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The Hamiltonian structure of general relativistic perfect fluids

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Abstract

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.

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Authors and Affiliations

  1. School of Mathematics, The Institute for Advanced Study, 08540, Princeton, NJ, USA

    David Bao

  2. Department of Mathematics, University of Houston-University Park, 77004, Houston, TX, USA

    David Bao

  3. Department of Mathematics, University of California, 94720, Berkeley, CA, USA

    Jerrold Marsden

  4. Space Systems Division, Lockheed Missiles & Space Company, Inc., 1111 Lockheed Way, 94086, Sunnyvale, CA, USA

    Ronald Walton

Authors
  1. David Bao
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  2. Jerrold Marsden
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  3. Ronald Walton
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Communicated by S.-T. Yau

Research supported in part by NSF grant MCS 81-08814(A02)

Research supported in part by NSF grant MCS 81-07086

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Bao, D., Marsden, J. & Walton, R. The Hamiltonian structure of general relativistic perfect fluids. Commun.Math. Phys. 99, 319–345 (1985). https://doi.org/10.1007/BF01240351

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