The Weiss variation of the gravitational action

  • Justin C. FengEmail author
  • Richard A. Matzner
Research Article


The Weiss variational principle in mechanics and classical field theory is a variational principle which allows displacements of the boundary. We review the Weiss variation in mechanics and classical field theory, and present a novel geometric derivation of the Weiss variation for the gravitational action: the Einstein–Hilbert action plus the Gibbons–Hawking–York boundary term. In particular, we use the first and second variation of area formulas (we present a derivation accessible to physicists in an “Appendix”) to interpret and vary the Gibbons–Hawking–York boundary term. The Weiss variation for the gravitational action is in principle known to the Relativity community, but the variation of area approach formalizes the derivation, and facilitates the discussion of time evolution in General Relativity. A potentially useful feature of the formalism presented in this article is that it avoids an explicit 3 \(+\) 1 decomposition in the bulk spacetime.


Weiss variation Gibbons–Hawking–York term Variation of area Hamilton–Jacobi theory 



This article is based on the dissertation work of J. C. Feng. We thank Mr. Mark Selover, Prof. E. C. G. Sudarshan and Prof. G. Bhamathi for their comments and encouragement. J. C. Feng also thanks Prof. Austin Gleeson, Prof. Philip J. Morrison, Prof. Richard D. Hazeltine, and Prof. Robert E. Gompf for their guidance and service as members of his dissertation committee. This work was partially supported by the National Science Foundation under Grant Number PHY-1620610.


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Authors and Affiliations

  1. 1.Theory Group, Department of PhysicsUniversity of Texas at AustinAustinUSA

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