In the light of modern quantum-field–theory investigations, it has become clear that all physical
observables should be associated with extended but finite spacetime domains [232, 231
]. Thus, observables
are always associated with open subsets of spacetime, whose closure is compact, i.e., they are quasi-local.
Quantities associated with spacetime points or with the whole spacetime are not observable in this
sense. In particular, global quantities, such as the total energy or electric charge, should be
considered as the limit of quasi-locally–defined quantities. Thus, the idea of quasi-locality is not
new in physics. Although in classical nongravitational physics this is not obligatory, we adopt
this view in talking about energy-momentum and angular momentum even of classical matter
fields in Minkowski spacetime. Originally, the introduction of these quasi-local quantities was
motivated by the analogous gravitational quasi-local quantities [488
, 492
]. Since, however, many of
the basic concepts and ideas behind the various gravitational quasi-local energy-momentum
and angular momentum definitions can be understood from the analogous nongravitational
quantities in Minkowski spacetime, we devote Section 2.2 to the discussion of them and their
properties.
To define the quasi-local conserved quantities in Minkowski spacetime, first observe that, for any Killing
vector , the 3-form
is closed, and hence, by the triviality of the third
de Rham cohomology group,
, it is exact: For some 2-form
we have
.
may be called a ‘superpotential’ for the conserved
current 3-form
. (However, note that while the superpotential for the gravitational energy-momentum
expressions of Section 3 is a local function of the general field variables, the existence of this
‘superpotential’ is a consequence of the field equations and the Killing nature of the vector field
. The
existence of globally-defined superpotentials that are local functions of the field variables can be proven
even without using the Poincaré lemma [535
].) If
is (the dual of) another superpotential for the
same current
, then by
and
the dual superpotential is
unique up to the addition of an exact 2-form. If, therefore,
is any closed orientable spacelike two-surface
in the Minkowski spacetime then the integral of
on
is free from this ambiguity. Thus,
if
is any smooth compact spacelike hypersurface with smooth two-boundary
, then
Obviously we can form the flux integral of the current on the hypersurface even if
is not a
Killing vector, even in general curved spacetime:
If , the orthogonal projection to
, then the part
of the
energy-momentum tensor is interpreted as the momentum density seen by the observer
.
Hence,
is the square of the mass density of the matter fields, where is the spatial metric in the plane
orthogonal to
. If
satisfies the dominant energy condition (i.e.,
is a future directed
nonspacelike vector for any future directed nonspacelike vector
, see, e.g., [240
]), then this is
non-negative, and hence,
Thus, even if there is a gauge-invariant and unambiguously-defined energy-momentum density
of the matter fields, it is not a priori clear how the various quasi-local quantities should be
introduced. We will see in the second part of this review that there are specific suggestions for the
gravitational quasi-local energy that are analogous to , others to
, and some to
.
In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not
necessarily flat) spacetime (see, e.g., [283, 558] and references therein) the configuration and
momentum variables,
and
, respectively, are fields on a connected three-manifold
,
which is interpreted as the typical leaf of a foliation
of the spacetime. The foliation can be
characterized on
by a function
, called the lapse. The evolution of the states in the
spacetime is described with respect to a vector field
(‘evolution vector field’ or
‘general time axis’), where
is the future-directed unit normal to the leaves of the foliation
and
is some vector field, called the shift, being tangent to the leaves. If the matter fields
have gauge freedom, then the dynamics of the system is constrained: Physical states can be
only those that are on the constraint surface, specified by the vanishing of certain functions
,
, of the canonical variables and their derivatives up
to some finite order, where
is the covariant derivative operator in
. Then the time
evolution of the states in the phase space is governed by the Hamiltonian, which has the form
However, if we want to recover the field equations for (which are partial differential
equations on the spacetime with smooth coefficients for the smooth field
) on the phase
space as the Hamilton equations and not some of their distributional generalizations,
then the functional differentiability of
must be required in the strong sense
of [534
].1
Nevertheless, the functional differentiability (and, in the asymptotically flat case, also the existence) of
requires some boundary conditions on the field variables, and may yield restrictions on the form of
. It may happen that, for a given
, only too restrictive boundary conditions would be able to
ensure the functional differentiability of the Hamiltonian, and, hence, the ‘quasi-local phase space’ defined
with these boundary conditions would contain only very few (or no) solutions of the field equations. In this
case,
should be modified. In fact, the boundary conditions are connected to the nature of the physical
situations considered. For example, in electrodynamics different boundary conditions must be imposed if the
boundary is to represent a conducting or an insulating surface. Unfortunately, no universal
principle or ‘canonical’ way of finding the ‘correct’ boundary term and the boundary conditions is
known.
In the asymptotically flat case, the value of the Hamiltonian on the constraint surface defines the total
energy-momentum and angular momentum, depending on the nature of , in which the total divergence
corresponds to the ambiguity of the superpotential 2-form
: An identically-conserved
quantity can always be added to the Hamiltonian (provided its functional differentiability is preserved).
The energy density and the momentum density of the matter fields can be recovered as the
functional derivative of
with respect to the lapse
and the shift
, respectively. In
principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising
achievements of [13
, 14
, 442] for the Klein–Gordon, Maxwell, and the Yang–Mills–Higgs fields, as
far as we know, such a systematic quasi-local Hamiltonian analysis of the matter fields is still
lacking.
