### 10.2 Kijowski’s approach

#### 10.2.1 The role of the boundary conditions

In the Brown-York approach the leading principle was the claim to have a well-defined variational principle. This led them to modify the Hilbert action to the trace--action and the boundary condition that the induced 3-metric on the boundary of the domain of the action is fixed.

However, as stressed by Kijowski [237239], 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 to free energy. Thus the different boundary conditions correspond to different physical situations, and, mathematically, to different phase spaces. 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 [239]). The boundary condition of Brown and York is only one of the possible boundary conditions.

#### 10.2.2 The analysis of the Hilbert action and the quasi-local internal and free energies

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 3-manifold and its boundary too, 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 Equation (67).) 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 2-metric on and the parts and of the canonical momentum above. Explicitly, with the notations of the previous 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 for the 2-metric and the freely specifiable parts and of the canonical momentum . Then, to define the ‘quasi-symmetries’ of the 2-surface, Kijowski suggests to embed first the 2-surface isometrically into an hyperplane of the Minkowski spacetime, and then define a world tube by dragging this 2-surface along the integral curves of the Killing vectors of the Minkowski spacetime. For example, to define the ‘quasi time translation’ of the 2-surface in the physical spacetime we must consider the time translation in the Minkowski spacetime of the 2-surface embedded in the hyperplane. This world tube gives an extrinsic curvature and vector potential . Finally, Kijowski’s choice for and is just and , respectively. In particular, to define the ‘quasi time translation’ he takes and , because this choice yields zero shift and constant lapse with value 1. The corresponding quasi-local quantity, the Kijowski energy, is

Here, as above, and are the trace of the extrinsic curvatures of in the physical spacetime corresponding to the outward pointing spacelike and the future pointing timelike unit normals to , which are orthogonal to each other. Obviously, is invariant with respect to the boost gauge transformations of the normals, because the ‘generator vector field’ of the energy is not linked to one of the normals of . A remarkable property of this procedure is that for round spheres in the Schwarzschild solution the choice , (i.e. the flat spacetime values) reproduces the lapse of the correct Schwarzschild time [237]. For round spheres (see Section 4.2.1) Equation (78) gives , which is precisely the standard round sphere expression (27). In particular [237], for the event horizon of the Schwarzschild solution it gives the expected value . However, there exist spacelike topological 2-spheres in the Minkowski spacetime for which is positive [292].

Kijowski considered another Legendre transformation on the 2-surface too, 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

In the special boost-gauge when the ‘free energy’ reduces to the Brown-York expression given by Equation (73). appears to have been re-discovered recently by Liu and Yau [253], 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 [238].