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

In 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 do not want to talk 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–theory investigations, it has become clear that all physical observables should be associated with extended but finite spacetime domains [232Jump To The Next Citation Point, 231Jump 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 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 [488Jump To The Next Citation Point, 492Jump 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 nongravitational quantities in Minkowski spacetime, we devote Section 2.2 to the discussion of them and their properties.

2.2.1 The definition of 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 ω := K Tef𝜀 abc e fabc is closed, and hence, by the triviality of the third de Rham cohomology group, 3 4 H (ℝ ) = 0, it is exact: For some 2-form ∪ [K ]ab we have KeT ef𝜀fabc = 3∇ [a ∪ [K ]bc]. 1 ∨cd := − 2 ∪ [K ]ab𝜀abcd may be called a ‘superpotential’ for the conserved current 3-form ωabc. (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 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 [535Jump To The Next Citation Point].) If ∪&tidle;[K ]ab is (the dual of) another superpotential for the same current ωabc, then by ∇ [a(∪ [K ]bc] − &tidle;∪[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 two-surface in the Minkowski spacetime then the integral of ∪[K ]ab on 𝒮 is free from this ambiguity. Thus, if Σ is any smooth compact spacelike hypersurface with smooth two-boundary 𝒮, then

∮ ∫ 1 ef 1 Q𝒮 [K ] := 2 ∪[K ]ab = KeT 3!𝜀fabc (2.5 ) 𝒮 Σ
depends only on 𝒮. Hence, it is independent of the actual Cauchy surface Σ of the domain of dependence D (Σ ) because all the spacelike Cauchy surfaces for D (Σ ) have the same common boundary 𝒮. Thus, Q [K ] 𝒮 can equivalently be interpreted as being associated with the whole domain of dependence D (Σ ), and, hence, it is quasi-local in the sense of [232Jump To The Next Citation Point, 231Jump To The Next Citation Point] above. It defines the linear maps P 𝒮 : T → ℝ and J 𝒮 : Ro → ℝ by a ab Q 𝒮[K ] =: TaP 𝒮 + MabJ 𝒮, i.e., they are elements of the corresponding dual spaces. Under Lorentz rotations of the Cartesian coordinates P a𝒮 and J𝒮ab transform as a Lorentz vector and anti-symmetric tensor, respectively. Under the translation xa-↦→ xa + ηa of the origin P a 𝒮 is unchanged, but ab J𝒮 transforms as ab ab [a b] J𝒮 ↦→ J𝒮 + 2 η P 𝒮. Thus, a P𝒮 and ab J𝒮 may be interpreted as the quasi-local energy-momentum and angular momentum of the matter fields associated with the spacelike two-surface 𝒮, or, equivalently, to D (Σ ). Then the quasi-local mass and Pauli–Lubanski spin are defined, respectively, by the usual formulae m2 := η P aP b 𝒮 ab 𝒮 𝒮 and Sa := 1𝜀a PbJ cd 𝒮 2 bcd 𝒮 𝒮. (If 2 m ⁄= 0, then the dimensionally-correct definition of the Pauli–Lubanski spin is -1 a m S𝒮.) As a consequence of the definitions, a b ηabP𝒮S 𝒮 = 0 holds, i.e., if a P𝒮 is timelike then a S𝒮 is spacelike or zero, but if P𝒮a is null (i.e., m2𝒮 = 0) then Sa𝒮 is spacelike or proportional to P a 𝒮.

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

∫ a ef 1 E Σ [ξ ] := ξeT 3!𝜀fabc. (2.6 ) Σ
Then, however, the integral EΣ [ξa] does depend on the hypersurface, because it is not connected with the spacetime symmetries. In particular, the vector field ξa can be chosen to be the unit timelike normal ta of Σ. Since the component μ := 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 EΣ [t ] as ‘the quasi-local energy of the matter fields seen by the fleet of observers being at rest with respect to Σ’. Thus, E Σ[ta] defines a different concept of the quasi-local energy: While that based on Q 𝒮[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 E Σ[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, E Σ[ξa] (for non-Killing ξa) depends on ta intrinsically and is a genuine three-hypersurface rather than a two-surface integral.

