The Hamiltonian formulation of the teleparallel equivalent of general relativity is considered. Definitions of energy, momentum and angular momentum of the gravitational field arise from the integral form of the constraint equations of the theory. In particular, the gravitational energy-momentum is given by the integral of scalar densities over a three-dimensional spacelike hypersurface. The definition for the gravitational energy is investigated in the context of the Kerr black hole. In the evaluation of the energy contained within the external event horizon of the Kerr black hole, we obtain a value strikingly close to the irreducible mass of the latter. The gravitational angular momentum is evaluated for the gravitational field of a thin, slowly rotating mass shell.

We establish the Hamiltonian formulation of the teleparallel equivalent of general relativity, without fixing the time gauge condition, by rigorously performing the Legendre transform. The time gauge condition, previously considered, restricts the teleparallel geometry to the three-dimensional spacelike hypersurface. Geometrically, the teleparallel geometry is now extended to the four-dimensional space-time.The resulting Hamiltonian formulation is structurally different from the standard ADM formulation in many aspects, the main one being that the dynamics is now governed by the Hamiltonian constraint H 0 and a set of primary constraints. The vector constraint H i is derived from the Hamiltonian constraint. The vanishing of the latter implies the vanishing of the vector constraint.

The expression of the gravitational energy-momentum defined in the context of the teleparallel equivalent of general relativity is extended to an arbitrary set of real-valued tetrad fields, by adding a suitable reference space subtraction term. The characterization of tetrad fields as reference frames is addressed in the context of the Kerr space-time. It is also pointed out that Einstein's version of the principle of equivalence does not preclude the existence of a definition for the gravitational energy-momentum density.

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.

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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.

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.