Communications in Mathematical Physics

, Volume 99, Issue 3, pp 319-345

First online:

The Hamiltonian structure of general relativistic perfect fluids

  • David BaoAffiliated withSchool of Mathematics, The Institute for Advanced StudyDepartment of Mathematics, University of Houston-University Park
  • , Jerrold MarsdenAffiliated withDepartment of Mathematics, University of California
  • , Ronald WaltonAffiliated withSpace Systems Division, Lockheed Missiles & Space Company, Inc.

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We show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift, are Hamiltonian relative to a certain Poisson structure. For the fluid variables, a Lie-Poisson structure associated to the dual of a semi-direct product Lie algebra is used, while the bracket for the gravitational variables has the usual canonical symplectic structure. The evolution is governed by a Hamiltonian which is equivalent to that obtained from a canonical analysis. The relationship of our Hamiltonian structure with other approaches in the literature, such as Clebsch potentials, Lagrangian to Eulerian transformations, and its use in clarifying linearization stability, are discussed.