11.3 The covariant approach
11.3.1 The covariant phase space methods
The traditional ADM approach to conserved quantities and the Hamiltonian analysis of general relativity is
based on the 3 + 1 decomposition of fields and geometry. Although the results and the content of a theory
may be covariant even if their form is not, the manifest spacetime covariance of a formalism may help to
find the (spacetime covariant) observables and conserved quantities, boundary conditions, etc. easily. No a
posteriori spacetime interpretation of the results is needed. Such a spacetime-covariant Hamiltonian
formalism was initiated by Nester [280
, 283
].
His basic idea is to use (tensor or Dirac spinor valued) differential forms as the basic field variables on
the spacetime manifold
. Thus his phase space is the collection of fields on the 4-manifold
,
endowed with the (generalized) symplectic structure of Kijowski and Tulczyjew [239]. He derives the
field equations from the Lagrangian 4-form, and for a fixed spacetime vector field
finds
a Hamiltonian 3-form
whose integral on a spacelike hypersurface takes the form
the sum of the familiar ADM constraints and a boundary term. The Hamiltonian is determined from the
requirement of the functional differentiability of
, i.e. that the variation
with respect to
the canonical variables should not contain any boundary term on an asymptotically flat
(see Sections 2.2.2, 3.2.1, and 3.2.2). For asymptotic translations the boundary term in the
Hamiltonian gives the ADM energy-momentum 4-vector. In tetrad variables
is essentially
Sparling’s 3-form [345], and the 2-component spinor version of
is essentially the
Nester-Witten 2-form contracted in the name index with the components of
(see also
Section 3.2.1).
The spirit of the first systematic investigations of the covariant phase space of the classical field
theories [122, 20, 146, 251
] is similar to that of Nester’s. These ideas were recast into the systematic
formalism by Wald and Iyer [389
, 215
, 216
], the so-called covariant Noether charge formalism (see
also [388, 251]). This formalism generalizes many of the previous approaches: The Lagrangian 4-form may
be any diffeomorphism invariant local expression of any finite order derivatives of the field variables. It gives
a systematic prescription for the Noether currents, the symplectic structure, the Hamiltonian etc. In
particular, the entropy of the stationary black holes turned out to be just a Noether charge derived from
Hilbert’s Lagrangian.
11.3.2 Covariant quasi-local Hamiltonians with explicit reference configurations
Update
The quasi-local Hamiltonian for a large class of geometric theories, allowing torsion and
non-metricity of the connection, was investigated by Chen, Nester, and Tung [109
, 107
, 285
] in the
covariant approach of Nester above [280, 283]. Starting with a Lagrangian 4-form for a first order
formulation of the theory and an arbitrary vector field
, they determine the general form of the
Hamiltonian 3-form
, including the boundary 2-form
. However, in the
variation of the corresponding Hamiltonian there will be boundary terms in general. To cancel
them, the boundary 2-form has to be modified. Introducing an explicit reference field
and canonical momentum
(which are solutions of the field equations), Chen, Nester, and
Tung suggest (in the differential form notation) either of the two 4-covariant boundary 2-forms
where the configuration variable
is some (tensor valued)
-form and
is the interior
product of the
-form
and the vector field
, i.e. in the abstract index formalism
. Then the boundary term in the variation
of the Hamiltonian is the
2-surface integral on
of
and
, respectively. Therefore,
the Hamiltonian is functionally differentiable with the boundary 2-form
if the configuration
variable
is fixed on
, but
should be used if
is fixed on
. Thus
the first boundary 2-form corresponds to a 4-covariant Dirichlet-type, while the second to a
4-covariant Neumann-type boundary condition. Obviously, the Hamiltonian evaluated in the reference
configuration
gives zero. Chen and Nester show [107
] that
and
are the
only boundary 2-forms for which the resulting boundary 2-form
in the variation
of the Hamiltonian 3-form vanishes on
, reflects the type of the boundary
conditions (i.e. which fields are fixed on the boundary), and is built from the configuration and
momentum variables 4-covariantly (‘uniqueness’). A further remarkable property of
and
is that the corresponding Hamiltonian 3-form can be derived directly from appropriate
Lagrangians. One possible choice for the vector field
is to be a Killing vector of the reference
geometry.
