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Conformally-Flatness and Static Space-Time

  • Osamu Kobayashi
  • Morio Obata
Part of the Progress in Mathematics book series (PM, volume 14)

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)
.

Keywords

Positive Function Constant Curvature Ricci Tensor Einstein Space Conformally Flat 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge Univ. Press, 1973.zbMATHCrossRefGoogle Scholar
  2. [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. [3]
    L. Lindblom, “Some properties of static general relativistic stellar models,” J. Math. Phys. 21 (1980), 1455–1459.CrossRefGoogle Scholar
  4. [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

Authors and Affiliations

  • Osamu Kobayashi
    • 1
    • 2
  • Morio Obata
    • 1
    • 2
  1. 1.Tokyo Metropolitan UniversityTokyo 158Japan
  2. 2.Faculty of Science and TechnologyKeio UniversityYokohama 223Japan

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