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Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. 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

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. 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 inline image and of an independent SO(3,1) (Lorentz) connection inline image1, or in terms of the tetrad fields only. The simplest realization, namely, a teleparallel theory constructed solely out of inline image, preserves the physical features of the theory. Given a set of tetrad fields, it is possible to construct the metric tensor inline image, the Christoffel symbols inline image and the torsion-free Levi-Civita connection inline image, to be defined below. It is also possible to construct the Weitzenböck connection inline image [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 inline image. At the point inline image its tetrad components are inline image. For the tetrad components inline image at inline image, it is easy to see that inline image, where inline image. 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 inline image, 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 inline image, where inline image is the flat space–time metric tensor, and the torsion tensor inline image. The tensor inline image is precisely the torsion of the Weitzenböck connection. Out of inline image one may construct the three Weitzenböck invariants: inline image, inline image and inline image, where inline image, 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 inline image, where inline image. 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 inline image, inline image and inline image 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 inline image, inline image and inline image 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 P4, 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 inline image 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 inline image (or a coframe one-form) and an arbitrary connection inline image. 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 inline image 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 inline image 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 inline image, 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 inline image, 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 inline image satisfies continuity (or balance) equations [24], which lead to conservation laws for inline image 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 inline image 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 inline image. 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 inline image and SO(3,1) (Lorentz) indices inline image run from 0 to 3. Time and space indices are indicated according to inline image. The tetrad fields are represented by inline image and the torsion tensor by inline image. The flat, tangent-space Minkowski space–time metric tensor raises and lowers tetrad indices, and is fixed by inline image. The frame components are given by the inverse tetrads inline image, although we may as well refer to inline image as the frame. The determinant of the tetrad field is represented by inline image.

The torsion tensor defined above is often related to the object of anholonomity inline image via inline image. 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 inline image. This torsion tensor is related to the antisymmetric part of the Weitzenböck connection inline image, which establishes the Weitzenböck space–time. The metric and torsion-free Christoffel symbols are denoted by inline image, and the associated torsion-free Levi-Civita connection inline image is defined by Eq. (7). These connections are related by Eq. (21) below.

2 The tetrad fields and reference frames

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. References

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

Given a worldline C of an observer, represented by inline image, the velocity of the observer along C is denoted by inline image. We identify the observer's velocity with the inline image component of inline image. Thus, inline image. The acceleration inline image of the observer is given by the absolute derivative of inline image along C,

  • display math(1)

where the covariant derivative is constructed out of the Christoffel symbols inline image. The last equality follows from

  • display math(2)

Thus, inline image yields the velocity and acceleration of an observer along the worldline. Therefore, a given set of tetrad fields, for which inline image describes a congruence of timelike curves, is adapted to a particular class of observers, namely, to observers characterized by the velocity field inline image, endowed with acceleration inline image. If inline image in the limit inline image, then inline image is adapted to static observers at spacelike infinity.

The geometrical characterization of tetrad fields as an observer's frame may be given by considering the acceleration of the frame along an arbitrary path inline image of the observer. The acceleration of the whole frame is determined by the absolute derivative of inline image along inline image. Thus, assuming that the observer carries an orthonormal tetrad frame inline image, the acceleration of the frame along the path is given by [35, 36]

  • display math(3)

where inline image is the antisymmetric acceleration tensor. According to Refs. [35, 36], in analogy with the Faraday tensor we can identify inline image, where a is the translational acceleration (inline image) and inline image is the frequency of rotation of the local spatial frame with respect to a non-rotating, Fermi–Walker transported frame. It follows from Eq. (3) that

  • display math(4)

The acceleration vector inline image defined by Eq. (1) may be projected on a frame in order to yield

  • display math(5)

Thus, inline image and inline image are not different translational accelerations of the frame. The expression of inline image given by Eq. (1) may be rewritten as

  • display math(6)

where inline image are the Christoffel symbols. We see that if inline image represents a geodesic trajectory, then the frame is in free fall and inline image. Therefore, we conclude that non-vanishing values of the latter quantities do represent inertial accelerations of the frame.

