### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

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 us 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 Poincaré group.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

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.

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.

### 2 The tetrad fields and reference frames

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

A set of tetrad fields is defined by four orthonormal, linearly independent vector fields in space–time, , which establish the local reference frame of an observer that moves along a trajectory *C*, represented by the worldline [32-34] (τ is the proper time of the observer). The components and are timelike and spacelike vectors, respectively; transforms as covariant vector fields under coordinate transformations, and as contravariant vector fields under SO(3,1) (Lorentz) transformations, i.e., , where the matrices are representations of the SO(3,1) group and satisfy . The metric tensor is obtained by the relation . The tetrad fields allow the projection of vectors and tensors in space–time in the local frame of an observer. In order to measure field quantities with magnitude and direction, an observer must project these quantities on the frame carried by the observer. The projection of a vector at a position , on a particular frame, is determined by .

### 4 The Hamiltonian formulation of the TEGR

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

The Hamiltonian formulation is of fundamental importance in the analysis of the structure of any physical theory. In field theory it reveals the existence of hyperbolic differential equations (time-evolution equations), of elliptic differential equations (constraint equations), of the dynamic and non-dynamic field quantities and of the radiating degrees of freedom of the theory. A well-defined physical theory must necessarily have a well-defined and consistent Hamiltonian formulation. The relevance of the Hamiltonian formulation of general relativity is clear from the work of Arnowitt, Deser and Misner (ADM) [18]. The ADM formulation is used in approaches to the quantization of the gravitational field, as well as in the establishment of the initial-value problem for configurations like binary black holes, with the purpose of investigating the time evolution of the system. With the use of numerical analysis and computational tools, the Hamiltonian formulation allows the investigation of the strong-field, non-linear nature of the gravitational field.

The Hamiltonian formulation of the TEGR is formulated by means of the following procedure. We start with the Lagrangian density (27) and make . The idea is to write the Lagrangian density *L* in the form , where *H* is recognized as the Hamiltonian density. The procedure requires the realization of the Legendre transform. As in the ADM formulation, this is a non-trivial step. The procedure demands the ability to identify the Lagrange multipliers as non-dynamic components of the tetrad fields, and the primary constraints out of the components of the momenta. The Hamiltonian formulation of the TEGR was first addressed in Ref. [16], where Schwinger's time gauge condition was imposed on the tetrad fields in order to simplify the calculations. The resulting Hamiltonian formulation is very similar to the ADM formulation. In particular, the constraints and constraint algebra resemble the corresponding expressions of the ADM formulation.

The full Hamiltonian formulation of the TEGR was established in Ref. [22], but a refined formulation was presented in Ref. [23]. The constraint algebra presented in the latter reference is similar to the algebra of the Poincaré group. The Hamiltonian formulation of unimodular gravity in the realm of the TEGR was investigated in Ref. [23]. We will dispense with the unimodular condition on the metric tensor, and follow Ref. [23] in this short presentation of the Hamiltonian formulation of the TEGR.

We first obtain the primary Hamiltonian . It is given by

- (45)

The quantity is defined by

- (46)

where

The constraints and are labelled with SO(3,1) indices, and consequently the gravitational energy–momentum and angular momentum densities, to be discussed in the next section, are also labelled with these indices. Moreover, the algebra given by Eqs. (52)–(54) is very much similar to the algebra of the Poincaré group. These features justify the use of the SO(3,1) indices in labelling the constraints.

### 5 Energy, momentum and angular momentum of the gravitational field

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

The definitions of the energy, momentum and angular momentum of the gravitational field constitute a long-standing problem in the theory of general relativity. These definitions are necessary in order to have a comprehensive understanding of the theory. The first approach to a solution of this problem consisted in the derivation of energy–momentum pseudo-tensors. However, the solution presented by this approach is not satisfactory for at least two important reasons. The first is that pseudo-tensors are not well defined with respect to coordinate transformations. As a consequence, the results obtained via pseudo-tensors are “valid” only in one coordinate system. The second reason is that there are several pseudo-tensors available in the literature, and there is no explanation as to why one pseudo-tensor is better than another one. The principle of equivalence is sometimes invoked to justify the non-existence of a well-defined expression for the gravitational energy density. The idea is that since one can transform an arbitrary metric tensor to the Minkowski metric tensor along any timelike worldline of an observer, a well-defined expression for the gravitational energy density cannot exist, since one may “remove” the gravitational field along this worldline. The problem with this argument is that the transformation in consideration can be carried out also along any spacelike trajectory, independently of whether the metric tensor obeys any field equations. The reduction of the metric tensor to the Minkowski metric tensor along any worldline is a feature of differential geometry, and is not a manifestation of any physical principle [45]. Moreover, this criticism does not apply to the teleparallel framework, because the torsion tensor cannot be made to vanish at a point in space–time by means of a coordinate transformation.

