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The Einstein Equations of Evolution -- A Geometric Approach

Fischer, Arthur E. and Marsden, Jerrold E. (1972) The Einstein Equations of Evolution -- A Geometric Approach. Journal of Mathematical Physics, 13 (4). pp. 546-568. ISSN 0022-2488. doi:10.1063/1.1666014.

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In this paper the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite‐dimensional geometry, we answer sharply the question: "In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?" By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body‐space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time‐dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics. We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the ``intrinsic shift vector field.'' The essence of our method is to extend the usual configuration space math = Riem(M) of Riemannian metrics to T×D×math, where T = C^∞(M,R) is the group of relativistic time translations and D = Diff(M) is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields (ξ_t,η_t) ∊ T×D. On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for math as a degenerate metric on D×math. On D×math, however, the metric is quadratic in the velocity variables. The groups T and D also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual ``constraint equations.'' A precise theorem is given for a remark of Misner that in an empty space‐time we must have H = 0. We study the relationship between the evolution equations for the time‐dependent metric gt and the Ricci flat condition of the reconstructed Lorentz metric gL. Finally, we make some remarks about a possible ``superphase space'' for general relativity and how our treatment on T×D×math is related to ordinary superspace and superphase space.

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Additional Information:© 1972. Received 12 July 1971 We thank P. Chernoff, D. Ebin, H. Kunzle, K. Kuchar, R. Sachs, and A. Taub for a variety of helpful suggestions. Partially supported by AEC Contract AT(04-3)-34. Partially supported by Navy Contract N00014-69-A-0200-1002 and NSF Contract GP-8257. Part of this author's work was done while at the University of Toronto, Toronto, Canada, and the Institute for Advanced study, Princeton, New Jersey.
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Issue or Number:4
Record Number:CaltechAUTHORS:20100805-131714334
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19297
Deposited By: Ruth Sustaita
Deposited On:05 Aug 2010 22:57
Last Modified:08 Nov 2021 23:51

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