Cawley's counterexample to Dirac's conjecture as a curved spacetime (Q1093194)

The Lagrangian of an example of a constraint dynamical system due to \textit{R. Cawley} [Phys. Rev. D 21, 2988-2990 (1980)] is translated into a \(2+1\)-dimensional metric \(ds^ 2=f(x,y)dt^ 2-2dxdy\), \(f(x,y)=1+2kxy- \lambda y^ 2\). The Einstein tensor \(G^{\mu v}\) for this metric is then matched to a matter energy-momentum tensor of the form \(T^{\mu v}=\phi u^{\mu}u^ v+p^{\mu v}\).