Since the tetrads are orthonormal vectors, we may rewrite Eq. (4) as inline image, where inline image. Now we take into account the identity

  • display math

where inline image is the metric-compatible, torsion-free Levi-Civita connection,

  • display math(7)

and express inline image as inline image. Finally, we consider the identity inline image, where inline image is the contorsion tensor defined by

  • display math(8)

and inline image (see Section 'Geometrical identities' below or Eq. (4) of Ref. [37]; the identity may be obtained by direct calculation). After simple manipulations, we finally obtain

  • display math(9)

The expression above is clearly not invariant under local SO(3,1) transformations, but is invariant under coordinate transformations. The values of inline image for given tetrad fields may be used to characterize the frame. We interpret inline image as the inertial accelerations along the trajectory inline image.

Therefore, given any set of tetrad fields for an arbitrary space–time, its geometrical interpretation may be obtained (i) either by suitably identifying the velocity inline image of the field of observers, together with the orientation in the three-dimensional space of the components inline image, or (ii) by the values of the acceleration tensor inline image, which characterize the inertial state of the frame. The condition inline image fixes only the three components inline image, inline image, inline image, because the component inline image is determined by the normalization condition inline image. In both cases, the fixation of the frame requires the fixation of six components of the tetrad fields.

Fermi–Walker transported frames define a standard of non-rotation for accelerated observers. These are frames for which the frequency of rotation inline image vanishes [34]. Suppose that a frame is given such that inline image. We may transform this frame into a Fermi–Walker transported frame by means of the following procedure. First we note that in terms of the torsion tensor the quantities inline image are written as

  • display math(10)

Under a local Lorentz transformation of the spatial components, we have

  • display math(11)
  • display math(12)

The coefficients inline image of the spatial components of the local Lorentz transformation are fixed by requiring inline image. It is possible to show that for given non-vanishing values of the quantities inline image, the condition inline image is achieved provided the coefficients inline image of the Lorentz transformation satisfy the equation [34]

  • display math(13)

Thus, given an arbitrary frame, it is possible, at least formally, to rotate the frame and obtain a Fermi–Walker transported frame. We note that the local Lorentz transformation (11) does not affect the component inline image.

3 The Lagrangian formulation of the TEGR

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. References

Teleparallel theories of gravity are constructed out of the tetrad fields inline image and the torsion tensor inline image. The class of theories whose field equations are second-order differential equations is established from a Lagrangian density constructed out of the Weitzenböck invariants inline image, inline image and inline image, where inline image. The equivalence of a particular teleparallel theory with Einstein's general relativity is verified by means of algebraic identities between the tetrad fields, the torsion tensor, the contorsion tensor and the SO(3,1) (Lorentz) connection inline image, which is the torsion-free, Levi-Civita connection. We will first present the identities, and then we discuss the field equations, the balance equations and the energy–momentum tensor of the gravitational field.

3.1 Geometrical identities

The most important identity relates the Levi-Civita connection inline image given by Eq. (7) with the contorsion tensor inline image, defined by

  • display math(14)

The identity reads

  • display math(15)

This identity may be obtained by direct calculation, or by means of the following procedure. Let us consider a four-dimensional pseudo-Riemannian manifold endowed with a set of tetrad fields inline image and an independent, arbitrary SO(3,1) connection inline image. These quantities define the torsion and curvature tensors according to

  • display math(16)
  • display math(17)

respectively (our notation is the same as in Ref. [38]). The equation that defines inline image can be solved for inline image. After a number of manipulations, where we take into account the antisymmetry inline image, it is possible to obtain the identity

  • display math(18)

where

  • display math(19)

It is possible to verify that the arbitrary connection inline image plays no role in the dynamics of tetrad fields in the TEGR, as we conclude from Eqs. (6) and (9) of Ref. [16]. Therefore, we dispense with this connection, and require it to vanish: inline image. As a consequence, inline image reduces to inline image, and Eq. (18) to Eq. (15).

