The Teleparallel Equivalent of General Relativity: History
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The teleparallel equivalent of general relativity (TEGR) is an alternative geometrical formulation of the relativistic theory of gravitation. A brief description of the  TEGR is presented. The building blocks of the theory and few main achievements are discussed.

• Teleparallelism
• TEGR
• Gravitational Energy-Momentum

The teleparallel equivalent of general relativity (TEGR) is an alternative geometrical
formulation of Einstein's general relativity[1][2]. The TEGR is formulated in terms of
the tetrad field and of the corresponding torsion tensor, which is the antisymmetric
part of the Weitzenböck connection, in contrast with Einstein's formulation of general
relativity, which is constructed out of the metric tensor and of the Riemann-Christoffel
curvature tensor. The field equations for the tetrad field in the TEGR are completely
equivalent to Einstein's equations for the metric tensor. A theory constructed out of
the tetrad field, such as the TEGR, allows the establishment of third rank tensors
(the torsion tensor, for instance), and these tensors allow, in turn, the definitions
of vector densities and total divergences. These mathematical quantities are suitable
for the construction of surface integrals, that are eventually identified with the
energy, momentum and 4-angular momentum of the gravitational field. In the metric
formulation of general relativity it is not easy to establish non-trivial third rank
tensors that yield well defined total divergences of vector fields. Thus, in the TEGR
it is possible to define the energy-momentum and 4-angular momentum of the
gravitational field. Morever, the definitions of the latter quantities satisfy the algebra of
the Poincare group in the phase space of the theory[2][1]. In particular[1], in the TEGR it
is possible to establish the definition of the centre of mass moment of the gravitational
field. The gravitational centre of mass density is a quantity that describes the regions
in space where the gravitational field is more intense, i.e., where geodesic particles
acquire stronger gravitational accelerations, compared with the geodesic motion in the
flat space-time.

In the TEGR the concept of frame, determined by the tetrad fields, is of great importance.
Observers are adapted to frames in space-time, and the frames are subject to inertial
and gravitational accelerations, in general. Inertial accelerations are those that cause
the deviation of the motion of free particles, for instance, from the geodesic behaviour.
As an example, a static frame is subject to inertial accelerations.

In the geometrical framework of teleparallelism, it is possible to establish the notion
of distant parallelism. This feature justifies the name of the geometrical formulation.
In order to understand the distant parallelism, one has first to fix a particular frame.
In a space–time endowed with a set of tetrad fields, two vectors at distant points in
a particular frame are called parallel if they have identical components with respect
to the local tetrads at the points considered. It is possible to show[2]  that the
Weitzenböck connection plays a major role in the establishment of the condition of
absolute parallelism in space-time.

All physical features and results that one obtains in the context of Einstein's general
relativity are also described in the TEGR. The latter approach to the relativistic theory
of gravitation further allows the consideration of additional concepts and definitions,
specially regarding the energy-momentum and 4-angular momentum of the gravitational field.
Among these applications, we mention the evaluation of the gravitational energy contained
within the external event horizon of a Kerr black hole (see[2]  and references therein). More
recently, the definition of the gravitational centre of mass moment has been applied to the
analysis of (non-linear) plane-fronted gravitational waves[3] [3].

### References

1. J. W. Maluf; The Teleparallel Equivalent of General Relativity and the Gravitational Centre of Mass. Universe 2016, 2, 19, 10.3390/universe2030019.
2. J. W. Maluf; The teleparallel equivalent of general relativity. Annalen der Phyzik 2013, 525, 339, 10.1002/andp.201200272.
3. J. W. Maluf, J. F. da Rocha-Neto, S. C. Ulhoa and F. L. Carneiro; The work energy-relation for particles on geodesics in the pp-wave sapce-times. JCAP 2019, 2019, JCAP03(2019)028, 10.1088/1475-7516/2019/03/028.
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