Communications in Mathematical Physics

, Volume 5, Issue 4, pp 237–256 | Cite as

Space-times containing perfect fluids and having a vanishing conformal divergence

  • L. C. Shepley
  • A. H. Taub


The solutions of the Einstein field equations are studied under the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the divergence of the conformal (Weyl) tensor vanishes, and (3a) either an equation of state exists such thatp=p (w),p being the pressure andw the rest energy density, or (3b) the rest particle density is conserved. Under assumptions (1), (2), and (3a) it is shown that the space-time is conformally flat and the metric is a Robertson-Walker metric. The flow is irrotational, shear-free, and geodesic. Under assumptions (1), (2), and (3b) it is shown that either the line element is static or the fluid has a very special caloric equation of state. Conditions for a static solution to exist are examined, and it is shown that the Schwarzschild interior solution satisfies these conditions as does the Einstein universe. The Schwarzschild interior and the Einstein universe are the only conformally flat, static solutions obeying (1), (2), and (3b).


Neural Network Energy Density Nonlinear Dynamics Static Solution Field Equation 
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Copyright information

© Springer-Verlag 1967

Authors and Affiliations

  • L. C. Shepley
    • 1
  • A. H. Taub
    • 1
  1. 1.University of CaliforniaBerkeley

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