizing tetrad fields and the torsion tensor. This approach provides distinct geometric insights into gravitational theory without altering the well-established dynamics of general relativity.

Key Features of TEGR

1. Tetrad Fields and Torsion Tensor:
   • TEGR is formulated using tetrad fields $e^a\,_\mu$ and the torsion tensor $T_{a\mu\nu}$. This framework allows the construction of the Weitzenböck connection, which characterizes geometries with vanishing curvature but non-zero torsion. Unlike traditional formulations relying on metric and curvature tensors, TEGR highlights the role of torsion in describing gravitational interactions.
2. Field Equations and Equivalence with General Relativity:
   • The equivalence with general relativity occurs at the level of field equations. The Lagrangian density of TEGR leads to second-order field equations identical to Einstein's equations when expressed in terms of the metric tensor.
   • The approach embraces both Riemannian and Weitzenböck geometries, providing a broader geometric framework consistent with traditional general relativity.
3. Gravitational Energy, Momentum, and Angular Momentum:
   • Within the TEGR framework, energy-momentum and angular momentum are derived as surface integrals of the field quantities. These definitions align with physical expectations and are supported by the Poincaré algebra in the phase space.
   • The expressions for gravitational energy and angular momentum are obtained from the constraint equations within the Hamiltonian formulation of TEGR, providing practical utility in calculating physical quantities for specific field configurations.

Implications and Future Prospects

The adoption of TEGR offers several theoretical and practical implications:

• Theoretical Insight: By reframing the gravitational interaction in terms of torsion rather than curvature, TEGR provides innovative insights that might pave the way for a deeper understanding of gravity, potentially useful for unifying theories.
• Gravitational Energy Localization: TEGR addresses fundamental issues related to the localization of gravitational energy and momentum, which have been traditionally challenging within the standard formulation of general relativity.
• Extensions and Modifications: The teleparallel framework allows straightforward modifications such as f(T) gravity theories, which have gained attention for addressing cosmological challenges like dark energy without resorting to additional fields or parameters.

Conclusion

The Teleparallel Equivalent of General Relativity provides a robust alternative to the traditional formulation of gravity, emphasizing the role of torsion and offering practical definitions for conserved quantities. The compatibility with Einstein's general relativity makes it an attractive candidate for exploring gravitational phenomena while adhering to classical predictions. As research in this field continues, it may significantly contribute to the ongoing development of quantum gravity and cosmology.