## 1 Introduction

The teleparallel equivalent of general relativity (TEGR) is an alternative geometrical formulation of Einstein's general relativity. It may be formulated either in terms of the tetrad fields and of an independent SO(3,1) (Lorentz) connection 1, or in terms of the tetrad fields only. The simplest realization, namely, a teleparallel theory constructed solely out of , preserves the physical features of the theory. Given a set of tetrad fields, it is possible to construct the metric tensor , the Christoffel symbols and the torsion-free Levi-Civita connection , to be defined below. It is also possible to construct the Weitzenböck connection [1]. The curvature tensor constructed out of the latter vanishes identically. In the realm of a theory constructed out of the tetrad fields only, it is possible to address geometrical issues of both the Weitzenböck and Riemannian geometries. Therefore, the tetrad theory of gravity is a geometrical framework more general than (and consistent with) the Riemannian geometry.

Teleparallelism is a geometrical setting where it is possible to establish the notion of distant parallelism. For this purpose, one has to fix a particular frame, but in the TEGR any frame is allowed in view of the field equations. In a space–time endowed with a set of tetrad fields, two vectors at distant points are called parallel [2] if they have identical components with respect to the local tetrads at the points considered. Thus, consider a vector field . At the point its tetrad components are . For the tetrad components at , it is easy to see that , where . The covariant derivative ∇ is constructed out of the Weitzenböck connection. Therefore, the vanishing of this covariant derivative defines a condition for absolute parallelism in space–time. Since , the tetrad fields constitute a set of auto-parallel fields. The covariant derivative is not covariant under local SO(3,1) (Lorentz) transformations. Geometrical quantities invariant under local Lorentz transformations can be freely rotated in every point of the space–time, and for such quantities it is not natural to establish the idea of distant parallelism. The lack of local SO(3,1) symmetry does not mean that a particular frame is distinguished. All physical frames are solutions of the field equations. The teleparallel geometry may be understood as a limiting case of the more general Riemann–Cartan geometries [3, 4], which are defined by arbitrary configurations of the curvature and torsion tensors.

The most simple geometrical quantities that are obtained from the tetrad fields are the metric tensor , where is the flat space–time metric tensor, and the torsion tensor . The tensor is precisely the torsion of the Weitzenböck connection. Out of one may construct the three Weitzenböck invariants: , and , where , and *A*, *B* and *C* are arbitrary numerical constants. Arbitrary values of the constants *A*, *B* and *C* lead to arbitrary teleparallel theories of gravity, defined by the Lagrangian density , where . In the period 1928–1931 Einstein became interested in teleparallel theories as a possible framework for unification. In 1929 Einstein noted that the field equations obtained from the theory for which , and are symmetric in the two free space–time indices, and that the resulting linearized theory describes the weak gravitational field. He allowed the three constants to acquire values slightly different from the values above, and pursued the formulation of a unified field theory of gravitation and electromagnetism. The extra six of the 16 degrees of freedom of the tetrad field would be identified with the electromagnetic fields. Lanczos noted that the invariant defined by , and is essentially equivalent to the Riemannian scalar curvature *R*, up to a total divergence. These facts are reported in the historical account by Sauer [5]. Einstein did not succeed in arriving at a faithful and consistent tensor-like description of the electromagnetic field equations in this approach. One of the difficulties of the unification program was the large freedom in the choice of the field equations. It was not possible to justify a uniquely determined set of acceptable equations, and for this reason Einstein abandoned the approach [5]. In this review we argue that the extra six degrees of freedom of the tetrad fields are taken to fix the reference frame in space–time. At the level of Hamiltonian field equations, they lead to six primary, first-class constraints, and also to the definition of the gravitational angular momentum.