These general ideas were applied to general relativity in the tetrad formalism (and also in the Dirac
spinor formulation of the theory [109, 105
], yielding a Hamiltonian which is slightly different from
Equation (87)) as well as in the usual metric formalism [105
, 108
]. In the latter it is the appropriate
projections to
of
or
in some coordinate system
that is
chosen to be fixed on
. Then the dual of the corresponding Dirichlet and Neumann boundary 2-forms,
respectively, will be
The first terms are analogous to Freud’s superpotential, while the second ones are analogous to Komar’s
superpotential. (Since the boundary 2-form contains
only in the form
, this is always
tensorial. If
is chosen to be vanishing, then the first term reduces to Freud’s superpotential.)
Because of the Komar-like term, the quasi-local quantities depend not only on the 2-surface data
(both in the physical spacetime and the reference configuration), but on the normal directional
derivative of
as well. The connection between the present expressions and the similar
previous results (pseudotensorial, tensorial, and quasi-local) is also discussed in [107
, 105]. In
particular, the expression based on the Dirichlet-type boundary 2-form (90) gives precisely the
Katz-Bicak-Lynden-Bell superpotential [230]. In the spinor formulation of these ideas the
vector field
would be built from a Dirac spinor (or a pair of Weyl spinors). The main
difficulty is, however, to find spinor fields representing both translational and boost-rotational
displacements [110]. In the absence of a prescription for the reference configuration (even though that
should be defined only on an open neighbourhood of the 2-surface) the construction is still
not complete, even if the vector field
is chosen to be a Killing vector of the reference
spacetime.
A nice application of the covariant expression is a derivation of the first law of black hole
thermodynamics [107
]. The quasi-local energy expressions have been evaluated for several specific
2-surfaces. For round spheres in the Schwarzschild spacetime both the 4-covariant Dirichlet and Neumann
boundary terms (with the Minkowski reference spacetime and
as the timelike Killing vector
) give
at infinity, but at the horizon the former gives
and the latter is
infinite [107]. The Dirichlet boundary term gives at the spatial infinity in the Kerr-anti-de-Sitter solution
the standard
and
values for the energy and angular momentum, respectively [191]. Also,
the center-of-mass is calculated both in the metric and the tetrad formulation of general relativity for the
eccentric Schwarzschild solution at the spatial infinity [286, 287], and it was found that the ‘Komar-like
term’ is needed to recover the correct, expected value. At the future null infinity of asymptotically flat
spacetimes it gives the Bondi-Sachs energy-momentum and the expression of Katz [229, 233] for the
angular momentum [192]. The general formulae are evaluated for the Kerr-Vaidya solution
too.
11.3.3 Covariant quasi-local Hamiltonians with general reference terms
Update
Anco and Tung investigated the possible boundary conditions and boundary terms in the quasi-local
Hamiltonian using the covariant Noether charge formalism both of general relativity (with the Hilbert
Lagrangian and tetrad variables) and of Yang-Mills-Higgs systems [7
, 8
]. (Some formulae of the journal
versions were recently corrected in the latest arXiv-versions.) They considered the world tube of a compact
spacelike hypersurface
with boundary
. Thus the spacetime domain they considered is the
same as in the Brown-York approach:
. Their evolution vector field
is assumed to be
tangent to the timelike boundary
of the domain
. They derived a criterion for the
existence of a well-defined quasi-local Hamiltonian. Dirichlet and Neumann-type boundary conditions are
imposed (i.e., in general relativity, the variations of the tetrad fields are restricted on
by requiring
the induced metric
to be fixed and the adaptation of the tetrad field to the boundary
to be preserved, and the tetrad components
of the extrinsic curvature of
to
be fixed, respectively). Then the general allowed boundary condition was shown to be just
a mixed Dirichlet-Neumann boundary condition. The corresponding boundary terms of the
Hamiltonian, written in the form
, were also determined [7]. The properties of the
co-vectors
and
(called the Dirichlet and Neumann symplectic vectors, respectively) were
investigated further in [8
]. Their part tangential to
is not boost gauge invariant, and to
evaluate them the boost gauge determined by the mean extrinsic curvature vector
is used
(see Section 4.1.2). Both
and
are calculated for various spheres in several special
spacetimes. In particular, for the round spheres of radius
in the
hypersurface in the
Reissner-Nordström solution
and
, and hence
the Dirichlet and Neumann ‘energies’ with respect to the static observer
are
and
, respectively. Thus
does not reproduce the standard round sphere expression, while
gives the standard round
sphere and correct ADM energies only if it is ‘renormalized’ by its own value in Minkowski
spacetime [8].
