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. more easily.
No a posteriori spacetime interpretation of the results is needed. Such a spacetime-covariant Hamiltonian
formalism was initiated by Nester [377, 380
].
His idea is to use (tensor or Dirac spinor valued) differential forms as the field variables on the spacetime
manifold . Thus, his phase space is the collection of fields on the four-manifold
, endowed
with the (generalized) symplectic structure of Kijowski and Tulczyjew [317]. 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 spirit of the first systematic investigations of the covariant phase space of the classical field
theories [158, 33, 197, 336] is similar to that of Nester’s. These ideas were recast into the systematic
formalism by Wald and Iyer [536
, 287
, 288
], the covariant Noether charge formalism (see also [535, 336]).
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 turns out to be just a Noether charge derived from
Hilbert’s Lagrangian.
The quasi-local Hamiltonian for a large class of geometric theories, allowing torsion and nonmetricity of the
connection, was investigated by Chen, Nester, and Tung [139, 136
, 382
] in the covariant approach of
Nester, above [377, 380]. 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 differential form notation) either of the two four-covariant
boundary 2-forms
The boundary term in the variation of the Hamiltonian with the boundary term (11.5
) and
(11.6
) is the two-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 four-covariant Dirichlet-type, while the second
corresponds to a four-covariant Neumann-type boundary condition. Obviously, the Hamiltonian evaluated
in the reference configuration
gives zero. Chen and Nester show [136
] 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
, which reflects the type
of boundary conditions (i.e., which fields are fixed on the boundary), and is built from the
configuration and momentum variables four-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 a
Killing vector of the reference geometry. This reference geometry is, however, not yet specified, in
general.
These general ideas were applied to general relativity in the tetrad formalism (and also in the Dirac
spinor formulation of the theory [139, 132], yielding a Hamiltonian, which is slightly different from
Eq. (11.4
)) as well as in the usual metric formalism [132
, 137
]. 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
will be, respectively,
A nice application of the covariant expression is a derivation of the first law of black hole
thermodynamics [136]. The quasi-local energy expressions have been evaluated for several specific
two-surfaces. For round spheres in the Schwarzschild spacetime, both the four-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 [136]. The Dirichlet boundary term gives, at spatial infinity in the Kerr–anti-de Sitter solution, the
standard
and
values for the energy and angular momentum, respectively [257]. The
center-of-mass is also calculated, both in the metric and the tetrad formulation of general relativity, for the
eccentric Schwarzschild solution at spatial infinity [389, 390], and it was found that the ‘Komar-like term’
is needed to recover the correct, expected value. At future null infinity of asymptotically flat
spacetimes it gives the Bondi–Sachs energy-momentum and the expression of Katz [305, 310] for the
angular momentum [258]. The general formulae are evaluated for the Kerr–Vaidya solution as
well.
The quasi-local energy-momentum is calculated on two-surfaces lying in intrinsically-flat spacelike
hypersurfaces in static spherically-symmetric spacetimes [138], and, in particular, for two-surfaces in the
slicing of the Schwarzschild solution in the Painlevé–Gullstrand coordinates. Though these
hypersurfaces are flat, and hence, the total (ADM type) energy is expected to be vanishing, the quasi-local
energy expression based on Eq. (11.7
) and a ‘naturally chosen’ frame field gives
. (N.B., the Cauchy
data on the
hypersurfaces do not satisfy the falloff conditions of Section 3.2.1. Though the
intrinsic metric is flat, the extrinsic curvature tends to zero only as
, while in the expression of the
ADM linear momentum a slightly faster than
falloff is needed. Thus, the vanishing of the
naïvly introduced ADM-type energy does not contradict the rigidity part of the positive energy
theorem.)
The null infinity limit of the quasi-local energy and the corresponding outgoing energy flux, based on
Eq. (11.5), are calculated in [563]. It is shown that, with Minkowski spacetime as a reference configuration,
and even with three different embeddings of the two-surface
into the reference spacetime,
the null infinity limit of these two quantities are just the standard Bondi energy and Bondi
mass-loss, respectively. A more detailed discussion of the general formulae for the quasi-local energy
flux, coming from Eqs. (11.5
) – (11.6
) and the two additional boundary expressions of [137
],
The quasi-local energy flux of spacetime perturbations on a stationary background is calculated by Tung
and Yu [531] using the covariant Noether charge formalism and the boundary terms above. As an example
they considered the Vaidya spacetime as a time-dependent perturbation of a stationary one with the
orthonormal frame field being adapted to the spherical symmetry. At null infinity they recovered the Bondi
mass-loss, while for the dynamical horizons they recovered the flux expression of Ashtekar and Krishnan
(see Section 13.3.2).
