Superenergy Tensors Research Articles

Do gravitational waves carry energy‐momentum and angular momentum?

AbstractWe show that gravitational waves which possess a non‐vanishing Riemann tensor Riklm ≠ 0 always carry energy‐momentum and angular momentum. Our proof uses canonical superenergy and supermomentum tensors for the gravitational field.

Positivity and conservation of superenergy tensors

Two essential properties of energy–momentum tensors Tμν are their positivity and conservation. This is mathematically formalized by, respectively, an energy condition, as the dominant energy condition, and the vanishing of their divergence ∇μTμν = 0. The classical Bel and Bel–Robinson superenergy tensors, generated from the Riemann and Weyl tensors, respectively, are rank-4 tensors. But they share these two properties with energy–momentum tensors: the dominant property (DP) and the divergence-free property in the absence of sources (vacuum). Senovilla [2, 3] defined a universal algebraic construction which generates a basic superenergy tensor T{A} from any arbitrary tensor A. In this construction, the seed tensor A is structured as an r-fold multivector, which can always be done. The most important feature of the basic superenergy tensors is that they satisfy automatically the DP, independently of the generating tensor A. In [8], we presented a more compact definition of T{A} using the r-direct Clifford algebra ⊗r \U0001d49eℓp,q. This form for the superenergy tensors allowed us to obtain an easy proof of the DP valid for any dimension. In this paper we include this proof. We explain which new elements appear when we consider the tensor T{A} generated by a non-degree-defined r-fold multivector A and how orthogonal Lorentz transformations and bilinear observables of spinor fields are included as particular cases of superenergy tensors. We find some sufficient conditions for the seed tensor A, which guarantee that the generated tensor T{A} is divergence-free. These sufficient conditions are satisfied by some physical fields, which are presented as examples.

Superenergy and supermomentum of Gödel universes

We review the canonical superenergy tensor and the canonical angular supermomentum tensor in general relativity and calculate them for spacetime homogeneous Gödel universes to show that both of these tensors do not, in general, vanish. We consider both an original dust-filled pressureless acausal Gödel model of 1949 and a scalar-field-filled causal Gödel model of Rebouças and Tiomno. For the acausal model, the nonvanishing 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 the Gödel universe rotates. However, the axial (totally antisymmetric) and vectorial parts of supermomentum tensors vanish. It is interesting that superenergetic quantities are sensitive to causality in a way that superenergy density gS00 of gravitation in the acausal model is positive, while superenergy density gS00 in the causal model is negative. That means superenergetic quantities might serve as criteria of causality in cosmology and prove useful.

Null cone preserving maps, causal tensors and algebraic Rainich theory

A rank-n tensor on a Lorentzian manifold whose contraction with n arbitrary causal future-directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general mathematical properties in arbitrary dimension N. Then, we prove that rank-2 tensors which map the null cone on itself are causal tensors. Previously it has been shown that, to any tensor field A on a Lorentzian manifold there is a corresponding ‘superenergy’ tensor field T{A} (defined as a quadratic sum over all Hodge duals of A) which always has the dominant property. Here we prove that, conversely, any symmetric rank-2 tensor with the dominant property can be written in a canonical way as a sum of N superenergy tensors of simple forms. We show that the square of any rank-2 superenergy tensor is proportional to the metric in dimension N ≤ 4, and that the square of the superenergy tensor of any simple form is proportional to the metric in arbitrary dimension. Conversely, we prove in arbitrary dimension that any symmetric rank-2 tensor T whose square is proportional to the metric must be a causal tensor and, up to sign, the superenergy of a simple p-form, and that the trace of T determines the rank p of the form. This generalizes, with respect to both the dimension N and the rank p, the classical algebraic Rainich conditions, which are necessary and sufficient conditions for a metric to originate algebraically in some physical field. Furthermore, it has the important geometric interpretation that the set of superenergy tensors of simple forms is precisely the set of tensors which leave the null cone invariant and preserve its time orientation. It also means that all involutory Lorentz transformations can be represented as superenergy tensors of simple forms, and that any rank-2 superenergy tensor is the sum of at most N conformally involutory Lorentz transformations. Non-symmetric null cone preserving maps are shown to have a symmetric part with the dominant property and are classified according to the null eigenvectors of the skew-symmetric part. We therefore obtain a complete classification of all conformal Lorentz transformations and singular null cone preserving maps on any Lorentzian manifold of any dimension.

