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 ${\cal T}_{abcx}{\cal T}^{abcy} = \delta_x^y {\cal T}_{abcd}{\cal T}^{abcd}/4$ and also the new conservation laws for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by {\it tensor} means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and also we are able to conclude that there are no {\it direct} higher dimensional analogues. In addition we demonstrate a simple way to show non-existense of such identities via counter examples; in particular we show that there is no non-trivial Bel tensor analogue of this simple Bel-Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalising dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the four dimensional Bel-Robinson tensor does have an analogue for a class of tensors in higher dimensions.Comment: 18 pages; TeX fil

Algebraic Rainich conditions for the tensor V

Algebraic conditions on the Ricci tensor in the Rainich-Misner-Wheeler unified field theory are known as the Rainich conditions. Penrose and more recently Bergqvist and Lankinen made an analogy from the Ricci tensor to the Bel-Robinson tensor $B_{\alpha\beta\mu\nu}$, a certain fourth rank tensor quadratic in the Weyl curvature, which also satisfies algebraic Rainich-like conditions. However, we found that not only does the tensor $B_{\alpha\beta\mu\nu}$ fulfill these conditions, but so also does our recently proposed tensor $V_{\alpha\beta\mu\nu}$, which has many of the desirable properties of $B_{\alpha\beta\mu\nu}$. For the quasilocal small sphere limit restriction, we found that there are only two fourth rank tensors $B_{\alpha\beta\mu\nu}$ and $V_{\alpha\beta\mu\nu}$ which form a basis for good energy expressions. Both of them have the completely trace free and causal properties, these two form necessary and sufficient conditions. Surprisingly either completely traceless or causal is enough to fulfill the algebraic Rainich conditions. Furthermore, relaxing the quasilocal restriction and considering the general fourth rank tensor, we found two remarkable results: (i) without any symmetry requirement, the algebraic Rainich conditions only require totally trace free; (ii) with a symmetry requirement, we recovered the same result as in the quasilocal small sphere limit.Comment: 17 page

Algebraic Rainich theory and antisymmetrisation 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 on a spacetime geometry for it to be an Einstein-Maxwell spacetime. One of the conditions is that $T^2$ 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 antisymmetrisation. Using antisymmetrisation 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 generalisation of the classical Rainich theory to five dimensions.Comment: 16 pages, LaTe

Two dimensional Sen connections and quasi-local energy-momentum

The recently constructed two dimensional Sen connection is applied in the problem of quasi-local energy-momentum in general relativity. First it is shown that, because of one of the two 2 dimensional Sen--Witten identities, Penrose's quasi-local charge integral can be expressed as a Nester--Witten integral.Then, to find the appropriate spinor propagation laws to the Nester--Witten integral, all the possible first order linear differential operators that can be constructed only from the irreducible chiral parts of the Sen operator alone are determined and examined. It is only the holomorphy or anti-holomorphy operator that can define acceptable propagation laws. The 2 dimensional Sen connection thus naturally defines a quasi-local energy-momentum, which is precisely that of Dougan and Mason. Then provided the dominant energy condition holds and the 2-sphere S is convex we show that the next statements are equivalent: i. the quasi-local mass (energy-momentum) associated with S is zero; ii.the Cauchy development $D(\Sigma)$ is a pp-wave geometry with pure radiation ($D(\Sigma)$ is flat), where $\Sigma$ is a spacelike hypersurface whose boundary is S; iii. there exist a Sen--constant spinor field (two spinor fields) on S. Thus the pp-wave Cauchy developments can be characterized by the geometry of a two rather than a three dimensional submanifold.Comment: 20 pages, Plain Tex, I

A Note on Matter Superenergy Tensors

We consider Bel-Robinson-like higher derivative conserved two-index tensors H_\mn in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine H_\mn so as to overcome non-commutativity of covariant derivatives, and maintain conservation, but they become model- and dimension- dependent, and generally lose their simple "BR" form.Comment: 3 page

Magnon softening in a ferromagnetic monolayer: a first-principles spin dynamics study

