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2.1 Energy-momentum and angular momentum density of matter fields

2.1.1 The symmetric energy-momentum tensor

It is a widely accepted view (appearing e.g. in excellent, standard textbooks on general relativity, too) that the canonical energy-momentum and spin tensors are well-defined and have relevance only in flat spacetime, and hence usually are underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus first we introduce these quantities for the matter fields in a general curved spacetime.

To specify the state of the matter fields operationally two kinds of devices are needed: The first measures the value of the fields, while the other measures the spatio-temporal location of the first. Correspondingly, the fields on the manifold M of events can be grouped into two sharply distinguished classes: The first contains the matter field variables, e.g. finitely many (r, s)-type tensor fields PN a1...ar b1...bs, whilst the other contains the fields specifying the spacetime geometry, i.e. the metric gab in Einstein’s theory. Suppose that the dynamics of the matter fields is governed by Hamilton’s principle specified by a Lagrangian L = L (gab,P , \~/ P ,..., \~/ ... \~/ P ) m m N e N e1 ek N: If I [gab,P ] m N is the volume integral of Lm on some open domain D with compact closure then the equations of motion are sum k EN b.a..... := dIm/dPN ab...... = n=0(- )n \~/ en ... \~/ e1(@Lm/@(\~ / e1 ... \~/ enPN ab......)) = 0, the Euler-Lagrange equations. The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by

--2--dIm- @Lm-- 1 e Tab := V~ ---dgab = 2 @gab - Lmgab + 2 \~/ (sabe + sbae - saeb- sbea - seab- seba) (1) |g|
where we introduced the so-called canonical spin tensor
( ) ea sum k sum n ie @Lm ac... fi+1...fng... h... s b := (- )dei \~/ ei-1... \~/ e1 ---------------c...-- D bei+1...end...h... \~/ fi+1... \~/ fnPN g.... (2) n=1 i=1 @( \~/ e1... \~/ enPN d...)
(The terminology will be justified in the next Section 2.2.) Here Dca1...apf1...fq db1...bqe1...ep is the (p + q + 1,p + q + 1)-type invariant tensor, built from the Kronecker deltas, appearing naturally in the expression of the Lie derivative of the (p,q)-type tensor fields in terms of the torsion free covariant derivatives: ´LKPa..b.... = \~/ KPa..b....- \~/ cKdDca..d.b.f...e.....Pe.f...... (For the general idea of deriving Tab and Equation (2View Equation), see e.g. Section 3 of [175Jump To The Next Citation Point].)

2.1.2 The canonical Noether current

Suppose that the Lagrangian is weakly diffeomorphism invariant in the sense that for any vector field Ka and the corresponding local 1-parameter family of diffeomorphisms f t one has

* ab ( * ab * * ) e (ftLm)(g ,PN , \~/ ePN ,...) - Lm ftg ,ftPN ,ft \~/ ePN ,... = \~/ eB t

for some 1-parameter family of vector fields Bet = Bet(gab,PN ,...). (Lm is called diffeomorphism invariant if \~/ Be = 0 e t, e.g. when L m is a scalar.) Let Ka be any smooth vector field on M. Then, calculating the divergence a \~/ a(LmK ) to determine the rate of change of the action functional Im along the integral curves of a K, by a tedious but straightforward computation one can derive the so-called Noether identity: EN ba......L´KPN a.b.....+ 12Tab´LKgab + \~/ eCe[K] = 0, where ´LK denotes the Lie derivative along Ka, and Ca[K], the so-called Noether current, is given explicitly by

