Dynamical laws of superenergy in general relativity

Abstract

The Bel and Bel-Robinson tensors were introduced nearly 50 years ago in an attempt to generalize to gravitation the energy-momentum tensor of electromagnetism. This generalization was successful from the mathematical point of view because these tensors share mathematical properties which are remarkably similar to those of the energy-momentum tensor of electromagnetism. However, the physical role of these tensors in general relativity has remained obscure and no interpretation has achieved wide acceptance. In principle, they cannot represent energy and the term superenergy has been coined for the hypothetical physical magnitude lying behind them. In this work, we try to shed light on the true physical meaning of superenergy by following the same procedure which enables us to give an interpretation of the electromagnetic energy. This procedure consists in performing an orthogonal splitting of the Bel and Bel-Robinson tensors and analyzing the different parts resulting from the splitting. In the electromagnetic case such splitting gives rise to the electromagnetic energy density, the Poynting vector and the electromagnetic stress tensor, each of them having a precise physical interpretation which is deduced from the dynamical laws of electromagnetism (Poynting theorem). The full orthogonal splitting of the Bel and Bel-Robinson tensors is more complex but, as expected, similarities with electromagnetism are present. Also the covariant divergence of the Bel tensor is analogous to the covariant divergence of the electromagnetic energy-momentum tensor and the orthogonal splitting of the former is found. The ensuing equations are to the superenergy what the Poynting theorem is to electromagnetism. Some consequences of these dynamical laws of superenergy are explored, among them the possibility of defining superenergy radiative states for the gravitational field.

Piecewise silence in discrete cosmological models

“…To determine whether a space-time is radiative is not straightforward in general. For our purposes, we exploit the well-known manifestly covariant and non-perturbative electromagnetic analogy for gravity [24], and say that the flux of gravitational radiation vanishes if the super-Poynting vector vanishes [25,44,45]. We present an argument that this condition implies that the energy carried between cells by the radiation also vanishes, despite the dimensionality of the super-Poynting vector being different from that of an energy flux.…”

“…Although the relationship between the super-Poynting vector and gravitational radiation has been considered many times in the literature [25,44,45], this is to the best of our knowledge the first time that it has been directly related to the energy flux of weak-field gravitational waves. From the physical point of view, equation (66) indicates that the super-Poynting flux can be interpreted as being proportional to the energy flux density, with a proportionality factor that depends on the frequency of the wave.…”

Stability of regular energy density in Palatini $$f(R)$$ f ( R ) gravity

“…To see effects of f (R) higher curvature terms in the formulation of structure scalars, we take GR explicit expressions of X αβ and Y αβ (developed by orthogonal splitting of Riemann tensor) [41,42]:…”

“…Herrera et al [41] used an orthogonal splitting of the Riemann tensor [42] to study the dynamical evolution of spherical collapse and presented a set of four structure scalars, i.e., Y T , X T , Y T F , and X T F . Furthermore, Herrera et al [43] explored the consequences of the cosmological constant for radiating spherical relativistic collapse by evaluating the shear and expansion evolution equations.…”

Nonlinear gravito-electromagnetism

“…This tensor enters into the super-momentum conservation equation for P a [33]. Further work on the super-energy-momentum tensor, including generalizations of the concept, may be found in [34][35][36].…”