We study the Fe/W(110) monolayer system through a combination of first principles calculations and atomistic spin dynamics simulations. We focus on the dispersion of the spin waves parallel to the [001] direction. Our results compare favorably with the experimental data of Prokop et al. [Phys. Rev. Lett. 102, 177206], and correctly capture a drastic softening of the magnon spectrum, with respect to bulk bcc Fe. The suggested shortcoming of the itinerant electron model, in particular that given by density functional theory, is refuted. We also demonstrate that finite temperature effects are significant, and that atomistic spin dynamics simulations represent a powerful tool with which to include these.Comment: v1: 11 pages, 3 figures. v2: double column, 5 pages, 3 figures, typos corrected, references adde

Two dimensional Sen connections in general relativity

The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric $\gamma_{AB}$ on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface $\$$. The curvature of the two dimensional Sen operator $\Delta_e$ is the pull back to $\$$ of the anti-self-dual part of the spacetime curvature while its `torsion' is a boost gauge invariant expression of the extrinsic curvatures of $\$$. The difference of the 2 dimensional Sen and the induced spin connections is the anti-self-dual part of the `torsion'. The irreducible parts of $\Delta_e$ are shown to be the familiar 2-surface twistor and the Weyl--Sen--Witten operators. Two Sen--Witten type identities are derived, the first is an identity between the 2 dimensional twistor and the Weyl--Sen--Witten operators and the integrand of Penrose's charge integral, while the second contains the `torsion' as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod's formula for the kinematical twistor.Comment: 14 pages, Plain Tex, no report numbe

Semimicroscopical description of the simplest photonuclear reactions accompanied by excitation of the giant dipole resonance in medium-heavy mass nuclei

A semimicroscopical approach is applied to describe photoabsorption and partial photonucleon reactions accompanied by the excitation of the giant dipole resonance (GDR). The approach is based on the continuum-RPA (CRPA) with a phenomenological description for the spreading effect. The phenomenological isoscalar part of the nuclear mean field, momentum-independent Landau-Migdal particle-hole interaction, and separable momentum-dependent forces are used as input quantities for the CRPA calculations. The experimental photoabsorption and partial $(n,\gamma)$-reaction cross sections in the vicinity of the GDR are satisfactorily described for $^{89}$Y, $^{140}$Ce and $^{208}$Pb target nuclei. The total direct-neutron-decay branching ratio for the GDR in $^{48}$Ca and $^{208}$Pb is also evaluated.Comment: 19 pages, 5 eps figure

The Chevreton Tensor and Einstein-Maxwell Spacetimes Conformal to Einstein Spaces

In this paper we characterize the source-free Einstein-Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure-radiation type and that it restricts the spacetimes to Petrov types \textbf{N} or \textbf{O}. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein-Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic field and the Weyl tensor must be aligned, and in the case that the electromagnetic field is null, the spacetime must be conformally Ricci-flat and all such solutions are known. In the non-null case, since the general solution is not known on closed form, we settle with giving the integrability conditions in the general case, but we do give new explicit examples of Einstein-Maxwell spacetimes that are conformal to Einstein spaces, and we also find examples where the vanishing of the Bach tensor does not imply that the spacetime is conformal to a $C$-space. The non-aligned Einstein-Maxwell spacetimes with vanishing Bach tensor are conformally $C$-spaces, but none of them are conformal to Einstein spaces.Comment: 22 pages. Corrected equation (12

On the Energy-Momentum Density of Gravitational Plane Waves

By embedding Einstein's original formulation of GR into a broader context we show that a dynamic covariant description of gravitational stress-energy emerges naturally from a variational principle. A tensor $T^G$ is constructed from a contraction of the Bel tensor with a symmetric covariant second degree tensor field $\Phi$ and has a form analogous to the stress-energy tensor of the Maxwell field in an arbitrary space-time. For plane-fronted gravitational waves helicity-2 polarised (graviton) states can be identified carrying non-zero energy and momentum.Comment: 10 pages, no figure