We consider the interpretation of tetrad fields as reference frames in spacetime. Reference frames may be characterized by an antisymmetric acceleration tensor, whose components are identified as the inertial accelerations of the frame (the translational acceleration and the frequency of rotation of the frame). This tensor is closely related to gravitoelectromagnetic field quantities. We construct the set of tetrad fields adapted to observers that are in free fall in the Schwarzschild spacetime, and show that the gravitational energy-momentum constructed out of this set of tetrad fields, in the framework of the teleparallel equivalent of general relatrivity, vanishes. This result is in agreement with the principle of equivalence, and may be taken as a condition for a viable definition of gravitational energy.

We redefine the gravitational angular momentum in the framework of the teleparallel equivalent of general relativity. In similarity to the gravitational energy-momentum, the new definition for the gravitational angular momentum is coordinate independent. By considering the Poisson brackets in the phase space of the theory, we find that the gravitational energy-momentum and angular momentum correspond to a representation of the Poincaré group. This result allows us to define Casimir type invariants for the gravitational field.

In the context of the teleparallel equivalent of general relativity we establish the Hamiltonian formulation of the unimodular theory of gravity. Here we do not carry out the usual 3 + 1 decomposition of the field quantities in terms of the lapse and shift functions, as in the ADM formalism. The corresponding Lagrange multiplier is the timelike component of the tetrad field. The dynamics is determined by the Hamiltonian constraint H ′ 0 and a set of primary constraints. The constraints are first class and satisfy an algebra that is similar to the algebra of the Poincaré group.

It is shown that in the passage of a short burst of non-linear plane gravitational wave, the kinetic energy of free particles may either decrease or increase. The decreasing or increasing of the kinetic energy depends crucially on the initial conditions (position and velocity) of the free particle. Therefore a plane gravitational wave may extract energy from a physical system.

The concept of gravitational pressure is naturally defined in the context of the teleparallel equivalent of general relativity. Together with the definition of gravitational energy, we investigate the thermodynamics of rotating black holes in the teleparallel framework. We obtain the value of the gravitational pressure over the external event horizon of the Kerr black hole, and write an expression for the thermodynamic relation TdS ¼ dE þ pdV, where the variations refer to the Penrose process for the Kerr black hole. We employ only the notions of gravitational energy and pressure that arise in teleparallel gravity, and do not make any consideration of the area or the variation of the area of the event horizon. However, our results are qualitatively similar to the standard expression of the literature.

We present a method to calculate the gravitational energy when asymptotic boundary conditions for the space-time are not given. This is the situation for most of the cosmological models. The expression for the gravitational energy is obtained in the context of the teleparallel equivalent of general relativity. We apply our method first to the Schwarzschild-de Sitter solution of Einstein's equation, and then to the Robertson-Walker Universe. We show that in the first case our method leads to an average energy density of the vacuum space-time, and in latter case the energy vanishes in the case of null curvature. June 21, 2018 4:18 WSPC/INSTRUCTION FILE GEPforCM 2 Ulhoa et. alThe motivation for finding an expression for the energy of the universe comes from the fact that this subject may help to answer some questions about the origin and evolution of the universe, with consequent improvements for the cosmological models 5 .Several investigations and results indicate that the total energy of the universe is zero. However these results are dubious since most of them are obtained by means of pseudotensors, which are coordinate-dependent expressions whose geometrical meaning is not clear. The results obtained by means of Komar's integrals, on the other hand, depend on the normalization of a Killing vector and in the case of cosmological systems, which are not asymptotically flat, there is no physically preferred choice for such vectors.In the framework of the Teleparallel Equivalent of General Relativity (TEGR) the gravitational energy is well defined for finite volumes of the three dimensional space, and in particular for asymptotically flat space-times. When asymptotic boundary condition are not available, for instance in the case of the Robertson-Walker model and of the de Sitter Universe, a new method has to be used. As remarked by Faddeev 9 , an expression for the gravitational energy must vanish only in the total absence of matter and gravitational field. Therefore in order to avoid the use of boundary conditions, it is possible, and perhaps mandatory, to use regularized expressions 10 , which do not require the establishment of asymptotic conditions. The purpose of this paper is to point out that the use of a regularized expression for the gravitational energy is quite suitable to cosmological models. This paper is organized as follows. In section 2, we introduce the formalism of teleparallel gravity. We show how to define the gravitational energy in this context and present the definition of the regularized expression for the gravitational energy, which can be applied for arbitrary field configurations. In section 3, we propose a method to evaluate the energy based on the regularized expression, since in cosmology the lack of spatial asymptotic conditions prevents the application of ordinary methods. We evaluate the energy for two configurations which have relevance in cosmology, the Schwarzschild-de Sitter's solution and the Robertson-Walker model for the universe. Finally we present some concluding remarks.Not...

We show that in the framework of the teleparallel equivalent of general relativity the gravitational energy-momentum of plane-fronted gravitational waves contained in an arbitrary three-dimensional volume V may be easily obtained and is nonpositive in the frame of static observers.

Gyratonic pp-waves are exact solutions of Einstein's equations that represent non-linear gravitational waves endowed with angular momentum. We consider gyratonic pp-waves that travel in the z direction and whose time dependence on the variable u = 1 √ 2 (z − t) is given by gaussians, so that the waves represent short bursts of gravitational radiation propagating in the z direction. We evaluate numerically the geodesics and velocities of free particles in the space-time of these waves, and find that after the passage of the waves both the kinetic energy and the angular momentum per unit mass of the particles are changed. Therefore there is a transfer of energy and angular momentum between the gravitational field and the free particles, so that the final values of the energy and angular momentum of the free particles may be smaller or larger in magnitude than the initial values.