The covariant derivative of the tetrad fields with respect to the Christoffel symbols and the Levi-Civita connection inline image is identically vanishing,

  • display math(20)

We lower the index a in the equation above, multiply all terms by inline image and obtain

  • display math(21)

The left-hand side is the Weitzenböck connection, which we denote as before by inline image. Taking into account Eq. (15), we obtain the identity

  • display math(22)

With the help of the identities above, we may write the scalar curvature density inline image constructed out of the Levi-Civita connection inline image, in terms of the tetrad fields and the torsion tensor inline image. Taking into account Eq. (15) in the expression of inline image, we obtain

  • display math(23)

where inline image.

Identity (23) is also obtained by means of Eq. (22). We first note that the curvature tensor constructed out of the Weitzenböck connection vanishes identically, inline image. Then we consider the standard form of the scalar curvature density in terms of the metric tensor and the Christoffel symbols, inline image, make use of Eq. (22) and eventually arrive at Eq. (23).

We note finally that since the left-hand side of Eq. (23) is invariant under local SO(3,1) transformations, the right-hand side of the equation (including the total divergence) is also invariant under the same transformations.

3.2 The field equations of the TEGR and the gravitational energy–momentum tensor

We introduce the tensor inline image defined by [16]

  • display math(24)

which yields the quadratic combination of the torsion tensor,

  • display math(25)

Thus, identity (23) may be rewritten as

  • display math(26)

Except for the total divergence, the quadratic scalar density inline image is equivalent to the scalar curvature density inline image.

Therefore, we define the Lagrangian density of the TEGR as [16, 24]

  • display math(27)

where inline image stands for the Lagrangian density of the matter fields and inline image or, in natural units, inline image (G is the gravitational constant). The absence in the Lagrangian density of the total divergence that arises in the right-hand side of Eq. (26) prevents the invariance of Eq. (27) under local, arbitrary SO(3,1) transformations. However, if the matrices of the local SO(3,1) transformations fall off sufficiently fast at spacelike infinity, then the action integral formed by the quadratic combination inline image is invariant under these special transformations [6]. We assume in this review that the Lagrangian density above is constructed for asymptotically flat space–times. The Lagrangian density for more general space–times may be constructed with suitable surface terms (just like in the ordinary metrical formulation of general relativity) that yield an invariant action integral S, whose variation inline image leads to the expected field equations.

The field equations derived from arbitrary variations of inline image with respect to inline image are given by [16, 24]

  • display math(28)

where inline image is defined by inline image. Although the Lagrangian density is not invariant under arbitrary SO(3,1) transformations, the field equations (28) are covariant under local transformations of the SO(3,1) group.

The theory defined by the Lagrangian density (27) is equivalent to Einstein's general relativity because it can be shown that the left-hand side of Eq. (28) is identically rewritten as inline image. In order to prove the identity, it is easier to start with the left-hand side of Eq. (28) and arrive at the latter expression. For this purpose, the following three identities are helpful. Let us define inline image. The identities

  • display math(29)

are useful in obtaining inline image by means of algebraic manipulations of the left-hand side of (28). We note that by means of these identities, one may always transform the standard form of Einstein's equations for a general (non-asymptotically flat) space–time into the field equations of the TEGR.

In Ref. [39] it is shown that the coupling of a Dirac spinor field with the gravitational field, in the framework of the Lagrangian density (27), is consistent. The coupling is established by considering inline image in the covariant derivative of the Dirac field. By using the resulting Dirac equation, it can be shown that the energy–momentum tensor for the Dirac field is symmetric.

The indices in the field equation (28) may be converted into space–time indices, and thus the left-hand side of the latter equation becomes proportional to inline image. Consequently, a metric tensor inline image that is a solution of Einstein's equations is also a solution of Eq. (28). For a given space–time metric, there exists an infinity of allowed frames. Therefore, all physical results derived from considerations of a space–time metric tensor, that is a solution of Einstein's equations, are valid in the present formulation of the TEGR. In particular, the coupling of the gravitational field with the electromagnetic field may be established in the standard way according to inline image, where inline image is the Faraday tensor. Thus, the electromagnetic field may couple to torsion, but in this context the concept of torsion is not the same as in the Einstein–Cartan theory (see the discussion in Section 'Final remarks'), where torsion is normally considered as an additional geometrical quantity in a metric theory.

