Time-dependent embeddings for Schwarzschild-like solutions to the gravitational field equations

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An explicit formula for embedding the Schwarzschild solution in a three-dimensional flat space with indefinite metric for arbitrary Kruskal timelike coordinate v is presented. The time development of the Schwarzschild solution can then be represented by a succession of spacelike surfaces, each corresponding to a different value of v. It is seen that the standard representation of the Schwarzschild metric, the Flamm paraboloid, is in fact the v = 0 special case of a similar time-dependent embedding in a three-dimensional Euclidean space with positive definite metric. However, this embedding is inadequate in that it is not defined for most values of v. Thus, the embedding in a space with indefinite metric is to be preferred. The results for the Schwarzschild case are found to be readily extended to all metrics of a certain class, and a general embedding formula for arbitrary v results. Embeddings for the Schwarzschild, de Sitter, and Reissner-Nordström metrics are then special cases of this general form. It is seen that all such solutions behave similarly as v gets large. This suggests an alternate interpretation of the oscillatory character of the Reissner-Nordström "wormhole."

Original languageEnglish (US)
Pages (from-to)1662-1667
Number of pages6
JournalJournal of Mathematical Physics
Issue number9
StatePublished - 1981
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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