Conformally-Flatness and Static Space-Time

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


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
$$\tilde g = - f{(x)^2}d{t^2} + x*g,$$
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.,
$$\mathop {Ric}\limits^ - \frac{1}{2}\tilde R\tilde g = (\mu + p)\eta \otimes \eta + p\tilde g,$$
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);
$$Ric - \frac{{Hess\;f}}{f} = \frac{1}{n}\left( {R - \frac{{\Delta f}}{f}} \right)g$$
and then, there are relations;
$$\mu = \frac{R}{2}$$
$$p = \frac{{n - 1}}{n}\left( {\frac{{\Delta f}}{f} - \frac{{n - 2}}{{2\left( {n - 1} \right)}}R} \right)$$


Positive Function Constant Curvature Ricci Tensor Einstein Space Conformally Flat 
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    S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge Univ. Press, 1973.zbMATHCrossRefGoogle Scholar
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    L. Lindblom, “Some properties of static general relativistic stellar models,” J. Math. Phys. 21 (1980), 1455–1459.CrossRefGoogle Scholar
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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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