Super-energy tensors

A simple and purely algebraic construction of super-energy(s-e) tensors for arbitrary fields is presented in any dimensions.These tensors have good mathematical and physical properties,andthey can be used in any theory having as basic arena an n-dimensionalmanifoldwith a metric of Lorentzian signature.In general, the completely timelike component of these s-e tensors has themathematical features of an energy density: they are positivedefinite and satisfy the dominant property, which provides s-eestimates useful for global results and helpful in other matters, such asthecausal propagation of the fields. Similarly, super-momentumvectors appear with the mathematical properties of s-e flux vectors.The classical Bel and Bel-Robinson tensors for the gravitational fields areincluded in our general definition. The energy-momentum and super-energytensorsof physical fields are also obtained, and the procedure will be illustratedbywriting down these tensors explicitly for the cases of scalar,electromagnetic,and Proca fields. Moreover, higher order (super)k-energytensors are defined and shown to be meaningful and in agreement for thedifferent physical fields. In flat spacetimes, they provide infinitely manyconserved quantities.In non-flat spacetimes, the fundamental question of the interchangeofs-e quantities between different fields is addressed, and answeredaffirmatively. Conserved s-e currents are found for any minimallycoupled scalar field whenever there is a Killing vector. Furthermore, theexchange of gravitational and electromagnetic super-energy is also shown bystudying the propagation of discontinuities. This seems to open the doorfornew types of conservation physical laws.

Review of some classical gravitational superenergy tensors usingcomputational techniques

We use computational algorithms recently developed by ourselves to study completelyfour index divergence-free quadratic in Riemann tensor polynomials in GR. Someresults are new and others reproduce and/or correct known ones.The algorithms are part of a Mathematica package calledTools of Tensor Calculus (TTC).

(SUPER)n-ENERGY FOR ARBITRARY FIELDS AND ITS INTERCHANGE: CONSERVED QUANTITIES

Inspired by classical work of Bel and Robinson, a natural purely algebraic construction of super-energy (s-e) tensors for arbitrary fields is presented, having good mathematical and physical properties. Remarkably, there appear quantities with mathematical characteristics of energy densities satisfying the dominant property, which provides s-e estimates useful for global results and helpful in other matters. For physical fields, higher order (super)n-energy tensors involving the field and its derivatives arise. In special relativity, they provide infinitely many conserved quantities. The interchange of s-e between different fields is shown. The discontinuity propagation law in Einstein–Maxwell fields is related to s-e tensors, providing quantities conserved along null hypersurfaces. Finally, conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector.

Positivity of General Superenergy Tensors

Several of the most important results in general relativity require or assume positivity properties of certain tensors. The positive energy theorem and the singularity theorems make assumptions about the energy-momentum tensor and Ricci tensor respectively. Positivity of the Bel–Robinson tensor is needed in the proof of the global stability of Minkowski spacetime. Senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor. For a Maxwell field or a scalar field the procedure yields the usual energy-momentum tensor, for the Weyl tensor and the Riemann tensor one obtains the Bel–Robinson tensor and Bel tensor respectively. In general, by considering any tensor as an r-fold n 1,…,n r )-form, one constructs a rank 2r superenergy tensor from it. By using spinor methods, we prove that the contraction of any such superenergy tensor with 2r future-pointing vectors is non-negative. We refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them. Some more examples are given and applications discussed.

