The total space $${\mathfrak M} \approx {\mathbb H}_1 \times S^1$$ M ≈ H 1 × S 1 of the canonical circle bundle over the 3-dimensional Heisenberg group $${\mathbb H}_1$$ H 1 is a space–time with the Lorentzian metric $$F_{\theta _0}$$ F θ 0 (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form $$\theta _0$$ θ 0 on $${\mathbb H}_1$$ H 1 . The matter and energy content of $$\mathfrak M$$ M is described by the energy-momentum tensor $${T}_{\mu \nu }$$ T μ ν (the trace-less Ricci tensor of $$F_{\theta _0}$$ F θ 0 ) as an effect of the non flat nature of Feferman’s metric $$F_{\theta _0}$$ F θ 0 . We study the gravitational field equations $$R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }$$ R μ ν - ( 1 / 2 ) R g μ ν = T μ ν on $${\mathfrak M}$$ M . We consider the first order perturbation $$g = F_{\theta _0} + \epsilon \, h$$ g = F θ 0 + ϵ h , $$\epsilon<< 1$$ ϵ < < 1 , and linearize the field equations about $$F_{\theta _0}$$ F θ 0 . We determine a Lorentzian metric g on $${\mathfrak M}$$ M which solves the linearized field equations corresponding to a diagonal perturbation h.