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,
[317
]).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
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
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
![]() This work is licensed under a Creative Commons License. E-mail us: |