Superenergy and angular supermomentum tensors in general relativity

We give a short information about the canonical angular supermomentum tensors of matter and gravitation, introduced recently by the author. These tensors can be considered as some substitutes for the nonexistent angular momentum tensors in general relativity. For completeness we recall some facts about canonical superenergy tensors introduced earlier by the author.

Super-energy tensor for space–times with vanishing scalar curvature

A four-index tensor is constructed with terms both quadratic in the Riemann tensor and linear in its second derivatives, which has zero divergence for space–times with vanishing scalar curvature. This tensor reduces in vacuum to the Bel–Robinson tensor. Furthermore, the completely timelike component referred to any observer is positive, and zero if and only if the space–time is flat (excluding some unphysical space–times). We also show that this tensor is the unique one that can be constructed with these properties. Such a tensor does not exist for general gravitational fields. Finally, we study this tensor in several examples: the Friedmann–Lemaı̂tre–Robertson–Walker space–times filled with radiation, the plane–fronted gravitational waves, and the Vaidya radiating metric.

Canonical superenergetic quantities for Friedman universes

Improved canonical superenergy tensors and integral superenergetic quantities which were introduced be the author some years ago are applied for Friedman universes. The analysis shows that the canonical superenergetic quantities are positive-definite in the considered case and have much better transformational and physical properties than the canonical energetic quantities.

Gravitational superenergy tensor

We provide a physical basis for the local gravitational superenergy tensor. Furthermore, our gravitoelectromagnetic deduction of the Bel-Robinson superenergy tensor permits the identification of the gravitational stress-energy tensor. This local gravitational analog of the Maxwell stress-energy tensor is illustrated for a plane gravitational wave.

Some Properties of the Bel and Bel-Robinson Tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the “pure matter” super-energy tensor and the “matter-gravity coupling” super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the “square” of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its “square” gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.

Canonical superenergy tensors in general relativity

We present the concept of superenergy tensors in the framework of general relativity (GR). These tensors were introduced constructively by the author years ago and they were obtained by a suitable averaging of the energy-momentum tensors or pseudotensors. Because in GR the Einstein canonical energy-momentum pseudotensorE tik of the gravitational field and the canonical energy-momentum complex\(_E K_i^k = \sqrt {|g|} (T_i^k + {}_Et_i^k )\), matter and gravitation,are physically distinguished, we confine this paper to thecanonical superenergy tensorgSik of the gravitational field Γkli and to the canonical total superenergy tensorSik=gSik+mSik of matter and gravitation only. These superenergy tensors can be obtained by the above-mentioned averaging of the pseudotensorEtik and complexEKik. We give the analytic forms of these two canonical superenergy tensors and show some of their possible applications in GR. The canonical superenergy tensorgSik of the gravitational field Γkli can be used as asubstitute for the nonexisting energy-momentum tensor of this field.

A formulation of linearised gravity

The formulation of linearised gravity in terms of the ‘electric’ and ‘magnetic’ gravitational fields is extended to take into account the presence of matter. The modes of radiation, the equations of motion and the potential in the static case are given. The relevant components of the superenergy tensor are calculated and a quantity named the superforce is introduced.

Gravitational Superenergy as a Generator of Canonical Transformation

The canonical transformation generated in linearized gravitation theory by the Robinson-Bel super-energy tensor is determined. The light this result sheds on the quantization program for the general theory of relativity is discussed.

Energy, superenergy and conservation laws in general relativity

After discussing the observability of the gravitational energy, we propose a gravitational-radiation Poynting pseudovector, constructed with metric coefficients and their first partial derivatives. By means of this Poynting pseudovector a definition is given for the difference of gravitational energy of a physical system between two different instants. Such a definition (of course relative to a frame of reference) has satisfactory invariance properties. Two applications are dealt with: the determination of the gravitational energy in the Schwarzschild universe and the determination of gravitational energy between two instants in a geodesic frame of reference. In the latter case it follows a proportionality between the difference of energy and the Bel superenergy tensor. The physical meaning of the superenergy is given.