If a a a Pb := δb − t tb, the orthogonal projection to Σ, 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 = μ2 + h jajb = μ2 − |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 nonspacelike vector for any future directed nonspacelike vector Va, see, e.g., [240Jump To The Next Citation Point]), then this is non-negative, and hence,

∫ ∘ ---------- 2 e 2 1-f M Σ := μ − |j | 3!t 𝜀fabc (2.7 ) Σ
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 Σ, even in general curved spacetime. However, although in Minkowski spacetime E Σ[K ] for the four translational Killing vectors gives the four components of the energy-momentum P a 𝒮, the mass M Σ is different from m 𝒮. In fact, while m 𝒮 is defined as the Lorentzian norm of a P 𝒮 with respect to the metric on the space of the translations, in the definition of M Σ the norm of the current T abtb is first taken with respect to the pointwise physical metric of the spacetime, and then its integral is taken. Nevertheless, because of more advantageous properties (see Section 2.2.3), we prefer to represent the quasi-local energy(-momentum and angular momentum) of the matter fields in the form Q [K ] 𝒮 instead of a E Σ[ξ ].

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 P0 𝒮, others to E [ta] Σ, and some to M Σ.

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, e.g., [283, 558Jump To The Next Citation Point] and references therein) the configuration and momentum variables, ϕA and πA, respectively, are fields on a connected three-manifold Σ, which is interpreted as the typical leaf of a foliation Σt of the spacetime. The foliation can be characterized on Σ by a function N, called the lapse. The evolution of the states in the spacetime is described with respect to a vector field a a a K = N t + N (‘evolution vector field’ or ‘general time axis’), where a t 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 A A Ci = Ci (ϕ ,De ϕ ,...,πA,De πA,...), i = 1,...,n, of the canonical variables and their derivatives up to some finite order, where De is the covariant derivative operator in Σ. Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form

∫ ( ) H [K ] = μN + jaN a + CiN i + DaZa dΣ. (2.8 ) Σ
Here d Σ is the induced volume element, the coefficients μ and ja are local functions of the canonical variables and their derivatives up to some finite order, the i N’s are functions on Σ, and a Z is a local function of the canonical variables and is a linear function of the lapse, the shift, the functions i N, 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 ϕA (which are partial differential equations on the spacetime with smooth coefficients for the smooth field ϕA) 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 [534Jump 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, Za 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 ∪ [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 a N, respectively. In principle, the whole analysis can be repeated quasi-locally too. However, apart from the promising achievements of [13Jump To The Next Citation Point, 14Jump To The Next Citation Point, 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.

2.2.3 Properties of the quasi-local quantities

Suppose that the matter fields satisfy the dominant energy condition. Then EΣ [ξa] is also non-negative for any nonspacelike ξa, and, obviously, E Σ[ta] is zero precisely when T ab = 0 on Σ, and hence, by the conservation laws (see, e.g., page 94 of [240Jump To The Next Citation Point]), on the whole domain of dependence D (Σ ). Obviously, M Σ = 0 if and only if a ab L := T tb is null on Σ. Then, by the dominant energy condition it is a future-pointing vector field on Σ, and ab LaT = 0 holds. Therefore, T ab on Σ has a null eigenvector with zero eigenvalue, i.e., its algebraic type on Σ is pure radiation.