Teleparallel gravity was reconsidered in 1976 by Cho [6, 7], who derived a tetrad theory of gravity as a gauge theory of the translation group, although the theory was not described in the geometrical framework of teleparallelism. Cho argued that the resulting Einstein–Cartan-type theory is the unique gauge theory of the Poincaré group *P*_{4}, if the Lagrangian density is constructed out of the lowest possible combinations of the field strengths [7]. At about the same time, teleparallel theories were investigated as gravity theories in the Weitzenböck space–time. The motivation for this renewed interest was the analysis by Hayashi [8] in 1977 on the gauge theory of the translation group in connection with the space–time torsion. Hayashi observed that a gravitational theory based on the Weitzenböck space–time may be interpreted as a gauge theory of the translation group, where the gauge field is identified as a part of the tetrad fields (Cho [7] made the same identification earlier in 1976). However, no specific theory was analysed by Hayashi. In 1979, Hayashi and Shirafuji [9] investigated in detail a general class of teleparallel theories. The theory was called “New General Relativity”, since it was a reconsideration of Einstein's previous approach. They again concluded that for a certain fixation of the constant parameters, the Lagrangian density reduces to the scalar curvature density of the Riemannian geometry. They established a one-parameter theory that deviates from the standard formulation of general relativity. In the same period, Hehl [10] and Nitsch [11] addressed a general class of gravity theories in the Riemann–Cartan geometry, the “Poincaré Gauge Theory of Gravity”, with the purpose of investigating the Yang–Mills-type structure of the field equations of gravity. These are theories with *a priori* independent connection and tetrad fields, which include teleparallel theories as particular cases, and one of these theories is equivalent to the standard general relativity.

Metric-affine theories of gravity are theories constructed out of a set of tetrad fields (or a coframe one-form) and an arbitrary connection . A metric-teleparallel theory belongs to a particular class of metric-affine theories where the Lagrangian density is given by a suitable invariant quadratic in the torsion tensor, constrained by the condition that the curvature tensor of the connection vanishes. One specific theory is equivalent to Einstein's general relativity in the sense that the field equations for the tetrad fields (or metric tensor) are precisely Einstein's equations. We will not address this formulation in the present review, because the connection introduces an additional geometrical structure. In the context of the TEGR, this connection plays no role in the dynamics of the tetrad fields, and consequently in the space–time geometry (see Sections 'Geometrical identities' and 'Final remarks').

The metric-teleparallel theory equivalent to the standard general relativity was critically analysed by Kopczyński [12], who concluded that the teleparallel field equations do not give full information about the teleparallel connection , and lead to a non-predictable behaviour of torsion. Nester [13] addressed the difficulties raised by Kopczyński, and found that they are not generic, but for certain special solutions there is a problematic gauge freedom. Nester also addressed the canonical analysis of the TEGR, with the purpose of obtaining a new proof of the positivity of the gravitational energy [14]. In the context of the teleparallel theory equivalent to the standard general relativity, Mielke [15] investigated a theory formulated in terms of Ashtekar's complex variables. In this approach, the field equations acquire a Yang–Mills-type structure with respect to a self-dual connection.

The Hamiltonian formulation of the TEGR was investigated in 1994 in Ref. [16]. In order to simplify the analysis, the canonical 3 + 1 decomposition and the constraint algebra were carried out under the imposition of Schwinger's time gauge condition [17]. The advantage of taking into account this gauge condition is that the resulting canonical structure and constraint algebra are structurally similar to the Arnowitt, Deser and Misner (ADM) Hamiltonian formulation [18] of the standard general relativity. This analysis was possible because the Lagrangian density and the field equations were written in a compact form, in terms of the tensor , which sometimes is called the superpotential, and which will be defined in Section 'The Lagrangian formulation of the TEGR'. The emergence of a scalar density as a total divergence of the trace of the torsion tensor, in the Hamiltonian constraint of the theory, motivated the interpretation of this term as the gravitational energy density. The integral of this term over the whole three-dimensional space yields the ADM energy, for suitable asymptotic boundary conditions, and a first covariant expression for the gravitational energy, in the realm of the TEGR, was presented in Ref. [19]. The torsion tensor cannot be made to vanish at a point in space–time by means of a coordinate transformation. Therefore, criticisms based on the principle of equivalence, which rest on the reduction of the metric tensor to the Minkowski metric tensor at any point in space–time by means of a coordinate transformation, do not apply to the definition of gravitational energy that arises in the TEGR. In the framework of the metrical formulation of general relativity it is not possible to construct any non-trivial scalar density that depends on the second-order derivatives of the metric tensor, that could be interpreted as the gravitational energy–momentum density. It is known that all gravitational energy–momentum pseudo-tensors depend on quantities that are badly behaved under coordinate transformations, since they depend on the coordinate system.

