Bel-Robinson Tensor Research Articles

[formula omitted] higher-spin supercurrents

We have constructed conserved N=2 higher-spin supercurrents for an arbitrary integer spin through the unconstrained superfield formulation of 4D,N=2 massless higher-spin theories in the harmonic superspace. The obtained supercurrents are gauge-invariant and determine consistent cubic interactions in the gauge superfield × supercurrent form for massless spin s1 and two massless spin s2N=2 supermultiplets. Such interactions and N=2 supercurrents exist only for s1≥2s2. These supercurrents are the N=2 supersymmetric extension of Berends-Burgers-van Dam higher-spin currents and generalize the linearized Bel-Robinson tensor. The relevant N=2 supercurrents can be considered as the descendants of the N=2 principle supercurrent, which has a simple universal structure. An important feature of our outcomes is that, ultimately, all N=2 supercurrents are built from the N=2 superfield strengths, which are constructed from the unconstrained analytical higher-spin prepotentials.

Causality in Scalar-Einstein Waves

A wavelike scalar-Einstein solution is found and indicating vectors constructed from the Bel-Robinson tensor are used to study which objects co-move with the wave and whether gravitational energy transfer is null. It is found that this Bel-Robinson energy criteria gives null energy transfer for both the vacuum plane wave and the scalar plane wave.

Revisiting the Bel–Robinson tensor

Revisiting the Bel–Robinson tensor

On the superstring-inspired quantum correction to the Starobinsky model of inflation

Superstring/M-theory is the theory of quantum gravity that can provide the UV-completion to viable inflation models. We modify the Starobinsky inflation model by adding the Bel-Robinson tensor Tμνλρ squared term proposed as the leading quantum correction inspired by superstring theory. The (R + 1/6m 2 R 2 - β/8m 6 T 2) model under consideration has two parameters: the inflaton mass m and the string-inspired positive parameter β. We derive the equations of motion in the Friedmann-Lemaitre-Robertson-Walker universe and investigate its solutions. We find the physical bounds on the value of the parameter β by demanding the absence of ghosts and consistency of the derived inflationary observables with the measurements of the cosmic microwave background radiation.

Gravitational entropy in Szekeres class I models

Developing a self-consistent notion of gravitational entropy in the context of cosmological structure formation has been so far an elusive task. Various theoretical proposals have been presented, initially based on Penrose’s Weyl curvature hypothesis, and variations of it. A more recent proposal by Clifton, Ellis, and Tavakol (CET) considered a novel approach by defining such entropy from a Gibbs equation constructed from an effective stress–energy tensor that emerges from the ‘square root’ algebraic decomposition of the Bel–Robinson tensor, the simplest divergence-less tensor related to the Weyl tensor. Since, so far all gravitational entropy proposals have been applied to highly restrictive and symmetric spacetimes, we probe in this paper the CET proposal for a class of much less idealized spacetimes (the Szekeres class I models) capable of describing the joint evolution of arrays of arbitrary number of structures: overdensities and voids, all placed on selected spatial locations in an asymptotic ΛCDM background. By using suitable covariant variables and their fluctuations, we find the necessary and sufficient conditions for a positive CET entropy production to be a negative sign of the product of the density and Hubble expansion fluctuations. To examine the viability of this theoretical result we examine numerically the CET entropy production for two elongated over dense regions surrounding a central spheroidal void, all evolving jointly from initial linear perturbations at the last scattering era into present day Mpc-size CDM structures. We show that CET entropy production is positive for all times after last scattering at the precise spatial locations where structure growth occurs and where the exact density growing mode is dominant. The present paper provides the least idealized (and most physically robust) probe of a gravitational entropy proposal in the context of structure formation.

