Superenergy Tensors Research Articles

Cylindrical gravitational waves: C-energy, super-energy and associated dynamical effects

The energy content of cylindrical gravitational wave spacetimes is analyzed by considering two local descriptions of energy associated with the gravitational field, namely those based on the C-energy and the Bel–Robinson super-energy tensor. A Poynting–Robertson-like effect on the motion of massive test particles, beyond the geodesic approximation, is discussed, allowing them to interact with the background field through an external force which accounts for the exchange of energy and momentum between particles and waves. In addition, the relative strains exerted on a bunch of particles displaced orthogonally to the direction of propagation of the wave are examined, providing invariant information on spacetime curvature effects caused by the passage of the wave. The explicit examples of monochromatic waves with either a single or two polarization states as well as pulses of gravitational radiation are discussed.

Canonical Superenergy and Angular Supermomentum Complexes in General Relativity and Some of their Applications

Many years ago we introduced into general relativity, GR, the canonical superenergy tensors, Sik, and the canonical angular supermomentum tensors, Sikl = (–)Skil, matter and gravitation. We have obtained these tensors by special averaging of the differences of the canonical energy-momentum and canonical angular momentum. The averaging was performed in Riemann normal coordinates, RNC(P); P is beginning of these coordinates. About four years ago we have observed that these tensors can also be obtained in other, simpler way, by using the canonical superenergy and angular super momentum complexes, KSki, and, KSikl = (–) KSkil, respectively. Such complexes can be introduced into GR in a natural way starting from canonical energy-momentum and angular momentum complexes. In this paper, at first, we define the canonical superenergy and angular supermomentum complexes in GR and then, we apply them to analyze a closed system, CS, Trautman's radiative spacetimes, TRS, and Friedman universes, FU. Finally, we compare these complexes and the results obtained with their help with the canonical superenergy and angular supermomentum tensors and results obtained with them in past. In Appendix, which is integral part of the paper, we summarize our old approach to canonical superenergy and angular supermomentum tensors.

On the energy content of electromagnetic and gravitational plane waves through super-energy tensors

The energy content of (exact) electromagnetic and gravitational plane waves is studied in terms of super-energy tensors (the Bel, Bel–Robinson and the—less familiar—Chevreton tensors) and natural observers. Starting from the case of single waves, the more interesting situation of colliding waves is then discussed, where the nonlinearities of the Einstein’s theory play an important role. The causality properties of the super-momentum four vectors associated with each of these tensors are also investigated when passing from the single-wave regions to the interaction region.

New conserved currents for vacuum space-times in dimension four with a Killing vector

A new family of conserved currents for vacuum space-times with a Killing vector is presented. The currents are constructed from the superenergy tensor of the Mars-Simon tensor and using the positivity properties of the former we find that the conserved charges associated to the currents have natural positivity properties in certain cases. Given the role played by the Mars-Simon tensor in local and semi-local characterisations of the Kerr solution, the currents presented in this work are useful to construct non-negative scalar quantities characterising Kerr initial data (known in the literature as non-Kerrness) which in addition are conserved charges.

Decay of solutions to the Maxwell equation on the Schwarzschild background

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current . The tensor vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field to the Coulomb solution on the Schwarzschild exterior.

Superenergy, conformal transformations, and Friedman universes

AbstractIn this paper we present the components of the canonical superenergy tensors for a dust flat Friedman universe. In order to simplify these calculations we have used the tranformational properties of the superenergy tensors under conformal change of the spacetime metric.

Conserved matter superenergy currents for orthogonally transitive Abelian G2 isometry groups

In a previous paper we showed that the electromagnetic superenergy tensor, the Chevreton tensor, gives rise to a conserved current when there is a hypersurface orthogonal Killing vector present. In addition, the current is proportional to the Killing vector. The aim of this paper is to extend this result to the case where we have a two-parameter Abelian isometry group that acts orthogonally transitive on non-null surfaces. It is shown that for four-dimensional Einstein–Maxwell theory with a source-free electromagnetic field, the corresponding superenergy currents lie in the orbits of the group and are conserved. A similar result is also shown to hold for the trace of the Chevreton tensor and for the Bach tensor, and also in Einstein–Klein–Gordon theory for the superenergy of the scalar field. This links up well with the fact that the Bel tensor has these properties and it gives further support to the possibility of constructing conserved mixed currents between the gravitational field and the matter fields.

The Tensors of the Averaged Relative Energy–Momentum and Angular Momentum in General Relativity and Some of 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.

Symmetric hyperbolic systems for a large class of fields in arbitrary dimension

Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r−1 arbitrary timelike vectors. The importance of the so-called “superenergy” tensors, which provide the necessary symmetric positive matrices, is emphasized and made explicit. Thereby, a unified treatment of many physical systems is achieved, as well as of the sometimes called “higher order” systems. The characteristics of these symmetric hyperbolic systems are always physical, and directly related to the null directions of the superenergy tensor, which are in particular principal null directions of the tensor field solutions. Generic energy estimates and inequalities are presented too. Examples are included, in particular a mixed gravitational-scalar field system at the level of the Bianchi equations.

