Manifolds and Lie Groups pp 197-206 | Cite as

# Conformally-Flatness and Static Space-Time

Chapter

## Abstract

A Lorentzian (n+1)-manifold \(\left( {{{\tilde M}^{n + 1}},\tilde g} \right)\) is called (globally) 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., 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 and then, there are relations; .

*static*[1], [2] if \(\tilde M\) is a product space ℝ x M of ℝ with an n-manifold M^{n}and the metric \(\tilde g\) has the form$$\tilde g = - f{(x)^2}d{t^2} + x*g,$$

(0.1)

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

(0.2)

*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)

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

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

(0.4)

## Keywords

Positive Function Constant Curvature Ricci Tensor Einstein Space Conformally Flat
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## References

- [1]S.W. Hawking and G.F.R. Ellis,
*The Large Scale Structure of Space-Time*, Cambridge Univ. Press, 1973.zbMATHCrossRefGoogle Scholar - [2]O. Kobayashi and M. Obata, “Certain mathematical problems on static models in general relativity,” to appear in Proc. Symp. Diff. Geom. and Partial Diff. Equ., Beijing, 1980.Google Scholar
- [3]L. Lindblom, “Some properties of static general relativistic stellar models,”
*J. Math. Phys.*21 (1980), 1455–1459.CrossRefGoogle Scholar - [4]S. Nishikawa and Y. Maeda, “Conformaily flat hypersurfaces in a conformai ly flat Riemannian manifold,”
*Tohoku Math. J.*26 (1974), 159–168.MathSciNetzbMATHCrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 1981