Asymptotic structure with a positive cosmological constant

This is the second of two papers (Fernández-Álvarez F and Senovilla J M M 2021 Class. Quantum Grav. 39 165011) that study the asymptotic structure of space–times with a non-negative cosmological constant Λ. This paper deals with the case Λ > 0. Our approach is founded on the ‘tidal energies’ built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel–Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in , where the first element of the triplet is a three-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on , and the third element is a traceless symmetric tensor field on . The full set of physically relevant properties of the space–time cannot be characterised at infinity without taking D ab into consideration, and our radiation criterion takes this fully into account. We similarly propose a no-incoming radiation criterion based also on the triplet and on radiant supermomenta deduced from the rescaled Bel–Robinson tensor too. We search for news tensors encoding the two degrees of freedom of gravitational radiation and argue that any news-like object must be associated to, and depends on, two-dimensional cross-sections of . We identify one component of news for every such cross-section and present a general strategy to find the second component, which depends on the particular physical situation. We put in connection the radiation condition and the news-like tensors with the directional structure of the gravitational field at infinity and the criterion of no-incoming radiation. We also introduce the concept of equipped by endowing the conformal boundary with a selected congruence of curves which may be determined by the algebraic structure of the asymptotic Weyl tensor. We also define a group of asymptotic symmetries preserving the new structures. We consider the limit Λ → 0, and apply all our results to selected exact solutions of Einstein field equations in order to illustrate their validity.

How appropriate are the gravitational entropy proposals for traversable wormholes?

In this paper we have examined the validity of some proposed definitions of gravitational entropy (GE) in the context of traversable wormhole solutions of the Einstein field equations. Here we have adopted two different proposals of GE and checked for their applicability in the case of these wormholes. The first one is the phenomenological approach proposed by Rudjord et al \cite{entropy1} and expanded by Romero et al in \cite{entropy2}, which is a purely geometric method of measuring gravitational entropy. The latter one is the Clifton-Ellis-Tavakol (CET) proposal \cite{CET} for the gravitational entropy which arises in relativistic thermodynamics and is based on the Bel-Robinson tensor, which represents the effective super-energy-momentum tensor of free gravitational fields. Considering some of the Lorentzian traversable wormholes along with the Brill solution for NUT wormholes and the AdS wormholes, we have evaluated the gravitational entropy for these systems. Incidentally, the application of the CET proposal can provide unique gravitational entropies for spacetimes of Petrov type D and N only, whereas the geometric method can be applied to almost every kind of spacetime, although it has no relation with thermodynamics. For any traversable wormhole to be physically realistic, it should have a viable GE. We found that the GE proposals do give us a consistent measure of GE in several of them. This means that the existence of a viable gravitational entropy strictly depends on its definition.

An investigation on gravitational entropy of cosmological models

In this paper, we investigate the entropy of the free gravitational field for a given epoch for some well-known isotropic and anisotropic cosmologies. We use the definition of gravitational entropy proposed by Clifton, Ellis and Tavakol, where the 2-index square root of the 4-index Bel–Robinson tensor is taken to be the energy– momentum tensor for the free gravity. We examine whether in the vicinity of the initial singularity, the ratio of energy density of free gravity to that of matter density goes to zero, validating Penrose conjecture on Weyl curvature. Whenever this is true, the gravitational entropy increases monotonically with time, leading to structure formation. For the models considered by us, we identify the conditions for which the Weyl curvature hypothesis is valid, and the assumptions under which it is validated, or otherwise.

Newman–Penrose scalars and black hole equations of state

In this work we explore the connections between Newman-Penrose scalars, including the Penrose-Rindler $\mathcal{K}$-curvature, with the equation of state of asymptotically Anti-de Sitter Reissner-Nordstr\"om black holes. After briefly reviewing the equation of state for these black holes from the point of view of both the Extended Phase Space and the Horizon Thermodynamics approaches, a geometric splitting is given for such an equation in terms of the non vanishing Newman-Penrose scalars which define the $\mathcal{K}$-curvature at the horizon. From this splitting, a possible thermodynamical interpretation is developed for such scalars in the context of the black hole thermodynamics approaches initially discussed. Afterwards, the square root of the Bel-Robinson tensor is employed to propose conditions at the horizons in terms of pressures or energy densities, which can be understood as alternative thermodynamical definitions of these surfaces.