The properties of the quasi-local quantities based on Q [K ] 𝒮 in Minkowski spacetime are, however, more interesting. Namely, assuming that the dominant energy condition is satisfied, one can prove [488Jump To The Next Citation Point, 492Jump To The Next Citation Point] that

  1. P𝒮a is a future directed nonspacelike vector, m2𝒮 ≥ 0;
  2. P a= 0 𝒮 if and only if T = 0 ab on D (Σ );
  3. 2 m 𝒮 = 0 if and only if the algebraic type of the matter on D (Σ ) is pure radiation, i.e., TabLb = 0 holds for some constant null vector La. Then Tab = τLaLb for some non-negative function τ. In this case P𝒮a= eLa-, where La-:= La 𝜗aa and e := ∫ τLa 1-𝜀 Σ 3! abcd;
  4. For 2 m 𝒮 = 0 the angular momentum has the form ab a b b a J 𝒮 = e L − eL, where ∫ ea := Σxaτ La31!𝜀abcd. 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 a E Σ[t ] and M Σ are integrals of functions on a hypersurface, they are obviously additive, e.g., for any two hypersurfaces Σ1 and Σ2 (having common points at most on their boundaries 𝒮1 and 𝒮2) one has E Σ1∪Σ2[ta] = E Σ1[ta] + EΣ2[ta]. On the other hand, the additivity of P𝒮a is a slightly more delicate problem. Namely, P a 𝒮1 and P a 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 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.

2.2.4 Global energy-momenta and angular momenta

If Σ 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 −4 𝒪 (r )) falloff of the energy-momentum tensor, where r is any spatial radial distance. However, the finiteness of the angular momentum and center-of-mass is not ensured by the − 4 𝒪(r ) falloff. Since the typical falloff of Tab – for the electromagnetic field, for example – is 𝒪 (r−4), 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) falloff, six global integral conditions for the leading terms of Tab must be imposed. At spatial infinity these integral conditions can be ensured by explicit parity conditions, and one can show that the ‘conservation equations’ ab T ;b = 0 (as evolution equations for the energy density and momentum density) preserve these falloff and parity conditions [497Jump 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 Tab = 𝒪(r− 4), 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

By the results of Section 2.2.4, the vanishing of the quasi-local mass, associated with a closed spacelike two-surface 𝒮, implies that the matter must be pure radiation on a four-dimensional globally hyperbolic domain D (Σ ). Thus, m 𝒮 = 0 characterizes ‘simple’, ‘elementary’ states of the matter fields. In the present section we review how these states on D (Σ ) can be characterized completely by data on the two-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 ϕ and the Yang–Mills fields, represented by the symmetric spinor fields ϕαAB, α = 1,...,N, where N is the dimension of the gauge group, the vanishing of the quasi-local mass is equivalent [498Jump To The Next Citation Point] to plane waves and the pp-wave solutions of Coleman [152], respectively. Then, the condition TabLb = 0 implies that these fields are completely determined on the whole D (Σ ) by their value on 𝒮 (in which case the spinor fields α ϕ AB are necessarily null: ϕ α = ϕ αOAOB AB, where ϕα are complex functions and OA is a constant spinor field such that L = O ¯O ′ a A A). Similarly, the null linear zero-rest-mass fields ϕ = ϕO O ...O AB...E A B E on D (Σ ) 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. two-surfaces foliating Σ, where La = ∇au, and, by the field equations, the complex functions ϕ and ϕ α turn out to be antiholomorphic [492Jump 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 (Σ ) can be characterized completely by appropriate data (the holographic data) on the ‘screen’ 𝒮.

These ‘quasi-local radiative modes’ can be used to map any continuous spinor field on D(Σ ) to a collection of holographic data. Indeed, the special radiative solutions of the form A ϕO (with fixed constant-spinor field OA), together with their complex conjugate, define a dense subspace in the space of all continuous spinor fields on Σ. 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 (Σ) by spacelike Cauchy surfaces Σt, then every spinor field on D (Σ ) can also be represented on 𝒮 by a time-dependent family of holographic data, as well [498Jump To The Next Citation Point]. This fact may be a specific manifestation in classical nongravitational physics of the holographic principle (see Section 13.4.2).

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