Conformally-Flatness and Static Space-Time

Abstract

A Lorentzian (n+1)-manifold $\left( {{{\tilde M}^{n + 1}},\tilde g} \right)$ is called (globally) static [1], [2] if $\tilde M$ is a product space ℝ x M of ℝ with an n-manifold Mⁿ and the metric $\tilde g$ has the form

$$\tilde g = - f{(x)^2}d{t^2} + x*g,$$

((0.1))

where $t:\tilde M \to$ and $x:\tilde M \to$ are the natural projections, g a Riemannian metric on M, and f a positive function on M. We consider Einstein’s equation on ($(\tilde M,\tilde g)$) with perfect fluid as a matter field, i.e.,

$$\mathop {Ric}\limits^ - \frac{1}{2}\tilde R\tilde g = (\mu + p)\eta \otimes \eta + p\tilde g,$$

((0.2))

where n is a l-form with $\tilde g\left( {\eta ,\eta } \right) = - 1$, whose associated vector field represents the flux of the fluid, and μ and p are functions on $\tilde M$ which are called the energy density and the pressure, respectively [l]. In other words, (0.2) says that, at each point of $\tilde M$, the Ricci tensor $\widetilde {Ric}$ has at most two distinct eigenvalues with multiplicities l and n. It is known [2] that under the condition (0.1), (0.2) is equivalent to the following equation on (M,g);

$$Ric - \frac{{Hess\;f}}{f} = \frac{1}{n}\left( {R - \frac{{\Delta f}}{f}} \right)g$$

((0.3))

and then, there are relations;

$$\mu = \frac{R}{2}$$

$$p = \frac{{n - 1}}{n}\left( {\frac{{\Delta f}}{f} - \frac{{n - 2}}{{2\left( {n - 1} \right)}}R} \right)$$

((0.4))

.