Knowlpedia

Theorem

Smooth splitting of globally hyperbolic spacetimes

Every globally hyperbolic spacetime has a smooth Cauchy temporal function and splits smoothly into time times a Cauchy hypersurface.

Statement

Every (M,g)(M,g) admits a smooth Cauchy temporal function t:M→Rt:M\to\mathbb R: its gradient is timelike and each level set

Σs=t−1(s)\Sigma_s=t^{-1}(s)

is a smooth spacelike .

Choosing one level Σ\Sigma gives a diffeomorphism

M≅R×ΣM\cong\mathbb R\times\Sigma

under which

g=−β dt2+ht,g=-\beta\,dt^2+h_t,

where β>0\beta>0 is smooth and hth_t is a smooth family of Riemannian metrics on Σ\Sigma. Conversely, a spacetime carrying a Cauchy temporal function is globally hyperbolic.

References
  1. Antonio N. Bernal and Miguel Sánchez, “Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes,” Communications in Mathematical Physics 257 (2005), 43–50. Journal record.