Narrow Results

In this paper, we find the energy–momentum distribution of stationary axisymmetric spacetimes in the context of teleparallel theory by using Möller prescription. The metric under consideration is the generalization of the Weyl metrics called the Lewis–Papapetrou metric. The class of stationary axisymmetric solutions of the Einstein field equations has been studied by Galtsov to include the gravitational effect of an external source. Such spacetimes are also astrophysically important as they describe the exterior of a body in equilibrium. The energy density turns out to be nonvanishing and well-defined and the momentum becomes constant except along θ-direction. It is interesting to mention that the results reduce to the already available results for the Weyl metrics when we take ω = 0.

We investigate the conservation law of energy–momentum for Randall–Sundrum models by the general displacement transform. The energy–momentum current has a superpotential and are therefore identically conserved. It is shown that for Randall–Sundrum solution, the momentum vanishes and most of the bulk energy is localized near the Planck brane. The energy density is ε = ε₀ e^-3k|y|.

This paper is devoted to discuss the energy–momentum for static axially symmetric spacetimes in the framework of teleparallel theory of gravity. For this purpose, we use the teleparallel versions of Einstein, Landau–Lifshitz, Bergmann and Möller prescriptions. A comparison of the results shows that the energy density is different but the momentum turns out to be constant in each prescription. This is exactly similar to the results available in literature using the framework of general relativity. It is mentioned here that Möller energy–momentum distribution is independent of the coupling constant λ. Finally, we calculate energy–momentum distribution for the Curzon metric, a special case of the above-mentioned spacetime.

The conserved quantities, that are, gravitational energy–momentum and its relevant quantities are investigated for cylindrical and spherical gravitational waves in the framework of teleparallel equivalent of General Relativity using the Hamiltonian approach. For both cylindrical and spherical gravitational waves, we obtain definite energy and constant momentum. The constant momentum shows consistency with the results available in General Relativity and teleparallel gravity. The angular momentum for cylindrical and spherical gravitational waves also turn out to be constant. Further, we evaluate their gravitational energy–momentum fluxes and gravitational pressure.

In this paper, we discuss the energy–momentum problem in the realm of teleparallel gravity. The energy–momentum distribution for a class of regular black holes coupled with a nonlinear electrodynamics source is investigated by using Hamiltonian approach of teleparallel theory. The generalized regular black hole contains two specific parameters α and β (a sort of dipole and quadrupole of nonlinear source) on which the energy distribution depends. It is interesting to mention here that our results exactly coincide with different energy–momentum prescriptions in general relativity.

An energy–momentum carried by electromagnetic field produced by two point-like charged particles is calculated. Integration region considered in the evaluation of the bound and emitted quantities produced by all points of world-lines up to the end points at which particles' trajectories puncture an observation hyperplane y⁰=t. Radiative part of the energy–momentum contains, apart from usual integrals of Larmor terms, also the sum of work done by Lorentz forces of point-like charges acting on one another. Therefore, the combination of wave motions (retarded Liénard–Wiechert solutions) leads to the interaction between the sources.

The energy distribution in the Locally Rotationally Symmetric (LRS) Bianchi type II space–time is obtained by considering the Møller energy–momentum definition in both Einstein's theory of general relativity and teleparallel theory of relativity. The energy distribution which includes both the matter and gravitational field is found to be zero in both of these different gravitation theories. This result agrees with previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee and Sen, Vargas, and Aydogdu and Salti. Our result — the total energy of the universe is zero — supports the view points of Albrow and Tryon.

We calculate the total energy of an exact spherically symmetric solutions, i.e. Schwarzschild and Reissner–Nordström, using the gravitational energy–momentum 3-form within the tetrad formulation of general relativity. We explain how the effect of the inertial makes the total energy unphysical. Therefore, we use the covariant teleparallel approach which makes the energy always physical one. We also show that the inertial has no effect on the calculation of momentum.

Unifying the massive spin-1 field with gravity requires the implementation of a regular vector field that satisfies the spin-1 Proca equation and is a fundamental part of the space–time metric. That vector field is one of the pair of vectors in the line element field $(X,-X)$, which is paramount to the existence of all Lorentzian metrics and Modified General Relativity (MGR). Symmetrization of the spin-1 Klein–Gordon equation in a curved Lorentzian space–time introduces the Lie derivative of the metric along the flow of one of the regular vectors in the line element field. The Proca equation in curved space–time can then be described geometrically in terms of the line element vector, the Lie derivative of the Lorentzian metric and the Ricci tensor, which unifies gravity and the spin-1 field. Related issues concerning charge conservation and the Lorenz constraint, singularities in a spherically symmetric curved space–time and geometrical implications of MGR to quantum theory are discussed. A geometrical unification of gravity with quantum field theory is presented.