Equation (28) may be rewritten as

  • display math(30)

where inline image and inline image is defined by

  • display math(31)

The tensor inline image is antisymmetric in the last two indices, inline image, and from this property it follows that inline image. Therefore,

  • display math(32)

In the standard metrical formulation of general relativity, there is no equation that is equivalent to (32). The equation above yields the continuity, or balance, equation

  • display math(33)

where the integration is carried out over a three-dimensional volume V, bounded by the surface S.

The tensors inline image and inline image appear on the same footing in Eqs. (32) and (33). We are led to interpret inline image as the gravitational energy–momentum tensor, and the quantity on the left-hand side of Eq. (33),

  • display math(34)

as the total energy–momentum contained within the volume V [20, 21]. In view of the field equation (30), inline image may be rewritten as

  • display math(35)

where inline image. The expression above is the definition for the gravitational energy–momentum presented in Refs. [20, 21], obtained in the framework of the vacuum field equations in Hamiltonian form. It is invariant under coordinate transformations of the three-dimensional space, under time reparametrizations and under global SO(3,1) transformations. In vacuum, Eq. (35) represents the gravitational energy–momentum vector inline image. We will reconsider the gravitational energy–momentum vector in Section 'Energy, momentum and angular momentum of the gravitational field', after the presentation of the Hamiltonian formulation. Expressions for the energy, momentum and angular momentum of the gravitational field arise in the context of the constraint equations of the Hamiltonian formulation of the theory, as we will see in Section 'Energy, momentum and angular momentum of the gravitational field'. The definition of the gravitational angular momentum, to be presented below, can only be obtained in the Hamiltonian framework.

We see that (32) is a true energy–momentum conservation equation. If we let inline image, the right-hand side of Eq. (33) goes to zero if the relevant field quantities fall off sufficiently fast at spacelike infinity. By inspecting the right-hand side of Eq. (33), we define [37]

  • display math(36)

as the gravitational energy–momentum flux and

  • display math(37)

as the energy–momentum flux of matter. Therefore, the inline image component of Eq. (33) yields

  • display math(38)

The expressions and definitions above are consequence of field equations (28) or (30) only. No consideration is made to action integrals, surface terms or boundaries.

The present formalism may be used to obtain the gravitational pressure on the external event horizon of the Kerr black hole [40], for instance. In vacuum, the conservation equation (33) is written as

  • display math(39)

Considering the field equation (30), the right-hand side of the equation above becomes

  • display math(40)

Restricting now the index a to inline image, where inline image, we find

  • display math(41)

where

  • display math(42)

The left-hand side of Eq. (41) represents the momentum of the field divided by time, and therefore it has dimension of force (inline image should not be confused with inline image given by Eq. (9)). Since on the right-hand side inline image is an element of area, we see that inline image represents the pressure along the (i) direction, over an element of area oriented along the j direction. In Cartesian coordinates the index inline image represents the directions inline image, respectively. In Ref. [40], the gravitational pressure on the external event horizon of the Kerr black hole has been evaluated in the analysis of the thermodynamic relation inline image.

3.3 f(T) theories of gravity

The teleparallel framework allows the formulation of an interesting class of alternative theories of gravity, known as f(T) theories, where “f” is a functional of inline image. One of the first attempts was the construction of a Born–Infeld-type theory, with the purpose of arriving at regular and singularity-free solutions of the field equations, just like in the Born–Infeld formulation of electrodynamics. This approach was carried out by Ferraro and Fiorini [41], who proposed the theory defined by the Lagrangian density

  • display math(43)

where λ is a Born–Infeld parameter that controls the scale at which the deformed solutions differ from the solutions of the standard theory (obtained in the limit inline image). Ferraro and Fiorini investigated black-hole solutions and the spatially flat Friedmann–Robertson–Walker cosmological model.