An expression for the gravitational energy–momentum in the TEGR as a surface integral, without the imposition of Schwinger's time gauge condition, was first presented in 1999 in Ref. [20], and investigated in detail in Ref. [21]. The full Hamiltonian formulation, together with the constraint algebra, was analysed in Ref. [22], and further refined in Ref. [23]. The gravitational energy–momentum vector satisfies continuity (or balance) equations [24], which lead to conservation laws for and to a definition of the gravitational energy–momentum tensor. These issues will presented in detail in this review.

Teleparallel gravity has been investigated by Aldrovandi and Pereira, as a gauge theory of the translation group. In similarity to Hayashi's approach [8], they identify the gravitational potential as a non-trivial part of the tetrad field, and gravity is described in the Weitzenböck space–time. Their approach is presented in Refs. [25, 26]. Teleparallel gravity has been readdressed by Obukhov and Pereira [27] in the geometrical framework of metric-affine theories, and further reconsidered by Obukhov and Rubilar [28, 29], with the purpose of investigating the transformation (covariance) properties and conserved currents in tetrad theories of gravity. They analysed the problem of consistently defining the gravitational energy–momentum and, in particular, the problem of regularization of the expression of the gravitational energy–momentum (see also Ref. [30]). This issue will also be addressed later on in the present geometrical framework.

This review aims at summarizing the work that has been developed since 1994 in the establishment of the TEGR, emphasizing the crucial role of tetrad fields as frames adapted to arbitrary observers in space–time. Accelerated frames are frames with torsion [31]. The tetrad fields describe at the same time the gravitational field and the frame. In particular, the torsion tensor plays an important role in the definition of the tensor of inertial accelerations on frames, a quantity that evidently is frame dependent. It is natural to consider the TEGR as an alternative description of the gravitational field, because the theory is constructed out of . We will argue that the frame dependence of quantities such as the gravitational energy–momentum vector is a physically consistent feature, since the concepts that are valid in the special theory of relativity are also valid in the general theory. There is no clear-cut division of the physical concepts in the special and general theories of relativity. The introduction of the gravitational field does not modify the frame dependence of the energy of a particle in special relativity (which is the zero component of a vector), and therefore the gravitational energy of a black hole, for instance, viewed as a particle at very large distances, should also be frame dependent. We will also briefly review the Hamiltonian formulation of the TEGR, which is of fundamental importance for a complete understanding of the theory.

**Notation**: space–time indices and SO(3,1) (Lorentz) indices run from 0 to 3. Time and space indices are indicated according to . The tetrad fields are represented by and the torsion tensor by . The flat, tangent-space Minkowski space–time metric tensor raises and lowers tetrad indices, and is fixed by . The frame components are given by the inverse tetrads , although we may as well refer to as the frame. The determinant of the tetrad field is represented by .

The torsion tensor defined above is often related to the object of anholonomity via . However, we assume that the space–time geometry is defined by the tetrad fields only, and in this case the only possible non-trivial definition for the torsion tensor is given by . This torsion tensor is related to the antisymmetric part of the Weitzenböck connection , which establishes the Weitzenböck space–time. The metric and torsion-free Christoffel symbols are denoted by , and the associated torsion-free Levi-Civita connection is defined by Eq. (7). These connections are related by Eq. (21) below.