Small sphere limit of the quasi-local energy with anti de-Sitter space reference

In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such spacetimes, the anti de-Sitter space is used as the reference for the quasi-local energy. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the quasi-local energy. The optimal embedding equation which identifies the critical points of the quasi-local energy is solved in order to evaluate the limit. Using the optimal embedding, we show that the limit recovers the stress-energy tensor of the matter field at $p$. For vacuum spacetimes, the quasi-local energy vanishes to a higher order. In this case, the limit of the quasi-local energy is related to the Bel-Robinson tensor at $p$.

Gravitational Energy in Newtonian Gravity: A Response to Dewar and Weatherall

The paper investigates the status of gravitational energy in Newtonian Gravity (NG), developing upon recent work by Dewar and Weatherall. The latter suggest that gravitational energy is a gauge quantity. This is potentially misleading: its gauge status crucially depends on the spacetime setting one adopts. In line with Møller-Nielsen’s plea for a motivational approach to symmetries, we supplement Dewar and Weatherall’s work by discussing gravitational energy–stress in Newtonian spacetime, Galilean spacetime, Maxwell-Huygens spacetime, and Newton–Cartan Theory (NCT). Although we ultimately concur with Dewar and Weatherall that the notion of gravitational energy is problematic in NCT, our analysis goes beyond their work. The absence of an explicit definition of gravitational energy–stress in NCT somewhat detracts from the force of Dewar and Weatherall’s argument. We fill this gap by examining the supposed gauge status of prima facie plausible candidates—NCT analogues of gravitational energy–stress pseudotensors, the Komar mass, and the Bel-Robinson tensor. Our paper further strengthens Dewar and Weatherall’s results. In addition, it sheds more light upon the subtle link between sufficiently rich inertial structure and the definability of gravitational energy in NG.

Conserved higher spin supercurrents for arbitrary spin massless supermultiplets and higher spin superfield cubic interactions

We give an explicit superspace construction of higher spin conserved supercurrents built out of 4D, mathcal{N}=1 massless supermultiplets of arbitrary spin. These supercurrents are gauge invariant and generate a large class of cubic interactions between a massless supermultiplet with superspin Y1 = s1 + 1/2 and two massless supermultiplets of arbitrary superspin Y2. These interactions are possible only for s1 ≥ 2Y2. At the equality, the supercurrent acquires its simplest form and defines the supersymmetric, higher spin extension of the linearized Bel-Robinson tensor.

Transferring energy in general relativity

A problem in general relativity is, how the gravitational field can transfer energy and momentum between different distant places. The issue is that matter stress tensor is locally conserved, with no explicit interaction with the free gravitational field, which is represented by the Weyl tensor. In this paper we explicitly construct an interaction tensor for free gravity and matter, that depicts the interplay between the energy momentum tensor of free gravity, which is taken to be the symmetric two index square root of Bel–Robinson tensor, and matter. This is examined both in the case of Coulomb-like Petrov type D spacetimes and radiation like Petrov type N spacetimes, where a unique square root exists. The first case generalises the and thought experiment regarding gravitational induction in Newtonian gravity to general relativistic scenarios, and the second gives a proposal for how gravitational radiation can transfer energy and momentum between distant objects separated by a vacuum.

Some remarks on the Bel-Robinson tensor and gravitational radiation

The asymptotic form of the Bel-Robinson tensor in the gravitational radiation-zone is obtained in terms of the mass quadrupole of the source.A comparison is made with the standard formula for the gravitational power emission. The problem of a fully covariant description of gravitational radiation in terms of this tensor is briefly discussed.

