Energy conditions in standard static spacetimes

Let (H, h) be a Riemannian manifold and letfH(0,) be a smooth function. The Lorentzian warped product (a,b) _f ×H, -a<b, with metricds ²=(–f ² dt ²)h is called a standard static spacetime. We investigate conditions on the warping functionf which guarantee that a standard static spacetime (a,b) _f ×H satisfies certain of the energy conditions from general relativity. Also, if (H, h) is a complete, finite volume Riemannian manifold with nonpositive Ricci curvature and the gradient off never vanishes onH, then (a, b) _f ×H cannot satisfy the null convergence condition.