Perhaps the most interesting application of f(T) theories is the attempt to explain the accelerating expansion of the universe. Presently there is a variety of theoretical models that propose an explanation of the cosmic expansion, suggested by recent cosmological observations of supernovas. Modified teleparallel gravity allows an alternative understanding of this important problem (see, for instance, Refs. [42]), without resorting to the dark-energy concept, to inflationary models, to unimodular gravity or to gravity theories with a cosmological constant. One relevant feature of these models is that the field equations of the theory are always second-order differential equations, irrespective of the functional form of f(T) (this feature is not shared by the corresponding f(R) models, where R is the scalar curvature). The Friedmann equations are slightly modified, and may be solved numerically in order to yield very interesting results. The field equations also allow the investigation of the existence of relativistic stars in the framework of f(T) theories [43], and of wormhole solutions in some viable models [44].

4 The Hamiltonian formulation of the TEGR

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. 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 inline image. The idea is to write the Lagrangian density L in the form inline image, 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.

In the present construction of the Hamiltonian formulation we deal directly with the space–time components of both the tetrad fields and the metric tensor. We do not carry out a 3 + 1 decomposition of the latter field quantities, i.e., the tetrad fields and the metric tensor are not projected on three-dimensional spacelike hypersurfaces. From the Lagrangian density (27) we obtain the momentum canonically conjugated to inline image. It reads

  • display math(44)

where the dot over inline image represents the time derivative. Given that inline image, we have inline image, which is a consequence of the fact that there is no time derivative of inline image. We refer the reader to Ref. [23] for all details of this analysis.

We first obtain the primary Hamiltonian inline image. It is given by

  • display math(45)

The quantity inline image is defined by

  • display math(46)

where

  • display math

The definition of the momenta inline image leads to primary constraints inline image,

  • display math(47)

and to inline image. Secondary constraints inline image arise from the time evolution of the primary constraints inline image, i.e., by requiring that inline image vanishes weakly. The constraints inline image do not yield secondary constraints. The full Hamiltonian density is given by

  • display math(48)

where inline image and inline image are Lagrange multipliers that are precisely determined by the evolution equations. The full expression of inline image may be presented in a simplified form as [23]

  • display math(49)

where inline image is defined by

  • display math(50)

It follows from Eq. (49) that inline image. It is important to observe, however, that the constraint inline image may also be rewritten as

  • display math(51)

where inline image is obtained from Eqs. (45) and (49). This form of inline image will be crucial in the following section.

The Poisson brackets of the constraints inline image and inline image are given by [23]

  • display math(52)
  • display math(53)
  • display math(54)

All Poisson brackets of the constraints inline image with both inline image and inline image vanish strongly. In view of the constraint algebra above, we see that the constraints inline image, inline image and inline image constitute a set of first-class constraints. Thus, the Hamiltonian formulation of the TEGR is mathematically well established, and the initial-value problem is well defined.

The constraints inline image and inline image 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.

The physical degrees of freedom of the theory may be counted in the following way. The pair of dynamical field quantities inline image displays inline image degrees of freedom. The 4 + 6 first-class constraints inline image generate symmetries of the action, and thus they reduce 10 + 10 = 20 degrees of freedom. Therefore, in the phase space of the theory there are four degrees of freedom, as expected. The action of the constraints on the tetrad fields and on the metric tensor is explicitly discussed in Ref. [23].

5 Energy, momentum and angular momentum of the gravitational field

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. 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 inline image, inline image, where inline image is the embedding momenta and inline image 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 inline image, inline image. The particle is subject to the potential inline image and has constant energy E. Denoting inline image, 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]

  • display math(55)

The action is extremized by varying the configuration space path and requiring inline image. We may simplify the integrand by writing inline image, 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 inline image as the momenta conjugated to inline image, we find inline image (where inline image), which leads to the constraint

  • display math(56)

The equation of motion obtained from the action integral has to be supplemented by the constraint equation inline image, 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 inline image, where inline image is given by Eq. (51). The first term on the right-hand side of Eq. (51) is inline image. We recall that the momentum inline image is a density and reads inline image, according to Eq. (44). In the metrical formulation of general relativity there does not exist any quantity of the type inline image, 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 inline image,

  • display math(57)

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

  • display math(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 inline image are inline image. If we assume that the tetrad fields satisfy asymptotic boundary conditions,

  • display math(59)

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

  • display math(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 inline image vanish. Indeed, all components inline image vanish if they are obtained from tetrad fields that satisfy (59), when we require the vanishing of the physical parameters.