This work is devoted to present and analyze an expression for the gravitational energy–momentum vector in the context of f(T) theories through field equations. Such theories are the analogous counterpart of the well-known f(R) theories, except using torsion instead of curvature. We obtain a general expression for the gravitational energy–momentum vector in this framework. Using the hypothesis of the isotropy of spacetime, we find the gravitational energy for a closed universe, since construction of real tetrads that do not constrain the functional form of the Lagrangian density was not possible for an open universe. Thus, we find a vanishing gravitational energy for the tetrad that we have used.

We investigate in this paper the Landau–Lifshitz energy distribution in the framework of $f(T)$ theory view as a modified version of Teleparallel theory. From some important Teleparallel theory results on the localization of energy, our investigations generalize the Landau–Lifshitz prescription from the computation of the energy–momentum complex to the framework of $f(T)$ gravity as it is done in the modified versions of General Relativity. We compute the energy density in the first step for three plane-symmetric metrics in vacuum. We find for the second metric that the energy density vanishes independently of $f(T)$ models. We find that the Teleparallel Landau–Lifshitz energy–momentum complex formulations for these metrics are different from those obtained in General Relativity for the same metrics. Second, the calculations are performed for the cosmic string spacetime metric. It results that the energy distribution depends on the mass $M$ and the radius $r$ of cosmic string and it is strongly affected by the parameter of the considered quadratic and cubic $f(T)$ models. Our investigation with this metric induces interesting results susceptible to be tested with some astrophysics hypothesis.

Modified General Relativity (MGR) is the natural extension of General Relativity (GR). MGR explicitly uses the smooth regular line element vector field $(X,-X)$, which exists in all Lorentzian spacetimes, to construct a connection-independent symmetric tensor that represents the energy–momentum of the gravitational field. It solves the problem of the nonlocalization of gravitational energy–momentum in GR, preserves the ontology of the Einstein equation, and maintains the equivalence principle. The line element field provides MGR with the extra freedom required to describe dark energy and dark matter. An extended Schwarzschild solution for the matter-free Einstein equation of MGR is developed, from which the Tully–Fisher relation is derived, and the gravitational energy density is calculated. The mass of the invisible matter halo of galaxy NGC 3198 calculated with MGR is identical to the result obtained from GR using a dark matter profile. Although dark matter in MGR is described geometrically, it has an equivalent representation as a particle with the property of a vector boson or a pair of fermions; the geometry of spacetime and the quantum nature of matter are linked together by the unit line element covectors that belong to both the Lorentzian metric and the spin-1 Klein–Gordon wave equation. The three classic tests of GR provide a comparison of the theories in the solar system and several parts of the cosmos. MGR provides the flexibility to describe inflation after the Big Bang and galactic anisotropies.

In this study, we develop the generalized Dirac-like four-momentum equation for rotating spin-1/2 particles in four-dimensional quaternionic algebra. The generalized quaternionic Dirac equation consists of the rotational energy and angular momentum of particle and antiparticle. Accordingly, we also discuss the four-vector form of quaternionic relativistic mass, moment of inertia and rotational energy-momentum in Euclidean space-time. The quaternionic four-angular momentum, (i.e. the rotational analogy of four-linear momentum) predicts the dual energy (rest mass energy and pure rotational energy) and dual momentum (linear-like momentum and pure rotational momentum). Further, the solutions of quaternionic rotational Dirac energy-momentum are obtained by using one-, two- and four-component of quaternionic spinor. We also demonstrate the solutions of quaternionic plane wave equation which gives the rotational frequency and wave propagation vector of Dirac particles and antiparticles in terms of quaternionic form.

Starting with the quaternionic Minkowski space–time and its four-vector representation, a rotational analogue of the quaternionic Dirac equation in the electromagnetic field is developed, which includes not only the energy solutions but also the angular momentum solutions for rotating Dirac fermions. The striking feature of the quaternionic approach is that it depicts a unified representation of energy–momentum in a single framework as the space–time. Furthermore, we establish the generalized Schrodinger–Pauli-like equation associated with generalized electric and magnetic dipole energies that corresponds to their dipole moments. As such, the present analysis deals with the invariant behavior of the extended quaternionic rotational Dirac equation under Lorentz–Poincaré, Gauge, duality, and CPT invariance.

The Lorentz-group of transformations usually consists of linear transformations of the coordinates, keeping as invariant the norm of the four-vector in (Minkowski) space-time. Besides those linear transformations, one may construct different forms of nonlinear transformations of the coordinates keeping unchanged that respective invariant. In this paper we explore nonlinear transformations of second-order which have a natural interpretation within the framework of Yamaleev's concept of the counterpart of rapidity (co-rapidity). The purpose of developed concept is to show that the formulae for energy and momentum of the relativistic particle become regular near the zero-mass and speed of light states. Furthermore, in a covariant formulation, the co-rapidity is presented as a four-vector which admits an extension of the Lorentz-group of transformations. In this paper we additionally show, that in the same way as the rapidity is related to the electromagnetic field, the co-rapidity is related to the field of strengths, which are given by a four-vector. The corresponding equations of such a field are also constructed.

In the context of the Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR), we compute the angular-momentum of an exact rotating solution.