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 [237, 239
], 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
spaces18.
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
])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 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
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
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