The averaged tensors of the relative energy-momentum and angular momentum in general relativity and some their applications

There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}. In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged relative energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, {\it depend on some fundamental length $L>0$}. The averaged relative energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to {\it coordinate independent} analysis (local and in special cases also global) of this field. We have applied the averaged relative energy-momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general) universes. The obtained results are very interesting, e.g., the averaged relative energy density is {\it positive definite} for the all Friedman universes.Comment: 30 pages, minor changes referring to Kasner universe

Energy-momentum and angular momentum of Goedel universes

We discuss the Einstein energy-momentum complex and the Bergmann-Thomson angular momentum complex in general relativity and calculate them for space-time homogeneous Goedel universes. The calculations are performed for a dust acausal model and for a scalar-field causal model. It is shown that the Einstein pseudotensor is traceless, not symmetric, the gravitational energy is "density" is negative and the gravitational Poynting vector vanishes. Significantly, the total (gravitational and matter) energy "density" fro the acausal model is zero while for the casual model it is negative.The Bergmann-Thomson angular momentum complex does not vanish for both G\"odel models.Comment: an amended version, 24 pages, accepted to PR

Superenergy and Supermomentum of Goedel Universes

We review the canonical superenergy tensor and the canonical angular supermomentum tensors in general relativity and calculate them for space-time homogeneous G\"odel universes to show that both of these tensors do not, in general, vanish. We consider both an original dust-filled pressureless acausal G\"odel model of 1949 and a scalar-field-filled causal G\"odel model of Rebou\c cas and Tiomno. For the acausal model, the non-vanishing components of superenergy of matter are different from those of gravitation. The angular supermomentum tensors of matter and gravitation do not vanish either which simply reflects the fact that G\"odel universe rotates. However, the axial (totally antisymmetric) and vectorial parts of supermomentum tensors vanish. It is interesting that superenergetic quantities are {\it sensitive} to causality in a way that superenergy density $_g S_{00}$ of gravitation in the acausal model is {\it positive}, while superenergy density $_g S_{00}$ in the causal model is {\it negative}. That means superenergetic quantities might serve as criterion of causality in cosmology and prove useful.Comment: an amended version, REVTEX, 26 pages, no figures, to appear in Classical and Quantum Gravit

Real null coframes in general relativity and GPS type coordinates

Based on work of Derrick, Coll, and Morales, we define a `symmetric' null coframe with {\it four real null covectors}. We show that this coframe is closely related to the GPS type coordinates recently introduced by Rovelli.Comment: Latex script, 9 pages, 4 figures; references added to work of Derrick, Coll, and Morales, 1 new figur

Gravitational energy in a small region for the modified Einstein and Landau-Lifshitz pseudotensors

The purpose of the classical Einstein and Landau-Lifshitz pseudotensors is for determining the gravitational energy. Neither of them can guarantee a positive energy in holonomic frames. In the small sphere approximation, it has been required that the quasilocal expression for the gravitational energy-momentum density should be proportional to the Bel-Robinson tensor $B_{\alpha\beta\mu\nu}$. However, we propose a new tensor $V_{\alpha\beta\mu\nu}$ which is the sum of certain tensors $S_{\alpha\beta\mu\nu}$ and $K_{\alpha\beta\mu\nu}$, it has certain properties so that it gives the same gravitational "energy-momentum" content as $B_{\alpha\beta\mu\nu}$ does. Moreover, we show that a modified Einstein pseudotensor turns out to be one of the Chen-Nester quasilocal expressions, while the modified Landau-Lifshitz pseudotensor becomes the Papapetrou pseudotensor; these two modified pseudotensors have positive gravitational energy in a small region.Comment:

Graviton-Graviton Scattering, Bel-Robinson and Energy (Pseudo)-Tensors

Motivated by recent work involving the graviton-graviton tree scattering amplitude, and its twin descriptions as the square of the Bel-Robinson tensor, B_{\m\n\a\b}, and as the "current-current interaction" square of gravitational energy pseudo-tensors t_{\a\b},we find an exact tensor-square root equality B_{\mn\a\b} = \pa^2_\mn t_{\a\b}, for a combination of Einstein and Landau-Lifschitz t_\ab, in Riemann normal coordinates. In the process, we relate, on-shell, the usual superpotential basis for classifying pseudo-tensors with one spanned by polynomials in the curvature.Comment: 7 page

Ideal gas sources for the Lemaitre-Tolman-Bondi metrics

New exact solutions emerge by replacing the dust source of the Lem\^aitre-Tolman-Bondi metrics with a viscous fluid satisfying the monatomic gas equation of state. The solutions have a consistent thermodynamical interpretation. The most general transport equation of Extended Irreversible Thermodynamics is satisfied, with phenomenological coefficients bearing a close resemblance to those characterizing a non relativistic Maxwell-Bolzmann gas.Comment: 7 pages, Plain TeX with IOP macros, important corrections to previous version, 3 figures (to appear in Classical and Quantum Gravity, June 1998

New positive small vacuum region gravitational energy expressions

We construct an infinite number of new holonomic quasi-local gravitational energy-momentum density pseudotensors with good limits asymptotically and in small regions, both materially and in vacuum. For small vacuum regions they are all a positive multiple of the Bel-Robinson tensor and consequently have positive energy.Comment: 4 page

Notes on f(T)f(T) Theories

The cosmological models based on teleparallel gravity with nonzero torsion are considered. To investigate the evolution of this theory, we consider the phase-space analysis of the $f(T)$ theory. It shows when the tension scalar can be written as an inverse function of $x$ where $x=\rho_{e}/(3m_{pl}^{2}H^{2})$ and $T=g(x)$, the system is an autonomous one. Furthermore,the $\omega_{e}-\omega'_{e}$ phase analysis is given out. We perform the dynamical analysis for the models $f(T)=\beta T\ln(T/T_{0})$ and $f(T)=\alpha m_{pl}^{2}(-T/m_{pl}^{2})^{n}$ particularly. We find that the universe will settle into de-Sitter phase for both models. And we have examined the evolution behavior of the power law form in the $\omega_{ep}-\omega'_{ep}$ plane.Comment: 13 pages, 2 figure