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2.2 Quasi-local energy-momentum and angular momentum of the matter fields

In the next Section 3 we will see that well-defined (i.e. gauge invariant) energy-momentum and angular momentum density cannot be associated with the gravitational ‘field’, and if we want to talk not only about global gravitational energy-momentum and angular momentum, then these quantities must be assigned to extended but finite spacetime domains.

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 [169Jump To The Next Citation Point168Jump To The Next Citation Point]. 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 [354Jump To The Next Citation Point358Jump To The Next Citation Point]. 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.

2.2.1 The definition of the quasi-local quantities

To define the quasi-local conserved quantities in Minkowski spacetime, first observe that for any Killing vector Ka (- K the 3-form w := K Tefe abc e fabc is closed, and hence, by the triviality of the third de Rham cohomology class, 3 4 H (R ) = 0, it is exact: For some 2-form U [K]ab we have KeT efefabc = 3 \~/ [a U [K]bc]. 1 \/ cd := - 2 U [K]abeabcd may be called a ‘superpotential’ for the conserved current 3-form wabc. (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 a K. The existence of globally defined superpotentials that are local functions of the field variables can be proven even without using the Poincaré lemma [388Jump To The Next Citation Point].) If ~ U [K]ab is (the dual of) another superpotential for the same current wabc, then by \~/ [a( U [K]bc] - ~ U [K]bc]) = 0 and H2(R4) = 0 the dual superpotential is unique up to the addition of an exact 2-form. If therefore S is any closed orientable spacelike 2-surface in the Minkowski spacetime then the integral of U [K]ab on S is free from this ambiguity. Thus if S is any smooth compact spacelike hypersurface with smooth 2-boundary S, then

gf integral Q [K] := 1 U [K] = K T ef 1e (5) S 2 S ab S e 3! fabc
depends only on S. Hence it is independent of the actual Cauchy surface S of the domain of dependence D(S) because all the spacelike Cauchy surfaces for D(S) have the same common boundary S. Thus QS [K] can equivalently be interpreted as being associated with the whole domain of dependence D(S), and hence quasi-local in the sense of [169Jump To The Next Citation Point168Jump To The Next Citation Point] above. It defines the linear maps : PS T --> R and : JS Ro --> R by : a ab QS [K] = Ta PS + Ma -bJS, i.e. they are elements of the corresponding dual spaces. Under Lorentz rotations of the Cartesian coordinates, a PS and ab JS transform as a Lorentz vector and anti-symmetric tensor, respectively, whilst under the translation xa-'--> xa-+ ja of the origin, P a S is unchanged while J ab'--> J ab + 2j[aPb] S S S. Thus P a S and J ab S may be interpreted as the quasi-local energy-momentum and angular momentum of the matter fields associated with the spacelike 2-surface S, or, equivalently, to D(S). Then the quasi-local mass and Pauli-Lubanski spin are defined, respectively, by the usual formulae a b m2S := ja bPS PS and a b cd SS := 12ea bcdPS JS. (If m2 /= 0, then the dimensionally correct definition of the Pauli-Lubanski spin is m1SaS.) As a consequence of the definitions j Pa Sb = 0 a b S S holds, i.e. if P a S is timelike then Sa S is spacelike or zero, but if a PS is null (i.e. 2 m S = 0) then a SS is spacelike or proportional to a P S.

Obviously, we can form the flux integral of the current T abqb on the hypersurface even if qa is not a Killing vector, even in general curved spacetime:

integral a ef 1 ES [q ] := qeT 3!efabc. (6) S
Then, however, the integral ES[qa] does depend on the hypersurface, because this is not connected with the spacetime symmetries. In particular, the vector field qa can be chosen to be the unit timelike normal ta of S. Since the component m := Tabtatb of the energy-momentum tensor is interpreted as the energy-density of the matter fields seen by the local observer a t, it would be legitimate to interpret the corresponding integral a ES[t ] as ‘the quasi-local energy of the matter fields seen by the fleet of observers being at rest with respect to S’. Thus ES[ta] defines a different concept of the quasi-local energy: While that based on QS [K] is linked to some absolute element, namely to the translational Killing symmetries of the spacetime and the constant timelike vector fields can be interpreted as the observers ‘measuring’ this energy, a ES[t ] is completely independent of any absolute element of the spacetime and is based exclusively on the arbitrarily chosen fleet of observers. Thus, while P aS is independent of the actual normal ta of S, ES[qa] (for non-Killing qa) depends on ta intrinsically and is a genuine 3-hypersurface rather than a 2-surface integral.