The universal ‘energy’ operator

The ‘positive square’ of any tensor is presented in a universal and unified manner, valid in Lorentzian manifolds of arbitrary dimension, and independently of any (anti)-symmetry properties of the tensor. For rank-m tensors, the positive square has rank 2m. Positive here means future, that is to say, satisfying the dominant property. The standard energy–momentum and super-energy tensors are recovered as appropriate parts of the general square. A richer structure of principal null directions arises.

Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank

We present a study of Rainich-like conditions for symmetric and trace-free tensors T . For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on T for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically, T must be the Bel–Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for T to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.

Super-energy and Killing-Yano tensors

Abstract In this paper we investigate a class of basic super-energy tensors, namely those constructed from Killing-Yano tensors, and give a generalization of super-energy tensors for cases when we start not with a single tensor, but with a pair of tensors.

Unique characterization of the Bel–Robinson tensor

We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel–Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.

Old and new results for superenergy tensors from dimensionally dependent tensor identities

It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel–Robinson tensor TabcxTabcy=δxy TabcdTabcd/4 and also the new conservation law for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by tensor means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and, moreover, we are able to conclude that there are no direct higher dimensional analogs. In addition we demonstrate a simple way to show the nonexistense of such identities via counter examples; in particular we show that there is no nontrivial Bel tensor analog of this simple Bel–Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalizing dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the 4-dimensional Bel–Robinson tensor does have an analog for a class of tensors in higher dimensions.

Conserved superenergy currents

We exploit once again the analogy between the energy–momentum tensor and the so-called ‘superenergy’ tensors in order to build conserved currents in the presence of Killing vectors. First of all, we derive the divergence-free property of the gravitational superenergy currents under very general circumstances, even if the superenergy tensor is not divergence-free itself. The associated conserved quantities are explicitly computed for the Reissner–Nordström and Schwarzschild solutions. The remaining cases, when the above currents are not conserved, lead to the possibility of an interchange of some superenergy quantities between the gravitational and other physical fields in such a manner that the total, mixed, current may be conserved. Actually, this possibility has been recently proved to hold for the Einstein–Klein–Gordon system of field equations. By using an adequate family of known exact solutions, we present explicit and completely non-obvious examples of such mixed conserved currents.

New electromagnetic conservation laws

The Chevreton superenergy tensor was introduced in 1964 as a counterpart, for electromagnetic fields, of the well-known Bel–Robinson tensor of the gravitational field. We here prove the unnoticed facts that, in the absence of electromagnetic currents, Chevreton's tensor (i) is completely symmetric, and (ii) has a trace-free divergence if the Einstein–Maxwell equations hold. It follows that the trace of the Chevreton tensor is a rank-2, symmetric, trace-free, conserved tensor, which is different from the energy–momentum tensor, and nonetheless can be constructed for any test Maxwell field or any Einstein–Maxwell spacetime.

CAUSAL TENSORS AND SIMPLE FORMS

A rank-r tensor on a Lorentzian manifold of dimension N is causal if the contraction with r arbitrary causal future-directed vectors is non-negative. General superenergy tensors1, such as the Bel and Bel-Robinson tensors, are examples of even ranked causal tensors1,2, and may therefore be useful when defining norms for geometric evolution equations3. We here show that any symmetric rank-2 causal tensor (energy-momentum tensors satisfying the dominant energy condition) can be written as a sum of at most N superenergy tensors of simple forms. If N=4 this can be expressed in an elegant way as the sum of four spinors squared. Since, for arbitrary N, the superenergy of any simple form is a self-map of the cone (its square is proportional to the metric) this leads to new representations and classifications of all conformal Lorentz transformations and to generalisations of the classical four dimensional Rainich-Misner-Wheeler (RMW) theory of determining the space-time physics from its geometry4. For non-simple forms more complicated equations are satisfied by the superenergy tensors, but also in this case we are able to generalise the RMW theory, and as an example the complete algebraic RMW theory in five dimensions is obtained5.

CAUSAL TENSORS AND RAINICH'S CONDITIONS

The mathematical properties of causal tensors,1 which are the tensors satisfying the dominant property (generalization of the dominant energy condition), were presented. The whole class of the so-called superenergy tensors2, including in particular the Bel-Robinson tensor, are causal tensors. Conversely, the superenergy tensors are the basic building blocks of all causal tensors1. Actually, the superenergy tensors of simple forms are intimately related to conformally Lorentz transformations as they leave the null cone invariant.1 This provides, in particular, a wide generalization of the algebraic Rainich conditions1 in arbitrary dimension and for many different fields. See 1,2 and references therein for details.

Algebraic Rainich theory and antisymmetrization in higher dimensions

The classical Rainich(–Misner–Wheeler) theory gives necessary and sufficient conditions on an energy–momentum tensor T to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations, these conditions can be expressed in terms of the Ricci tensor, thus providing conditions for a spacetime geometry to be an Einstein–Maxwell spacetime. One of the conditions is that T2 is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple p-form. Here we examine algebraic Rainich conditions for general p-forms in higher dimensions and their relations to identities by antisymmetrization. Using antisymmetrization techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse: that the identities are sufficient to determine the form. As an example we obtain the complete generalization of the classical Rainich theory to five dimensions.

Do gravitational waves carry energy-momentum and angular momentum?

In the paper we show that the real gravitational waves which have $R_{iklm}\not= 0$ always carry energy-momentum and angular momentum. Our proof uses canonical superenergy and supermomentum tensors for gravitational field.