An important step towards the concept of gravitational energy–momentum was provided by the work of Arnowitt, Deser and Misner (ADM) [18]. In this framework, the total gravitational energy–momentum is given by surface integrals, constructed out of the components of the metric tensor at spatial infinity, and is valid only for asymptotically flat space–times. The ADM energy–momentum first appeared in the construction of the Hamiltonian formulation of general relativity. It must be present in the total Hamiltonian of the theory (the integral of the Hamiltonian and vector constraints, multiplied by the lapse and shift functions, respectively), so that the total Hamiltonian has well-defined functional derivatives with respect to the phase-space variables. In this case, the total Hamiltonian generates the correct equations of motion [46]. It is important to mention that the search for the gravitational energy density within the Hamiltonian formulation of general relativity was suggested to follow from a canonical transformation of the phase-space variables of the ADM formulation to new variables that would be classified as (i) embedding variables and (ii) the true gravitational degrees of freedom [47]. After this transformation, one would expect the constraints to be written as , , where is the embedding momenta and is the gravitational energy density and energy flux carried by the true gravitational degrees of freedom [48]. However, this suggestion has never been implemented in the metrical ADM formulation.

It is clear from the analyses of the pseudo-tensors and of the total gravitational energy–momentum provided by the ADM approach that the gravitational energy density must be given by the second-order derivatives of the metric tensor. However, there does not exist a non-trivial, covariant expression constructed out of the metric tensor that yields, at the same time, a scalar density that may be interpreted as the gravitational energy density, and the total ADM energy, when integrated over the whole three-dimensional space. This is a limitation of the metrical formulation of general relativity. It turns out that such an expression exists in a theory formulated in terms of the torsion tensor.

In the TEGR, the field equations of the theory (Euler–Lagrange and first-class constraint equations) are interpreted as equations that define the energy, momentum and angular momentum of the gravitational field. We already verified that, in the context of the Euler–Lagrange field equations, we may obtain definitions (34) and (35), together with the balance equations (33), (38) and (40), which establish the conservation of the gravitational energy–momentum. In the Hamiltonian framework, a similar feature takes place. The interpretation of a constraint equation as an energy equation for a physical system is not a specific feature of the TEGR. It occurs, for instance, in the consideration of Jacobi's action [49] for a parametrized non-relativistic particle. In order to make clear this feature, let us consider a particle of mass *m* described in the configuration space by generalized coordinates , . The particle is subject to the potential and has constant energy *E*. Denoting , where *t* is a monotonically increasing parameter between the (fixed) initial and end points of the path, the Jacobi action integral for this particle can be written as [50]

- (55)

The action is extremized by varying the configuration space path and requiring . We may simplify the integrand by writing , which shows that the action is invariant under reparametrizations of the time parameter *t*. Thus, in Jacobi's formulation of the action principle, it is the energy *E* of the particle that is fixed, not its initial and final instants of time. In view of the time reparametrization of the action integral, the Hamiltonian constructed out of the Lagrangian above vanishes identically, which is a feature of reparametrization-invariant theories. If we denote as the momenta conjugated to , we find (where ), which leads to the constraint

- (56)

The equation of motion obtained from the action integral has to be supplemented by the constraint equation , in order to be equivalent with Newton's equation of motion with fixed energy *E* [50]. Therefore, we see that the constraint equation defines the energy of the particle. This is exactly the feature that takes place in the TEGR: the definitions of the energy–momentum and angular momentum of the gravitational field arise from the constraint equations of the theory [21, 51]. These definitions are viable as long as they yield consistent values in the consideration of relevant and well-understood gravitational field configurations.

