In the light of modern quantum field theoretical investigations it has become clear that all physical
observables should be associated with extended but finite spacetime domains [169, 168
]. 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 apparently in classical non-gravitational 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 [354
, 358
]. 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 non-gravitational quantities
in Minkowski spacetime, we devote the present section 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 class,
, 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 the next 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 [388
].) 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 2-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 2-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
non-spacelike vector for any future directed non-spacelike vector
, see for example [175
]), 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 the present 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 for example [212, 396] and references therein) the configuration and
momentum variables,
and
, respectively, are fields on a connected 3-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 [387
]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 [7
, 8
, 327] 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 non-spacelike
, and, obviously,
is zero precisely when
on
, and hence, by
the conservation laws (see for example Page 94 of [175
]), 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 [354
, 358
] 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,
i.e. for example 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 2-surfaces belong to one and the same vector space. If there were no
natural connection between the Killing vectors on different 2-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
) fall-off of the energy-momentum tensor, where
is any spatial radial distance. However, the
finiteness of the angular momentum and centre-of-mass is not ensured by the
fall-off. Since the
typical fall-off of
- for example for the electromagnetic field - is
, we may not impose faster
than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in
addition to the
fall-off, six global integral conditions for the leading terms of
must be imposed. At the 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 fall-off and parity
conditions [364
].
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.
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 [365
] to plane waves and the pp-wave solutions of
Coleman [118], respectively. Then the condition
implies that these fields are completely
determined on the whole
by their value on
(whenever 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
2-surfaces
foliating
, where
, and by the field equations the complex functions
and
turn out to be anti-holomorphic [358
]. 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 so-called 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, too [365
]. This fact may be a
specific manifestation in the classical non-gravitational physics of the holographic principle (see
Section 13.4.2).
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