Killing pairs constructed from a recurrent vector field
type: Item
instance of: scholarly article
title: Killing pairs constructed from a recurrent vector field
zbMATH Open document ID: 0830.53065
DOI: 10.1007/BF02105321
author: C. D. Collinson
author: E. G. L. R. Vaz
published in: General Relativity and Gravitation
publication date: 1996-01-28T00:00:00Z
review text: For a space-time \((M,g)\) the authors investigate the possibility of the existence of vector fields \(X\) and \(Y\) on \(M\) with the property that, for any geodesic with (affinely parametrised) tangent vector \(V_1\) the quantity \(\Phi = g(X,V) g(Y,V)^{-1}\) is constant (presumably where defined) along the geodesic. They quote a previous result of the first author which gives necessary and sufficient conditions on \(X\), \(Y\) and \(g\) for this to be the case. They are that (in a standard notation) \(X_{(a;b)} = X_{(a} P_{b)}\) and \(Y_{(a;b)} = Y_{(a} P_{b)}\) for some vector field \(P\) on \(M\). The conditions they require of \(\Phi\) are trivially satisfied if \(X = \alpha X^1\) and \(Y = \alpha Y^1\) for \(\alpha : M \to \mathbb{R}\) and Killing vectors \(X^1\) and \(Y^1\) of \(g\). The authors search here for non-trivial examples of such Killing pairs \(\{X, Y\}\) which leads to the above condition on \(\Phi\). In particular they consider the case when one member of the sought-for pair, say \(X\), is recurrent \((X_{a;b} = X_a \varphi_b\) for some vector field \(\varphi\) on \(M)\). It then turns out that, for \(\Phi\) not to be trivial \(X\) must be null and \(Y\) cannot be recurrent. The authors then consider the subclass of all spacetime metrics admitting a null recurrent vector field and which are represented by metrics of the form \[ ds^2 + dv^2 + dx^2 = 2 du dt + 2C dt^2 \] where \(C(u,v,x,t) = uf(t) + g(v,x,t)\) and which admit the recurrent null vector field \(X\) with components in this coordinate system given by \(X^a = \delta^a\). Conditions on \(f\) and \(g\) are then found for which a (non-trivial) Killing pair exists.
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 83C20
Mathematics Subject Classification ID: 83C40
zbMATH DE Number: 798695
zbMATH Keywords: conservation laws
zbMATH Keywords: first integral
zbMATH Keywords: Killing pairs
reviewed by: G. S. Hall
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02105321
OpenAlex ID: W1975574437
description: scientific article; zbMATH DE number 798695