Invariant conformal vectors in static spacetimes (Q1174416)

The line element of a static spacetime is locally written in the form of some kind of space-time decomposition as \[ ds^ 2=\alpha^ 2(x^ k)[- dt^ 2+g_{ij}(x^ k)dx^ idx^ j],\;\;(i,j,k=1,2,3).\leqno(1) \] Physicists call it by different nomenclatures. However, mathematicians call the spacetimes with line elements \(ds^ 2\) and \(ds'{}^ 2\) as conformally related if \[ ds^ 2=\alpha^ 2ds'{}^ 2, \quad ds'{}^ 2=- dt^ 2+g_{ij}dx^ idx^ j,\leqno(2) \] where \(\alpha\) is some scalar point function. Moreover, the spacetime (2) is said to be conformally related to itself if \({\mathfrak L}_ vg_{ab}=2\phi g_{ab}\), \((a,b=0,1,2,3)\), where \({\mathfrak L}_ v\) is the operator of Lie - derivation, \(\phi\) is the scalar given by \(\phi={\mathfrak L}_ v(\ln \alpha)\) in the present context, a being the norm of the Killing vector \(\xi\) of (1). The authors study the above form of static spacetime and show that the invariant conformal vectors \(V\) of (1) (the ones which commute with the static Killing vectors \(\xi\) of (1)) are generated in the form \(V=\alpha\xi+v\) by the Killing vectors \(v\) of the metric \(g_{ij}\), and thereby infer that the problem of finding the invariant conformal vectors \(V\) is, in fact, reduced to the simpler one of finding the isometries of a three-dimensional metric \(g_{ij}\). Next, considering the perfect fluid case with energy density \(\rho\), pressure \(p\), the authors write down the Einstein's field equations in terms of \(\alpha\), \(g_{ij}\), \(\rho\) and \(p\) explicitly and deduce their integrability conditions. Finally, the authors look for proper invariant conformal vectors (which are neither Killing nor homothetic ones) in the spacetimes and are lead to the conclusion that the only static perfect fluid spacetimes admitting proper invariant conformal vectors are either the Schwarzschild interior solution or a straightforward generalization of the spherically symmetric line element due to \textit{H. A. Buchdahl} and \textit{W. J. Land} [J. Aust. Math. Soc. 8 (1968)].