#### 5.1 Gravitational energy–momentum

Let us consider the expression of the constraint equation , where is given by Eq. (51). The first term on the right-hand side of Eq. (51) is . We recall that the momentum is a density and reads , according to Eq. (44). In the metrical formulation of general relativity there does not exist any quantity of the type , i.e., a non-trivial total divergence. The emergence of a density in the form of a total divergence is the motivation to consider the integral form of the constraint equation ,

- (57)

as an equation for the gravitational energy–momentum. This is exactly the argument presented in Refs. [19, 21]. Therefore, we define

- (58)

as the total gravitational energy–momentum. This is precisely the expression (35), obtained in the realm of the Lagrangian formulation. We recall that Eq. (58) was first presented in Ref. [20]. The components of the vector are . If we assume that the tetrad fields satisfy asymptotic boundary conditions,

- (59)

at spatial infinity, i.e., in the limit , then the total gravitational energy is the ADM energy [21],

- (60)

Definition (58) has been applied to several configurations of the gravitational field, and all results are consistent. In this review we will reconsider only one major result that follows from Eq. (58). In the next section we will review the application of Eq. (58) to the Kerr space–time. It is important, however, to address the problem of regularization of definition (58).

Most sets of tetrad fields that are adapted to ordinary observers satisfy the asymptotic boundary conditions (59). It is clear that when we enforce the vanishing of the physical parameters of the metric tensor, such as mass, angular momentum and charge, the space–time in consideration is reduced to the flat space–time, and in this case one expects that the torsion tensor components vanish. Indeed, all components vanish if they are obtained from tetrad fields that satisfy (59), when we require the vanishing of the physical parameters.

The evaluation of definition (58) is carried out in the configuration space. The definition is (i) invariant under general coordinate transformations of the three-dimensional space, (ii) invariant under time reparametrizations and (iii) covariant under global SO(3,1) transformations. The non-covariance of Eq. (58) under the local SO(3,1) group reflects the frame dependence of the definition. In the TEGR each set of tetrad fields is interpreted as a reference frame in space–time. Integral quantities like cannot be covariant under local SO(3,1) transformations.

Invariance of the field quantities under local SO(3,1) (Lorentz) transformations implies that the measurement of these quantities is the same in inertial and accelerated frames. This is not an expected feature of concepts such as energy, momentum and angular momentum. The energy is always the zero component of an energy–momentum vector. It cannot be invariant under any type of SO(3,1) transformation.

Finally, we mention that the evaluation of in a freely falling frame in the Schwarzschild space–time leads to a vanishing gravitational energy–momentum, i.e., [33]. This result is in agreement with the standard description of the principle of equivalence, since the local effects of gravity are not measured by an observer in free fall. Such an observer cannot measure its own gravitational acceleration. The tetrad fields that establish the frame of an observer in free fall are related to stationary frames, for instance, by a frame transformation, not by a coordinate transformation.

#### 5.2 Gravitational angular momentum

In the TEGR the definition of the gravitational angular momentum is also obtained from the constraint equations of the theory, in similarity to the definition of the gravitational energy–momentum discussed in Section 'Gravitational energy–momentum'. The primary constraints in the Hamiltonian density yield the equations , or

- (63)

Therefore, we define the gravitational angular momentum density as

- (64)

and the total angular momentum of the gravitational field, contained within a volume *V* of the three-dimensional space, according to [51, 54]

- (65)

The expression above may be calculated from the field quantities in the configuration space of the theory. In contrast to the expression of the gravitational energy–momentum, Eq. (65) does not arise in the form of a total divergence.

In the Newtonian description of classical mechanics, the angular momentum of the source is frame dependent. This feature also holds in relativistic mechanics. If the angular momentum of the source in general is frame dependent, it is reasonable to consider that the angular momentum of the field is frame dependent as well. Differently from other definitions of gravitational angular momentum, that are formulated in terms of surface integrals at spacelike infinity and depend only on the asymptotic behaviour of the metric tensor, the definition considered here naturally depends on the frame, since it is covariant under global SO(3,1) transformations of the tetrad fields. In the present framework, observers that are in rotational motion around the rotating source measure the angular momentum of the gravitational field differently from static observers. Rotating and static observers also obtain different values for the angular momentum of the source. In Newtonian mechanics, the angular momentum of the source, in the frame of observers that co-rotate with the source, vanishes.

We use definition (64) to calculate the components of . It is possible to verify that only two components are non-vanishing. These components eventually arise as total divergences. We find [54]

- (71)

and

- (72)

In order to obtain the total angular momentum of the gravitational field, in the frame determined by Eq. (69), we evaluate the integral of Eq. (71) as a surface integral, such that the surface of integration *S*, determined by the condition constant, is located at spacelike infinity. We obtain

- (73)

We may then verify whether, for a given space–time metric tensor, the total gravitational angular momentum is finite, vanishes or diverges. If is given in spherical coordinates and if the following asymptotic behaviour is verified,

- (74)

then expression (73) will be finite.

