Killing pairs constructed from a recurrent vector field
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.
scientific article; zbMATH DE number 798695