Biframe bundle geometry and an extension of RMW theory: Application to a charged perfect fluid

Hammon, K. S.

doi:10.1007/bf00760228

Cited by 1 publication

(1 citation statement)

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“…These are the classical algebraic Rainich conditions [23,31] (see also [15,17,19,26]). They are necessary and sufficient conditions for a spacetime metric to originate algebraically (via Einstein's equations) in a Maxwell field, i.e.…”

Section: Symmetric Null-cone Preserving Maps and Algebraic Rainich Co...mentioning

confidence: 99%

Null cone preserving maps, causal tensors and algebraic Rainich theory

Bergqvist

Senovilla

2001

Class. Quantum Grav.

153

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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 generalises, both with respect to the dimension N and the rank 1 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. Nonsymmetric 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 skewsymmetric 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.2

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