If a a a Pb := db - t tb, the orthogonal projection to S, then the part a a bc j := P b T tc of the energy-momentum tensor is interpreted as the momentum density seen by the observer ta. Hence

(t Tab)(tT cd)g = m2 + h jajb = m2 - |ja| 2 a c bd ab

is the square of the mass density of the matter fields, where hab is the spatial metric in the plane orthogonal to ta. If Tab satisfies the dominant energy condition (i.e. T abVb is a future directed non-spacelike vector for any future directed non-spacelike vector Va, see for example [175Jump To The Next Citation Point]), then this is non-negative, and hence

integral V~ ---------- 2 e 2 1-f MS := m - |j | 3!t efabc (7) S
can also be interpreted as the quasi-local mass of the matter fields seen by the fleet of observers being at rest with respect to S, even in general curved spacetime. However, although in Minkowski spacetime ES[K] for the four translational Killing vectors gives the four components of the energy-momentum a P S, the mass MS is different from mS. In fact, while mS is defined as the Lorentzian norm of a PS with respect to the metric on the space of the translations, in the definition of MS first the norm of the current T abtb is taken with respect to the pointwise physical metric of the spacetime, and then its integral is taken. Nevertheless, because of the more advantageous properties (see Section 2.2.3 below), we prefer to represent the quasi-local energy(-momentum and angular momentum) of the matter fields in the form QS[K] instead of a ES[q ].

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 P0S, others to ES[ta] and some to MS.

2.2.2 Hamiltonian introduction of the quasi-local quantities

In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see for example [212396Jump To The Next Citation Point] and references therein) the configuration and momentum variables, fA and pA, respectively, are fields on a connected 3-manifold S, which is interpreted as the typical leaf of a foliation S t of the spacetime. The foliation can be characterized on S by a function N, called the lapse. The evolution of the states in the spacetime is described with respect to a vector field Ka = N ta + N a (‘evolution vector field’ or ‘general time axis’), where ta is the future directed unit normal to the leaves of the foliation and N a 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 Ci = Ci(fA, DefA, ...,pA,DepA, ...), i = 1,...,n, of the canonical variables and their derivatives up to some finite order, where De is the covariant derivative operator in S. Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form

integral H [K] = (mN + j N a + C N i + D Za)dS. (8) S a i a
Here dS is the induced volume element, the coefficients m and ja are local functions of the canonical variables and their derivatives up to some finite order, the i N’s are functions on S, and Za is a local function of the canonical variables, the lapse, the shift, the functions N i, and their derivatives up to some finite order. The part CiN i of the Hamiltonian generates gauge motions in the phase space, and the functions N i are interpreted as the freely specifiable ‘gauge generators’.

However, if we want to recover the field equations for fA (which are partial differential equations on the spacetime with smooth coefficients for the smooth field fA) on the phase space as the Hamilton equations and not some of their distributional generalizations, then the functional differentiability of H[K] must be required in the strong sense of [387Jump To The Next Citation Point]1. Nevertheless, the functional differentiability (and, in the asymptotically flat case, also the existence) of H[K] requires some boundary conditions on the field variables, and may yield restrictions on the form of Za. It may happen that for a given Za 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 a Z 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 Ka, in which the total divergence DaZa corresponds to the ambiguity of the superpotential 2-form U [K]ab: 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 H[K] with respect to the lapse N and the shift N a, respectively. In principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising achievements of [7Jump To The Next Citation Point8Jump To The Next Citation Point327] 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.

2.2.3 Properties of the quasi-local quantities

Suppose that the matter fields satisfy the dominant energy condition. Then a ES[q ] is also non-negative for any non-spacelike qa, and, obviously, ES[ta] is zero precisely when T ab = 0 on S, and hence, by the conservation laws (see for example Page 94 of [175Jump To The Next Citation Point]), on the whole domain of dependence D(S). Obviously, M = 0 S if and only if La := T abt b is null on S. Then by the dominant energy condition it is a future pointing vector field on S, and ab LaT = 0 holds. Therefore, ab T on S has a null eigenvector with zero eigenvalue, i.e. its algebraic type on S is pure radiation.

