J. York

Title: From the action to the thin sandwich theorem and beyond

Abstract: Two forms of the variational principle belonging to the Einstein equations are fruitful. One is the Hilbert action modified by boundary terms which I introduced in 1972. This principle is manifestly coordinate-free. The other is the Arnowitt-Deser-Misner canonical action which I have improved recently and which led to the Brown-York boundary total stress-energy tensor. I pass from the Hilbert to the canonical form using geometric methods. Identification of dynamical variables and arbitrary multipliers associated with constraints enables a correct variation procedure to be carried out. It becomes clear that Einstein's equations are both underdetermined and overdetermined. Both aspects are present in the canonical constraint ``algebra'' and suggest two viewpoints about how to approach the initial value conditions ("overdetermination") and how to use the variables in the theory with arbitrary time evolution ("underdetermination"). The initial conditions are set from each of the two viewpoints as differing, but equivalent, semi-linear elliptic systems. The propagation of these constraints by a first order symmetric hyperbolic system - a version of the Bianchi identities - is demonstrated. There are a number of very important open problems in the hyperbolic sector of Einstein's equations. Some of these are discussed.

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