However, the tetrad fields do not always have the same asymptotic behaviour of the metric tensor. When this is the case, we may have inline image even for the flat space–time. One example is given by the following set of tetrad fields:

  • display math(61)

The tetrad fields above yield the line element for the flat space–time in spherical coordinates. From this set of tetrad fields we obtain three non-vanishing components: inline image, inline image and inline image. By transforming Eq. (61) into Cartesian coordinates, we clearly see that the latter does not display the boundary conditions given in Eq. (59) [52].

We will denote the set of flat tetrads that displays the feature above as inline image, and the momenta constructed out of inline image by inline image. The regularized form of the gravitational energy–momentum inline image is defined by [52]

  • display math(62)

This definition guarantees that the energy–momentum of the flat space–time always vanishes. The tetrad fields inline image are obtained from the physical fields inline image by just requiring the vanishing of the parameters (inline image). We remark that regularized expressions like Eq. (62) are useful in the investigation of cosmological models, when one does not dispose of asymptotic boundary conditions [53].

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 inline image 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.

It is worthwhile to recall a simple physical situation in which the frame dependence of the gravitational energy–momentum takes place. For this purpose we consider a black hole of mass m and an observer that is very distant from the black hole. The black hole will appear to this observer as a particle of mass m, with energy inline image. The parameter m is the rest mass of the black hole, i.e., the mass of the black hole in the frame where the black hole is at rest. If, however, the black hole is moving at velocity v with respect to the observer, then its total gravitational energy will be inline image, where inline image. The gravitational energy is indeed the zero component of the gravitational energy–momentum vector. This example is a consequence of the special theory of relativity, and demonstrates the frame dependence of the gravitational energy–momentum. The frame dependence is not restricted to observers at spacelike infinity. It holds for observers located everywhere in the three-dimensional space.

Finally, we mention that the evaluation of inline image in a freely falling frame in the Schwarzschild space–time leads to a vanishing gravitational energy–momentum, i.e., inline image [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 inline image in the Hamiltonian density yield the equations inline image, or

  • display math(63)

Therefore, we define the gravitational angular momentum density as

  • display math(64)

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

  • display math(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.

Let us consider a general line element for a space–time with axial symmetry,

  • display math(66)

where all metric components depend on the spherical coordinates r and θ: inline image. Here we will adopt inline image. One relevant frame is determined by the set of tetrad fields adapted to stationary observers. This frame is established by the conditions

  • display math(67)

in spherical coordinates. We also choose the inline image component to be oriented asymptotically (inline image) with the unit vector inline image along the z axis, namely,

  • display math(68)

The resulting frame in inline image coordinates reads [54]

  • display math(69)

The functions inline image and D are defined by

  • display math(70)

where inline image.

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

  • display math(71)

and

  • display math(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 inline image constant, is located at spacelike infinity. We obtain

  • display math(73)

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

  • display math(74)

then expression (73) will be finite.

In Ref. [54] a specific model for a rotating neutron star was investigated in detail. It was found that the angular momentum of the field is given in terms of the angular momentum of the source inline image according to the equation inline image. This result seems to be general, and is also verified for the Kerr space–time, in the frame of static observers, where we find inline image.

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

  • display math(75)

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

With the purpose of analysing the frame dependence of the gravitational angular momentum, let us consider the general (and simple) form of the line element that describes a rotating neutron star, for instance. This line element is given by [54]

  • display math(76)

where α, β and Ω are functions of the radial coordinate r only. Again we adopt inline image. The radius of the star is denoted by R. These quantities are defined for the interior (inline image) and for the exterior (inline image) of the star. In the exterior region we have inline image. Instead of adopting the frame determined by Eq. (67), let us consider a frame that satisfies Schwinger's time gauge condition,

  • display math(77)

together with condition (68). In inline image coordinates, the frame reads [54]

  • display math(78)

This frame is adapted to the field of observers whose velocity inline image is given by

  • display math(79)

Here inline image is the dragging velocity of inertial frames that rotate under the action of the neutron star. The expression above for inline image describes the velocity field of observers that are dragged in circular motion around the star. It turns out that the angular momentum of the gravitational field calculated out of (78) vanishes [54],

  • display math(80)

since inline image. In fact, all components of inline image vanish. This class of observers is known in the literature as zero angular momentum observers (ZAMOs). They follow trajectories with constant radial coordinate r and with angular velocity given by the dragging velocity of inertial frames.