Anco continued the investigation of the Dirichlet Hamiltonian in [6], which takes the form
Here the 2-surface
is assumed to be mean convex, whenever the boost gauge freedom in the
gauge potential
can be, and, indeed, is fixed by using the globally defined
orthonormal vector basis
in the normal bundle obtained by normalizing the mean
curvature basis
. The vector field
is still arbitrary, and
is an arbitrary
function of the metric
on the 2-boundary
, i.e. of the boundary data. This
is
actually assumed to have the structure
for
as an arbitrary function of
.
This Hamiltonian is functionally differentiable, gives the correct Einstein equations and, for
solutions, its value e.g. with
is the general expression of the quasi-local energy of Brown
and York. (Compare Equation (92) with Equation (86), or rather with Equations (72, 73,
74).)
However, to rule out the dependence of this notion of quasi-local energy on the completely freely
specifiable vector field
(i.e. on three arbitrary functions on
), Anco makes
dynamical by
linking it to the vector field
. Namely, let
, where
and
are constant,
is the area of
, and extend this
from
to
in
a smooth way. Then Anco proves that, keeping the 2-metric
and
fixed on
,
is a correct Hamiltonian for the Einstein equations, where
is an arbitrary function of
. For
with the choice
the boundary term reduces to the Hawking energy, and for
it is the Epp
and Kijowski-Liu-Yau energies depending on the choice of
(i.e. the definition of the reference term).
For general
choosing the reference term
appropriately Anco gives a 1-parameter generalization
of the Hawking and the Epp-Kijowski-Liu-Yau-type quasi-local energies (called the ‘mean
curvature masses’). Also, he defines a family of quasi-local angular momenta. Using the positivity of
the Kijowski-Liu-Yau energy (
) it is shown that the higher power (
) mean
curvature masses are bounded from below. Although these masses seem to have the correct large
sphere limit at spatial infinity, for general convex 2-surfaces in Minkowski spacetime they do not
vanish.
11.3.4 Pseudotensors and quasi-local quantities
As we discussed briefly in Section 3.3.1, many, apparently different pseudotensors and
-gauge
dependent energy-momentum density expressions can be recovered from a single differential form defined on
the bundle
of linear frames over the spacetime manifold: The corresponding superpotentials are the
pull-backs to
of the various forms of the Nester-Witten 2-from
from
along the various
local sections of the bundle [142, 266, 352, 353]. Thus the different pseudotensors are simply the gauge
dependent manifestations of the same geometric object on the bundle
in the different gauges. Since,
however,
is the unique extension of the Nester-Witten 2-form
on the principal bundle
of normalized spin frames
(given in Equation (12)), and the latter has been proven to be connected
naturally to the gravitational energy-momentum, the pseudotensors appear to describe the
same physics as the spinorial expressions, though in a slightly old fashioned form. That this is
indeed the case was demonstrated clearly by Chang, Nester, and Chen [104, 108, 285], by
showing an intimate connection between the covariant quasi-local Hamiltonian expressions and the
pseudotensors. Writing the Hamiltonian
in the form of the sum of the constraints and a
boundary term, in a given coordinate system the integrand of this boundary term may be the
superpotential of any of the pseudotensors. Then the requirement of the functional differentiability of
gives the boundary conditions for the basic variables at
. For example, for the Freud
superpotential (for Einstein’s pseudotensor) what is fixed on the boundary
is a certain piece of
.