A gravitational energy–momentum and the thermodynamic description of gravity

A proposal for the gravitational energy–momentum tensor, known in the literature as the square root of Bel–Robinson tensor (SQBR), is analyzed in detail. Being constructed exclusively from the Weyl part of the Riemann tensor, such tensor encapsulates the geometric properties of free gravitational fields in terms of optical scalars of null congruences: making use of the general decomposition of any energy–momentum tensor, we explore the thermodynamic interpretation of such geometric quantities. While the matter energy–momentum is identically conserved due to Einstein’s field equations, the SQBR is not necessarily conserved and dissipative terms could arise in its vacuum continuity equation. We discuss the possible physical interpretations of such mathematical properties.

Area deficits and the Bel–Robinson tensor

The first law of causal diamonds relates the area deficit of a small ball relative to flat space to the matter energy density it contains. At second order in the Riemann normal coordinate expansion, this energy density should receive contributions from the gravitational field itself. In this work, we study the second-order area deficit of the ball in the absence of matter and analyze its relation to possible notions of gravitational energy. In the small ball limit, a reasonable expectation for any proposed gravitational energy functional is that it evaluate to the Bel–Robinson energy density W in vacuum spacetimes. A direct calculation of the area deficit reveals a result that is not simply proportional to W. We discuss how the deviation from W is related to ambiguities in defining the shape of the ball in curved space, and provide several proposals for fixing these shape ambiguities.

Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at $p$. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.

Gravitational entropy of a Schwarzschild-type black hole

In this study, Clifton, Ellis and Tavakol’s gravitational entropy proposal is used to determine the entropy of free gravitational fields due to a spherical symmetric Schwarzschild-type black hole which is considered in the framework of f(R) gravity. In order to obtain the gravitational entropy, we calculate the Weyl tensor of the black hole to determine the Bel-Robinson tensor, giving the super energy density. By using the super energy density, we obtain the gravitational energy density and the gravitational temperature to calculate the gravitational entropy of the f(R) gravity black hole. This proposal can reproduce the Bekenstein-Hawking value in general relativity limit, but cannot reproduce it in the f(R) gravity case.

Plebański–Demiański solution of general relativity and its expressions quadratic and cubic in curvature: Analogies to electromagnetism

Analogies between gravitation and electromagnetism have been known since the 1950s. Here, we examine a fairly general type D solution — the exact seven parameter solution of Plebański–Demiański (PD) — to demonstrate these analogies for a physically meaningful spacetime. The two quadratic curvature invariants B2 - E2 and E⋅B are evaluated analytically. In the asymptotically flat case, the leading terms of E and B can be interpreted as gravitoelectric mass and gravitoelectric current of the PD solution, respectively, if there are no gravitomagnetic monopoles present. Furthermore, the square of the Bel–Robinson tensor reads (B2 + E2)2 for the PD solution, reminiscent of the square of the energy density in electrodynamics. By analogy to the energy–momentum 3-form of the electromagnetic field, we provide an alternative way to derive the recently introduced Bel–Robinson 3-form, from which the Bel–Robinson tensor can be calculated. We also determine the Kummer tensor, a tensor cubic in curvature, for a general type D solution for the first time, and calculate the pieces of its irreducible decomposition. The calculations are carried out in two coordinate systems: In the original polynomial PD coordinates and in a modified Boyer–Lindquist-like version introduced by Griffiths and Podolský (GP) allowing for a more straightforward physical interpretation of the free parameters.

On the conservation of superenergy and its applications

In this work we present a geometric identity involving the Bel–Robinson tensor which is formally similar to the Sparling identity (which involves the Einstein tensor through the Einstein 3-form). In our identity the Bel–Robinson tensor enters through the Bel–Robinson 3-form which, we believe, is introduced in the literature for the first time. The meaning of this identity is that it is possible to formulate a generic conservation law for the quantity represented by the Bel–Robinson tensor (superenergy). We also show how one can use the Bel–Robinson 3-form to estimate the components of the Bel–Robinson tensor which are computed with respect to the causal elements of a frame. This estimate could be useful in a global existence proof of the solutions of a theory of gravitation in dimension four.