Suppose that the matter fields satisfy the dominant energy condition. Then is also non-negative
for any nonspacelike
, and, obviously,
is zero precisely when
on
, and
hence, by the conservation laws (see, e.g., page 94 of [240
]), on the whole domain of dependence
. Obviously,
if and only if
is null on
. Then, by the dominant
energy condition it is a future-pointing vector field on
, and
holds. Therefore,
on
has a null eigenvector with zero eigenvalue, i.e., its algebraic type on
is pure
radiation.
The properties of the quasi-local quantities based on in Minkowski spacetime are, however,
more interesting. Namely, assuming that the dominant energy condition is satisfied, one can
prove [488
, 492
] that
Therefore, the vanishing of the quasi-local energy-momentum characterizes the ‘vacuum state’ of the classical matter fields completely, and the vanishing of the quasi-local mass is equivalent to special configurations representing pure radiation.
Since and
are integrals of functions on a hypersurface, they are obviously additive, e.g.,
for any two hypersurfaces
and
(having common points at most on their boundaries
and
) one has
. On the other hand, the additivity of
is a slightly more
delicate problem. Namely,
and
are elements of the dual space of the translations, and hence, we
can add them and, as in the previous case, we obtain additivity. However, this additivity comes
from the absolute parallelism of the Minkowski spacetime: The quasi-local energy-momenta of
the different two-surfaces belong to one and the same vector space. If there were no natural
connection between the Killing vectors on different two-surfaces, then the energy-momenta
would belong to different vector spaces, and they could not be added. We will see that the
quasi-local quantities discussed in Sections 7, 8, and 9 belong to vector spaces dual to their own
‘quasi-Killing vectors’, and there is no natural way of adding the energy-momenta of different
surfaces.
If extends either to spatial or future null infinity, then, as is well known, the existence of the limit of
the quasi-local energy-momentum can be ensured by slightly faster than
(for example by
) falloff of the energy-momentum tensor, where
is any spatial radial distance. However, the
finiteness of the angular momentum and center-of-mass is not ensured by the
falloff. Since the
typical falloff of
– for the electromagnetic field, for example – is
, we may not
impose faster than this, because otherwise we would exclude the electromagnetic field from our
investigations. Thus, in addition to the
falloff, six global integral conditions for the leading
terms of
must be imposed. At spatial infinity these integral conditions can be ensured by
explicit parity conditions, and one can show that the ‘conservation equations’
(as
evolution equations for the energy density and momentum density) preserve these falloff and parity
conditions [497
].
Although quasi-locally the vanishing of the mass does not imply the vanishing of the matter fields
themselves (the matter fields must be pure radiative field configurations with plane wave fronts), the
vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the
mass, the fields must be plane waves, furthermore, by , they must be asymptotically
vanishing at the same time. However, a plane-wave configuration can be asymptotically vanishing only if it
is vanishing.
By the results of Section 2.2.4, the vanishing of the quasi-local mass, associated with a closed spacelike
two-surface , implies that the matter must be pure radiation on a four-dimensional globally hyperbolic
domain
. Thus,
characterizes ‘simple’, ‘elementary’ states of the matter fields. In the
present section we review how these states on
can be characterized completely by data on the
two-surface
, and how these states can be used to formulate a classical version of the holographic
principle.
For the (real or complex) linear massless scalar field and the Yang–Mills fields, represented by the
symmetric spinor fields
,
, where
is the dimension of the gauge group, the
vanishing of the quasi-local mass is equivalent [498
] to plane waves and the pp-wave solutions of
Coleman [152], respectively. Then, the condition
implies that these fields are completely
determined on the whole
by their value on
(in which case the spinor fields
are
necessarily null:
, where
are complex functions and
is a constant spinor field
such that
). Similarly, the null linear zero-rest-mass fields
on
with any spin and constant spinor
are completely determined by their value on
.
Technically, these results are based on the unique complex analytic structure of the
two-surfaces foliating
, where
, and, by the field equations, the complex functions
and
turn out to be antiholomorphic [492
]. Assuming, for the sake of simplicity, that
is
future and past convex in the sense of Section 4.1.3 below, the independent boundary data for
such a pure radiative solution consist of a constant spinor field on
and a real function
with one, and another with two, variables. Therefore, the pure radiative modes on
can be characterized completely by appropriate data (the holographic data) on the ‘screen’
.
These ‘quasi-local radiative modes’ can be used to map any continuous spinor field on to a
collection of holographic data. Indeed, the special radiative solutions of the form
(with fixed
constant-spinor field
), together with their complex conjugate, define a dense subspace in the space of
all continuous spinor fields on
. Thus, every such spinor field can be expanded by the special radiative
solutions, and hence, can also be represented by the corresponding family of holographic data. Therefore, if
we fix a foliation of
by spacelike Cauchy surfaces
, then every spinor field on
can also
be represented on
by a time-dependent family of holographic data, as well [498
]. This fact may be a
specific manifestation in classical nongravitational physics of the holographic principle (see
Section 13.4.2).
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Living Rev. Relativity 12, (2009), 4
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