The properties of the quasi-local quantities based on QS [K] in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [354Jump To The Next Citation Point358Jump To The Next Citation Point] that

  1. a PS is a future directed nonspacelike vector, m2S > 0;
  2. PSa= 0 if and only if Tab = 0 on D(S);
  3. m2 = 0 S if and only if the algebraic type of the matter on D(S) is pure radiation, i.e. b TabL = 0 holds for some constant null vector a L. Then Tab = tLaLb for some non-negative function t, whenever a P S = eLa-, where La := Lahaa and integral e := StLa 13!eabcd;
  4. For m2 = 0 S the angular momentum has the form Ja b= eaLb - ebLa- S, where a integral a a 1 e := Sx tL 3!eabcd. Thus, in particular, the Pauli-Lubanski spin is zero.

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 ES[ta] and MS are integrals of functions on a hypersurface, they are obviously additive, i.e. for example for any two hypersurfaces S1 and S2 (having common points at most on their boundaries S1 and S2) one has a a a ES1U S2[t ] = ES1[t ] + ES2[t ]. On the other hand, the additivity of a P S is a slightly more delicate problem. Namely, a PS1 and a P S2 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.

2.2.4 Global energy-momenta and angular momenta

If S 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 O(r -3) (for example by O(r -4)) fall-off of the energy-momentum tensor, where r is any spatial radial distance. However, the finiteness of the angular momentum and centre-of-mass is not ensured by the O(r - 4) fall-off. Since the typical fall-off of Tab - for example for the electromagnetic field - is -4 O(r ), we may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations. Thus, in addition to the O(r -4) fall-off, six global integral conditions for the leading terms of Tab 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’ T ab = 0 ;b (as evolution equations for the energy density and momentum density) preserve these fall-off and parity conditions [364Jump To The Next Citation Point].

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 - 4 Tab = O(r ) they must be asymptotically vanishing at the same time. However, a plane wave configuration can be asymptotically vanishing only if it is vanishing.

2.2.5 Quasi-local radiative modes and a classical version of the holography for matter fields

UpdateJump To The Next Update Information By the results of the previous Section 2.2.4 the vanishing of the quasi-local mass, associated with a closed spacelike 2-surface S, implies that the matter must be pure radiation on a 4-dimensional globally hyperbolic domain D(S). Thus mS = 0 characterizes ‘simple’, ‘elementary’ states of the matter fields. In the present section we review how these states on D(S) can be characterized completely by data on the 2-surface S, 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 f and the Yang-Mills fields, represented by the symmetric spinor fields a fAB, a = 1,...,N, where N is the dimension of the gauge group, the vanishing of the quasi-local mass is equivalent [365Jump To The Next Citation Point] to plane waves and the pp-wave solutions of Coleman [118], respectively. Then the condition TabLb = 0 implies that these fields are completely determined on the whole D(S) by their value on S (whenever the spinor fields fa AB are necessarily null: a a f AB = f OAOB, where a f are complex functions and OA is a constant spinor field such that La = OAOA'). Similarly, the null linear zero-rest-mass fields fAB...E = fOAOB ...OE on D(S) with any spin and constant spinor OA are completely determined by their value on S. Technically, these results are based on the unique complex analytic structure of the u = const. 2-surfaces foliating S, where L = \~/ u a a, and by the field equations the complex functions f and fa turn out to be anti-holomorphic [358Jump To The Next Citation Point]. Assuming, for the sake of simplicity, that S 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 S and a real function with one and another with two variables. Therefore, the pure radiative modes on D(S) can be characterized completely by appropriate data (the so-called holographic data) on the ‘screen’ S.

These ‘quasi-local radiative modes’ can be used to map any continuous spinor field on D(S) to a collection of holographic data. Indeed, the special radiative solutions of the form fOA (with fixed constant spinor field OA) together with their complex conjugate define a dense subspace in the space of all continuous spinor fields on S. 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 D(S) by spacelike Cauchy surfaces St, then every spinor field on D(S) can also be represented on S by a time dependent family of holographic data, too [365Jump To The Next Citation Point]. 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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