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General Relativity and Gravitation

, Volume 25, Issue 9, pp 939–962 | Cite as

The generic condition is generic

  • John K. Beem
  • Steven G. Harris
Research Articles

Abstract

We consider the generic condition for vectors—both null and non-null—at a fixed pointp of a spacetime, and ask just how generic this condition is. In a general spacetime, if the curvature is not zero at the pointp, then the generic condition is found to be generic in the mathematical sense that it holds on an open dense set of vectors atp; more specifically, if there are as many as five non-null vectors in general position atp which fail to satisfy the generic condition, then the curvature vanishes atp. If the Riemann tensor is restricted to special forms, then stronger statements hold: An Einstein spacetime with three linearly independent nongeneric timelike vectors atp is flat atp. A Petrov type D spacetime may not have any nongeneric timelike vectors except possibly those lying in the plane of the two principal null directions; if any of the non-null vectors in such a plane are nongeneric, then so are all the vectors of that plane, as well as the plane orthogonal to it.

Keywords

Generic Condition Special Form Differential Geometry General Position Strong Statement 
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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Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • John K. Beem
    • 1
  • Steven G. Harris
    • 2
  1. 1.Mathematics DepartmentUniversity of Missouri-ColumbiaColumbiaUSA
  2. 2.Department of MathematicsSaint Louis UniversitySt. LouisUSA

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