The quasi-local energy-momentum, based on Eq. (11.7) in the tetrad approach to general relativity, is
calculated for arbitrary two-surfaces
lying in the hypersurfaces of the homogeneity in all the Bianchi
cosmological models in [391] (see also [340]). In these calculations the tetrad field was chosen to be the
geometrically distinguished triad, being invariant with respect to the global action of the isometry group,
and the future-pointing unit timelike normal of the hypersurfaces; while the vector field
was chosen to
have constant components in this frame. For class A models (i.e., for I, II, VI
, VII
, VIII and IX
Bianchi types) this is zero, and for class B models (III, IV, V, VI
and VII
Bianchi models) the
quasi-local energy is negative, and the energy is proportional to the volume of the domain that is
bounded by
. (Here a sign error in the previous calculations, reported in [134, 387, 385], is
corrected.) The apparent contradiction of the nonpositivity of the energy in the present context
and the non-negativity of the energy in general small-sphere calculations indicates that the
geometrically distinguished tetrad field in the Bianchi models does not reduce to the ‘natural’
approximate translational Killing fields near a point. Another interpretation of the vanishing
and negativity of the quasi-local energy, different from this and those in Section 4.3, is also
given.
Instead of the specific boundary terms, So considered a two-parameter family of boundary terms [464],
which generalized the special expressions (11.5) – (11.6
) and (11.9
) – (11.10
). The main idea
behind this generalization is that one cannot, in general, expect to be able to control only, for
example, either the configuration or the momentum variables, rather only a combination of
them. Hence, the boundary condition is not purely of a Dirichlet or Neumann type, but rather
a more general mixed one. It is shown that, with an appropriate value for these parameters,
the resulting energy expression for small spheres is positive definite, even in the holonomic
description.
In the general covariant quasi-local Hamiltonians Chen, Nester and Tung left the reference configuration and the boundary conditions unspecified, and hence their construction was not complete. These have been specified in [386]. The key ideas are as follow.
First, because of its correct, advantagous properties (especially its asymptotic behaviour in
asymptotically flat spacetimes), Nester, Chen, Liu and Sun choose (11.7) a priori as their Hamiltonian
boundary term. Their reference configuration is chosen to be the Minkowski spacetime, and the generator
vector field is the general Killing vector (depending on ten parameters).
Next, to match the physical and the reference geometries, they require the two full 4-dimensional
metrics to coincide at the points of the two-surface (rather than only the induced two-metrics on
).
This condition leaves two unspecified functions in the quasi-local quantities. To find the ‘best matched’ such
embedding of
into the Minkowski spacetime, Nester, Chen, Liu and Sun propose to choose the one that
extremize the quasi-local mass.
This, and some other related strategies have been used to compute quasi-local energy in various spherically symmetric configurations in [135, 341, 561, 562].
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 [13, 14
]. (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. In general relativity, the variations of the tetrad fields are restricted on
by requiring in the
first case that the induced metric
is fixed and the adaptation of the tetrad field to the boundary
is preserved, while in the second case that the tetrad components
of the extrinsic
curvature of
is fixed. 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 [13]. The properties of the
co-vectors
and
(called the Dirichlet and Neumann symplectic vectors, respectively) were
investigated further in [14
]. 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 [14].
Anco continued the investigation of the Dirichlet Hamiltonian in [11], which takes the form (see also
Eqs. (8.1) and (10.20
))
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
dynamic 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 two-metric
and
fixed on
,
The boundary condition on closed untrapped spacelike two-surfaces that make the covariant
Hamiltonian functionally differentiable were investigated by Tung [526, 527
]. He showed that such a
boundary condition might be the following: the area 2-form and the mean curvature vector of
are fixed,
and the evolution vector field
is proportional to the dual mean curvature vector, where
the factor of proportionality is a function of the area 2-form. Then, requiring that the value
of the Hamiltonian reproduce the ADM energy, he recovers the Hawking energy. If, however,
is allowed to have a part tangential to
, and
is required to be fixed (up to
total
-divergences), then, though the value of the Hamiltonian is still proportional to the
Hawking energy, the factor of proportionality depends on the angular momentum, given by
(11.3
), as well. With this choice the vector field
becomes a generalization of the Kodama
vector field [321] (see also Section 4.2.1). The results of [527, 528] are extensions of those in
[526].
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 pullbacks to
of the various forms of the Nester–Witten 2-from
from
along the various local sections of the bundle [192, 358, 486, 487]. 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 Eq. (3.10
)), 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 [131, 137, 382] 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
.
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Living Rev. Relativity 12, (2009), 4
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