( ) Ce[K] = Be + heaKa + se[ab] + sa[be] + sb[ae] \~/ aKb. (3)
Here e B is the derivative of e B t with respect to t at t = 0, which may depend on Ka and its derivatives, and hab, the so-called canonical energy-momentum tensor, is defined by
sum k sum n ( @L ) hab := - Lmdab - (-)idae \~/ ei -1 ... \~/ e1----------m--------- \~/ b\ ~/ ei+1 ... \~/ enPN cd....... (4) n=1 i=1 i @( \~/ e1 ... \~/ enPN cd......)
Note that, apart from the term e B, the current e C [K] does not depend on higher than the first derivative of Ka, and the canonical energy-momentum and spin tensors could be introduced as the coefficients of Ka and its first derivative, respectively, in Ce[K]. (For the original introduction of these concepts, see [56Jump To The Next Citation Point57Jump To The Next Citation Point323Jump To The Next Citation Point]. If the torsion Qcab is not vanishing, then in the Noether identity there is a further term, 1 ab c 2S c´LKQ ab, where the so-called dynamical spin tensor ab S c is defined by V~ --- |g|Sabc := 2dIm/dQcab, and the Noether current has a slightly different structure [193Jump To The Next Citation Point194Jump To The Next Citation Point].) Obviously, Ce[K] is not uniquely determined by the Noether identity, because that contains only its divergence, and any identically conserved current may be added to it. In fact, Be t may be chosen to be an arbitrary non-zero (but divergence free) vector field even for diffeomorphism invariant Lagrangians. Thus, to be more precise, if e B = 0, then we call the specific combination (3View Equation) the canonical Noether current. Other choices for the Noether current may contain higher derivatives of Ka, too (see e.g. [228Jump To The Next Citation Point]), but there is a specific one containing Ka algebraically (see the Points 3 and 4 below). However, Ca[K] is sensitive to total divergences added to the Lagrangian, and, if the matter fields have gauge freedom, then in general it is not gauge invariant even if the Lagrangian is. On the other hand, ab T is gauge invariant and is independent of total divergences added to Lm because it is the variational derivative of the gauge invariant action with respect to the metric. Provided the field equations are satisfied, the Noether identity implies [56Jump To The Next Citation Point57Jump To The Next Citation Point323Jump To The Next Citation Point193Jump To The Next Citation Point194Jump To The Next Citation Point] that
  1. ab \~/ aT = 0,
  2. ab ab c[ab] a[bc] b[ac] T = h + \~/ c(s + s + s ),
  3. Ca[K] = TabKb + \~/ c((sa[cb]- sc[ab] - sb[ac])Kb), where the second term on the right is an identically conserved (i.e. divergence free) current, and
  4. a C [K] is conserved if a K is a Killing vector.

Hence ab T Kb is also conserved and can equally be considered as a Noether current. (For a formally different, but essentially equivalent introduction of the Noether current and identity, see [389Jump To The Next Citation Point215Jump To The Next Citation Point141Jump To The Next Citation Point].)

The interpretation of the conserved currents Ca[K] and TabKb depends on the nature of the Killing vector Ka. In Minkowski spacetime the 10-dimensional Lie algebra K of the Killing vectors is well known to split to the semidirect sum of a 4-dimensional commutative ideal, T, and the quotient K/T, where the latter is isomorphic to so(1,3). The ideal T is spanned by the constant Killing vectors, in which a constant orthonormal frame field {Ea } a on M, a- = 0,...,3, forms a basis. (Thus the underlined Roman indices a --, b -, … are concrete, name indices.) By a b gabE aE b = jab := diag(1, -1, -1,- 1) the ideal T inherits a natural Lorentzian vector space structure. Having chosen an origin o (- M, the quotient K/T can be identified as the Lie algebra Ro of the boost-rotation Killing vectors that vanish at o. Thus K has a ‘4 + 6’ decomposition into translations and boost-rotations, where the translations are canonically defined but the boost-rotations depend on the choice of the origin o (- M. In the coordinate system a {x } adapted to a {E a} (i.e. for which the 1-form basis dual to a {Ea } has the form a a ha = \~/ ax) the general form of the Killing vectors (or rather 1-forms) is Ka = Tahaa + Ma -b(xahba - xb-haa) for some constants Ta- and Ma- b = -Mb a. Then the corresponding canonical Noether current is Ce[K] = Ee (heaT - (hea xb - hebxa - 2se-[ab])M ) e a ab, and the coefficients of the translation and the boost-rotation parameters Ta- and Ma- b are interpreted as the density of the energy-momentum and the sum of the orbital and spin angular momenta, respectively. Since, however, the difference Ca[K] - T abKb is identically conserved and TabKb has more advantageous properties, it is T abKb that is used to represent the energy-momentum and angular momentum density of the matter fields.

Since in the de-Sitter and anti-de-Sitter spacetimes the (ten dimensional) Lie algebra of the Killing vector fields, so(1,4) and so(2,3), respectively, are semisimple, there is no such natural notion of translations, and hence no natural ‘4 + 6’ decomposition of the ten conserved currents into energy-momentum and (relativistic) angular momentum density.

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