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 (i) to modify the Hilbert action to the trace-χ-action and (ii) to the boundary condition that the induced three-metric on the boundary of the domain D of the action is fixed.

However, as stressed by Kijowski [315Jump To The Next Citation Point, 317Jump To The Next Citation Point, 229Jump To The Next Citation Point], 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, [317Jump To The Next Citation Point]).19 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 D above), and defining the Hamiltonian by the standard Legendre transformation on the typical compact spacelike three-manifold Σ and its boundary 𝒮 = ∂ Σ as well, Kijowski arrived at a variation formula involving the value on 𝒮 of the variation of the canonical momentum, ∘ --- &tidle;πab := − -1-- |γ|(Θab − Θ γab) 16πG, conjugate to γab. (Apart from a numerical coefficient and the subtraction term, this is essentially the surface stress-energy tensor ab τ given by Eq. (10.3View Equation).) 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 𝒮.

The first Legendre transformation that he considered gave a Hamiltonian whose variation involves the variation of the induced two-metric q ab on 𝒮 and the parts &tidle;πabtt a b and &tidle;πabt Πc a b of the canonical momentum above. Explicitly, with the notation of Section 10.1, the latter two are ab π tatb = k∕(16πG ) and ab π taqbc = Ac∕(16 πG ), respectively. (ab π is the de-densitized ab &tidle;π.) Then, however, the lapse and the shift on the boundary 𝒮 will not be independent. As Kijowski shows, they are determined by the boundary conditions of the two-metric and the freely specifiable parts k and Ac of the canonical momentum πab. Then, to define the ‘quasi-symmetries’ of the two-surface, Kijowski suggests that one embed first the two-surface isometrically into an 0 x = const. 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 x0 = const. hyperplane. This world tube gives an extrinsic curvature 0 kab and vector potential 0 A c. Finally, Kijowski’s choices for k and Ac are just k0 and A0c, respectively. In particular, to define ‘quasi time translation’ he takes πabtatb = k0 ∕(16πG ) and πabtaΠcb = 0, because this choice yields zero shift and constant lapse with value one. The corresponding quasi-local quantity, the Kijowski energy, is

1 ∮ (k0)2 − (k2 − l2) EK (𝒮) := ------ --------0-------d𝒮. (10.14 ) 16πG 𝒮 k
Here, as above, k and l are the trace of the extrinsic curvatures of 𝒮 in the physical spacetime corresponding to the outward-pointing spacelike and the future pointing timelike unit normals to 𝒮, which are orthogonal to each other. Obviously, EK (𝒮) 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 𝒮. A remarkable property of this procedure is that, for round spheres in the Schwarzschild solution, the choice πabtatb = k0∕(16 πG ), πabtaqbc = 0 (i.e., the flat spacetime values) reproduces the lapse of the correct Schwarzschild time [315Jump To The Next Citation Point]. For round spheres (see Section 4.2.1) Eq. (10.14View Equation) gives -r- 2G [1 − exp (− 2α )], which is precisely the standard round sphere expression (4.8View Equation). In particular [315], for the event horizon of the Schwarzschild solution it gives the expected value m ∕G. However, there exist spacelike topological two-spheres 𝒮 in the Minkowski spacetime for which E (𝒮 ) K is positive [401Jump To The Next Citation Point].

Kijowski considered another Legendre transformation on the two-surface as well, and in the variation of the resulting Hamiltonian only the value on 𝒮 of the variation of the metric γab appears. Thus, in this phase space the components of γab can be specified freely on 𝒮, and Kijowski calls the value of the resulting Hamiltonian the ‘free energy’. Its form is

1 ∮ ( √ -------) FK (𝒮 ) :=----- k0 − k2 − l2 d𝒮. (10.15 ) 8 πG 𝒮
In the special boost-gauge when l = 0 the ‘free energy’ FK(𝒮 ) reduces to the Brown–York expression E (𝒮 ) BY given by Eq. (10.9View Equation). F (𝒮) K appears to have been rediscovered recently by Liu and Yau [338Jump To The Next Citation Point], and we discuss the properties of FK (𝒮) 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 𝒮 are given in [316].
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