Gravitational field equations on Fefferman space-times

The total space $\mathfrak{M} \approx \mathbb{H}_1 \times S^1$ of the canonical circle bundle over the 3-dimensional Heisenberg group $\mathbb{H}_1$ is a space–time with the Lorentzian metric $F_{\theta_0}$ (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form $\theta_0$ on $\mathbb{H}_1$. The matter and energy content of $\mathfrak{M}$ is described by the energy-momentum tensor $\buildrel{\circ}\over{T}_{\mu \nu}$ (the trace-less Ricci tensor of $F_{\theta_0}$) as an effect of the non flat nature of Fefferman’smetric $F_{\theta_0}$. We study the gravitational field equations $R_{\mu \nu} −(1/2) R g_{\mu \nu} = \buildrel{\circ}\over{T}_{\mu \nu}$ on $\mathfrak{M}$. We consider the first order perturbation $g = F_{\theta_0} + \epsilon h$, $\epsilon << 1$, and linearize the field equations about $F_{\theta_0}$. We determine a Lorentzian metric $g$ on $\mathfrak{M}$ which solves the linearized field equations corresponding to a diagonal perturbation $h$.