1,060 research outputs found
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.Comment: extended version, 13 pages, shorter version published in
Class.Quant.Gra
Algebraic Rainich theory and antisymmetrisation in higher dimensions
The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient
conditions on an energy-momentum tensor 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 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 -form. Here we examine algebraic Rainich
conditions for general -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 is a pp-wave geometry with pure
radiation ( is flat), where 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
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 , 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
fulfill these conditions, but so also does our recently
proposed tensor , which has many of the desirable
properties of . For the quasilocal small sphere limit
restriction, we found that there are only two fourth rank tensors
and 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
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 when 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 the possibility of
constructing conserved mixed currents between the gravitational field and the
matter fields.Comment: 15 page
Quasi-local mass in the covariant Newtonian space-time
In general relativity, quasi-local energy-momentum expressions have been
constructed from various formulae. However, Newtonian theory of gravity gives a
well known and an unique quasi-local mass expression (surface integration).
Since geometrical formulation of Newtonian gravity has been established in the
covariant Newtonian space-time, it provides a covariant approximation from
relativistic to Newtonian theories. By using this approximation, we calculate
Komar integral, Brown-York quasi-local energy and Dougan-Mason quasi-local mass
in the covariant Newtonian space-time. It turns out that Komar integral
naturally gives the Newtonian quasi-local mass expression, however, further
conditions (spherical symmetry) need to be made for Brown-York and Dougan-Mason
expressions.Comment: Submit to Class. Quantum Gra
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 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 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 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
Dynamical laws of superenergy in General Relativity
The Bel and Bel-Robinson tensors were introduced nearly fifty 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 {\em energy} and the term {\em 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 {\em 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 analysing the different parts
resulting from the splitting. In the electromagnetic case such splitting gives
rise to the electromagnetic {\em energy density}, the Poynting vector and the
electromagnetic stress tensor, each of them having a precise physical
interpretation which is deduced from the {\em 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 {\em
equations} are to the superenergy what the Poynting theorem is to
electromagnetism. See paper for full abstract.Comment: 27 pages, no figures. Typos corrected, section 9 suppressed and more
acknowledgments added. To appear in Classical and Quantum Gravit
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