The quantity *L*^{(0)(3)} is interpreted as the gravitational center of mass moment. It vanished for the rotating neutron star investigated in Ref. [54]. The model determined by (66) is arbitrary in the sense that the metric tensor depends arbitrarily on θ. In view of the axial symmetry of the model, it is natural that the gravitational center of mass vanishes along the *x* and *y* directions, but, because of the θ dependence of the metric tensor, the integral of (72) does not vanish in general.

In similarity to the definition of the regularized gravitational energy–momentum, we may also establish a definition for the regularized gravitational angular momentum. In view of Eq. (62), we may extend the definition of the gravitational angular momentum as

- (75)

where is defined exactly as in Section 'Gravitational energy–momentum'.

The result given by Eq. (80) shows that observers that are in rotational motion around the rotating source measure the gravitational angular momentum differently from static observers. An explanation for this result must take into account the angular momentum of the source, which is different for observers at rest and for those that rotate around the source. In the Newtonian theory, the angular momentum of the source, in the frame of observers that rotate at the same angular frequency, vanishes. We know that this feature holds for a rigid body in Newtonian mechanics, where the angular momentum depends not only on the frame, but also on the origin of the frame. Observers whose angular velocity around the rotating source is the same as the dragging velocity do not measure this dragging velocity (and possibly other dragging effects), and therefore for these observers the gravitational angular momentum vanishes.

#### 5.3 The algebra of the Poincaré group

### 6 The Kerr space–time

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

The definitions of the energy, momentum and angular momentum obtained in the TEGR may be applied to any configuration of the gravitational field, and also to cosmological models. In this review we discuss one relevant application, and consider the Kerr space–time. We will evaluate the energy contained within the external event horizon of the Kerr black hole. This is a configuration where the concept of localization of gravitational energy is very clear. No form of energy can escape from the external event horizon of the black hole, not even in the Penrose process of extraction of energy from black holes, for instance. This energy is related to the irreducible mass of the black hole. This subject has already been investigated in the context of several definitions of gravitational energy. We will assume . In spherical coordinates, the Kerr space–time is established by the line element

- (82)

with the following definitions:

- (83)

We consider initially a stationary Kerr black hole with mass *m* and angular momentum per unit mass . In the Penrose process [55] of extraction of energy of rotating black holes, the initial mass *m* and angular momentum *J* of the black hole vary by and , respectively, such that , where is the angular velocity of the external event horizon of the black hole, , and is the radius of the external event horizon: . In the Penrose process, the variation of the area *A* of the black hole satisfies . In the final stage of an idealized process, the mass of the black hole becomes the irreducible mass [56], defined by the relation , and the Kerr black hole becomes a Schwarzschild black hole. The irreducible mass is given by . An analysis of various gravitational energy expressions for the Schwarzschild and Kerr black holes has been carried out in Ref. [57]. Considering all known expressions for gravitational energy, it was concluded that the energy contained within the event horizon of the Schwarzschild black hole is 2*m*. One would expect that in the final stage of the Penrose process, the energy contained within the external event horizon of the Kerr black hole (which becomes a non-rotating black hole) would be . However, none of the expressions analysed in Ref. [57] yield . The present definition for the gravitational energy yields an expression for the energy contained within the external event horizon of the Kerr black hole that is strikingly close to . Let us first establish the frame.

The frame must be defined such that the radial coordinate *r* runs from to infinity, i.e., the frame must be defined in the whole region outside the external event horizon, and consequently inside the ergosphere of the black hole. The ergosphere is defined by the region between the external event horizon, characterized by , and . The values of determine the external boundary of the ergosphere. We know that it is not possible to establish a static frame inside the ergosphere, because in this region all observers are necessarily dragged in circular motion by the gravitational field. The four-velocity of observers that circulate around the black hole, outside the external horizon, under the action of the gravitational field of the Kerr space–time, is given by

- (84)

where all functions are defined in Eq. (83). It is possible to show that if we restrict the radial coordinate to , the component of Eq. (84) becomes . The quantity is the dragging velocity of inertial frames.

The tetrad fields (i) that are adapted to observers whose four-velocities are given by Eq. (84), i.e., for which , and consequently defined in the region , (ii) whose components in Cartesian coordinates are asymptotically oriented along the unit vectors , , and (iii) that are asymptotically flat, are given by

- (85)

where

These tetrad fields are the unique configuration that satisfies the above conditions, since six conditions are imposed on . It satisfies Schwinger's time gauge condition . Therefore, we may evaluate the gravitational energy contained within any surface *S* determined by the condition , and in particular for . Expression (85) is precisely the same set of tetrad fields (Eq. (4.9)) considered in Ref. [21]. This frame allows observers to reach the vicinity of the external event horizon of the Kerr black hole.

