In the Brown–York approach the leading principle was the claim to have a well-defined variational principle. This led them (i) to modify the Hilbert action to the trace--action and (ii) to the boundary condition that the induced three-metric on the boundary of the domain of the action is fixed.
However, as stressed by Kijowski [315, 317, 229], the boundary conditions have much deeper content. For example in thermodynamics the different definitions of the energy (internal energy, enthalpy, free energy, etc.) are connected with different boundary conditions. Fixing the pressure corresponds to enthalpy, but fixing the temperature corresponds to free energy. Thus, the different boundary conditions correspond to different physical situations, and, mathematically, to different phase spaces.18 Therefore, to relax the a priori boundary conditions, Kijowski abandoned the variational principle and concentrated on the equations of motions. However, to treat all possible boundary conditions on an equal footing he used the enlarged phase space of Tulczyjew (see, for example, ).19 The boundary condition of Brown and York is only one of the possible boundary conditions.
Starting with the variation of Hilbert’s Lagrangian (in fact, the corresponding Hamilton–Jacobi principal function on a domain above), and defining the Hamiltonian by the standard Legendre transformation on the typical compact spacelike three-manifold and its boundary as well, Kijowski arrived at a variation formula involving the value on of the variation of the canonical momentum, , conjugate to . (Apart from a numerical coefficient and the subtraction term, this is essentially the surface stress-energy tensor given by Eq. (10.3).) Since, however, it is not clear whether or not the initial + boundary value problem for the Einstein equations with fixed canonical momenta (i.e., extrinsic curvature) is well posed, he did not consider the resulting Hamiltonian as the appropriate one, and made further Legendre transformations on the boundary .
The first Legendre transformation that he considered gave a Hamiltonian whose variation involves the variation of the induced two-metric on and the parts and of the canonical momentum above. Explicitly, with the notation of Section 10.1, the latter two are and , respectively. ( is the de-densitized .) Then, however, the lapse and the shift on the boundary will not be independent. As Kijowski shows, they are determined by the boundary conditions of the two-metric and the freely specifiable parts and of the canonical momentum . Then, to define the ‘quasi-symmetries’ of the two-surface, Kijowski suggests that one embed first the two-surface isometrically into an hyperplane of the Minkowski spacetime, and then define a world tube by dragging this two-surface along the integral curves of the Killing vectors of the Minkowski spacetime. For example, to define ‘quasi time translation’ of the two-surface in the physical spacetime we must consider the time translation in the Minkowski spacetime of the two-surface embedded in the hyperplane. This world tube gives an extrinsic curvature and vector potential . Finally, Kijowski’s choices for and are just and , respectively. In particular, to define ‘quasi time translation’ he takes and , because this choice yields zero shift and constant lapse with value one. The corresponding quasi-local quantity, the Kijowski energy, is. For round spheres (see Section 4.2.1) Eq. (10.14) gives , which is precisely the standard round sphere expression (4.8). In particular , for the event horizon of the Schwarzschild solution it gives the expected value . However, there exist spacelike topological two-spheres in the Minkowski spacetime for which is positive .
Kijowski considered another Legendre transformation on the two-surface as well, and in the variation of the resulting Hamiltonian only the value on of the variation of the metric appears. Thus, in this phase space the components of can be specified freely on , and Kijowski calls the value of the resulting Hamiltonian the ‘free energy’. Its form is, and we discuss the properties of further in Section 10.4. A more detailed discussion of the possible quasi-local Hamiltonians and the strategies to define the appropriate ‘quasi-symmetries’ of are given in .
Living Rev. Relativity 12, (2009), 4
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