Gravitational Field Equations on Fefferman Space–Times

• Published in: Complex analysis and operator theory 2017-12, Vol.11 (8), p.1685-1713
• Main Authors: Barletta, Elisabetta, Dragomir, Sorin, Jacobowitz, Howard
• Format: Article
• Language: English
• Subjects: Analysis; Gravitation; Mathematical analysis; Mathematics; Mathematics and Statistics; Operator Theory; Perturbation
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-017-0670-8

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