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 inline image 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

A very interesting property of definitions (58) and (65) for the gravitational energy–momentum inline image and angular momentum inline image, respectively, is that these quantities satisfy the algebra of the Poincaré group in the phase space of the theory. The functional derivatives of inline image and inline image are

  • display math

In view of the relations above, it is not difficult to arrive at the following Poisson brackets:

  • display math(81)

It is known that quantities that satisfy the Poincaré algebra are intimately related to energy–momentum and angular momentum. Therefore, the Poincaré algebra for inline image and inline image confirms the consistency of the definitions.

6 The Kerr space–time

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. 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 inline image. In spherical coordinates, the Kerr space–time is established by the line element

  • display math(82)

with the following definitions:

  • display math(83)

We consider initially a stationary Kerr black hole with mass m and angular momentum per unit mass inline image. 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 inline image and inline image, respectively, such that inline image, where inline image is the angular velocity of the external event horizon of the black hole, inline image, and inline image is the radius of the external event horizon: inline image. In the Penrose process, the variation of the area A of the black hole satisfies inline image. In the final stage of an idealized process, the mass of the black hole becomes the irreducible mass inline image [56], defined by the relation inline image, and the Kerr black hole becomes a Schwarzschild black hole. The irreducible mass is given by inline image. 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 2m. 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 inline image. However, none of the expressions analysed in Ref. [57] yield inline image. 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 inline image. Let us first establish the frame.

The frame must be defined such that the radial coordinate r runs from inline image 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 inline image, and inline image. The values of inline image 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

  • display math(84)

where all functions are defined in Eq. (83). It is possible to show that if we restrict the radial coordinate to inline image, the inline image component of Eq. (84) becomes inline image. The quantity inline image 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 inline image, and consequently defined in the region inline image, (ii) whose inline image components in Cartesian coordinates are asymptotically oriented along the unit vectors inline image, inline image, inline image and (iii) that are asymptotically flat, are given by

  • display math(85)

where

  • display math
  • display math

These tetrad fields are the unique configuration that satisfies the above conditions, since six conditions are imposed on inline image. It satisfies Schwinger's time gauge condition inline image. Therefore, we may evaluate the gravitational energy contained within any surface S determined by the condition inline image, and in particular for inline image. 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 inline image component of Eq. (58),

  • display math(86)

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

  • display math(87)

The quantities p and k are defined by

  • display math

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

image

Figure 1. Gravitational energy contained within the external event horizon of the Kerr black hole, as function of the angular momentum per unit mass. The upper figure represents inline image, and the lower one inline image. For a static or rotating black hole, the maximum value of these quantities is 2.

Download figure to PowerPoint

In Fig. 1 we have plotted (i) inline image, where E is given by Eq. (87), and (ii) inline image, as functions of inline image. The curves are parametrized by k, which varies from 0 to 1. The upper curve represents Eq. (87) and the lower curve represents inline image. The almost coincidence between the two expressions is striking, and is one major achievement of definition (58). It shows that Eq. (87) is in very close agreement with inline image, as expected. The result also supports the idea of localization of the gravitational energy.

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

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. References

In this review we have described general relativity in terms of the tetrad fields inline image and of the torsion tensor inline image. 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.

Acknowledgement

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. References

The author is grateful to J. F. da Rocha-Neto, S. C. Ulhoa and F. F. Faria for the participation and essential contribution in the papers that allowed him to prepare this review, and also to S. C. Ulhoa for a careful reading of the manuscript.

  1. 1

    inline image are SO(3,1) or Lorentz indices, and inline image are space–time indices.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 The tetrad fields and reference frames
  5. 3 The Lagrangian formulation of the TEGR
  6. 4 The Hamiltonian formulation of the TEGR
  7. 5 Energy, momentum and angular momentum of the gravitational field
  8. 6 The Kerr space–time
  9. 7 Final remarks
  10. Acknowledgement
  11. References