The energy contained within the external event horizon of the black hole is calculated by means of the component of Eq. (58),

- (86)

Here *S* is a surface of constant radius determined by the condition . After a number of algebraic calculations, we obtain [21]

- (87)

The quantities *p* and *k* are defined by

The dimensionless parameter *k* above should not be confused with in Eq. (27). Equation (87) is functionally different from . However, the two expressions are very similar, as we can verify in Fig. 1.

We conclude that the definition of gravitational energy is physically acceptable. The definition must be evaluated in the frame adapted to distinguished observers in space–time.

### 7 Final remarks

- Top of page
- Abstract
- 1 Introduction
- 2 The tetrad fields and reference frames
- 3 The Lagrangian formulation of the TEGR
- 4 The Hamiltonian formulation of the TEGR
- 5 Energy, momentum and angular momentum of the gravitational field
- 6 The Kerr space–time
- 7 Final remarks
- Acknowledgement
- References

In this review we have described general relativity in terms of the tetrad fields and of the torsion tensor . The tetrad fields constitute the frame adapted to observers in space–time. All observers are allowed, and to each one there is a frame adapted to its worldline. This alternative description does not imply an alternative dynamics for the metric tensor. The tetrad fields satisfy field equations that are strictly equivalent to Einstein's equations. In this geometrical description, the tetrad fields yield several new definitions that cannot be established in the ordinary metrical formulation. The field equations lead to an actual conservation equation, and to consistent definitions of the energy, momentum and angular momentum of the gravitational field. In the analysis of some standard configurations of the gravitational field, these definitions lead to results that are consistent with the physical configuration. The definitions are not invariant or covariant under local SO(3,1) transformations, but only covariant under global transformations. Invariance of field quantities under local SO(3,1) transformations implies that the measurement of these quantities is the same in inertial and accelerated frames. This invariance is not a natural feature of concepts such as energy, momentum and angular momentum. Energy is always the zero component of an energy–momentum vector.

Although teleparallel gravity was first addressed by Hayashi and Shirafuji [9] in a geometrical framework similar to the one adopted here, it may be considered as a limiting case of the more general framework of metric-affine theories of gravity. In this context, the gravitational field is described both by the tetrad fields and an independent affine connection, and the theory exhibits explicit invariance under local SO(3,1) transformations. However, one has to deal with Lagrange multipliers that enforce the vanishing of the curvature tensor of the connection, and one also has field equations for the zero-curvature connection [10, 11, 16, 27]. The geometrical framework is more intricate, and it is not clear that the initial-value problem is well established for all field quantities. There is an ambiguity in the determination of the Lagrange multipliers [27]. Moreover, in our opinion local Lorentz invariance is not a natural feature of distant parallelism, or teleparallelism.

The TEGR is geometrically different from Einstein–Cartan-type theories. The latter are theories with both metric and torsion as independent field quantities, and the torsion may or may not propagate in space–time. In these theories, torsion is an additional geometrical entity related to spinning matter. In the TEGR, torsion plays a relevant role in both the kinematic and dynamical descriptions of the gravitational field, as we have seen.

Since the TEGR is formulated in terms of tetrad fields, one may construct the space–time curvature tensor. Of course the curvature tensor is of utmost importance in the ordinary metrical formulation of general relativity. The curvature tensor is non-vanishing in general, but it does not play a major role in the formulation of the TEGR. We argued in the Introduction that a theory formulated in terms of the tetrad fields is geometrically more rich than the metrical formulation, because one may dispose of the concepts of both the Weitzenböck and Riemannian geometries. The torsion tensor depends on first-order derivatives of the tetrad fields, and is geometrically simpler than the curvature tensor, which depends on second-order derivatives.

We have presented the Hamiltonian formulation and the constraint algebra of the theory. All constraints are first-class constraints. Therefore, the time evolution of all field quantities is well defined. As a consequence of the Hamiltonian formulation, the initial-value problem in the realm of the TEGR is mathematically and physically consistent.

In summary, the TEGR is a simple and consistent description of the gravitational field. It embodies all physical features of the standard metrical formulation, and allows definitions for the energy, momentum and angular momentum of the gravitational field that satisfy the algebra of the Poincaré group in the phase space of the theory.