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 that the canonical energy-momentum and spin tensors are well defined and have relevance only in flat spacetime, and, hence, are usually underestimated and abandoned. However, it is only the analog of these canonical quantities that can be associated with gravity itself. Thus, we first 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 a1...ar ΦN b1...bs, whilst the second 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 ab Lm = Lm (g ,ΦN ,∇eΦN ,...,∇e1 ...∇ekΦN ). If ab Im[g ,ΦN ] is the action functional, i.e., the volume integral of Lm on some open domain D with compact closure, then the equations of motion are

N b... --1----δIm-- ∑k n (--------∂Lm---------) E a...:= ∘ ---δΦ a... = (− ) ∇en ...∇e1 ∂(∇ ...∇ Φ a...) = 0, |g| Nb... n=0 e1 en Nb...
the Euler–Lagrange equations. (Here, of course, δI ∕δΦ a... m Nb... denotes the formal variational derivative of I m with respect to the field variable a... ΦN b....) The symmetric (or dynamical) energy-momentum tensor is defined (and is given explicitly) by
2 δIm ∂Lm 1 e Tab := ∘----δgab = 2∂gab-− Lmgab + 2∇ (σabe + σbae − σaeb − σbea − σeab − σeba), (2.1 ) |g|
where we introduced the canonical spin tensor
∑k ∑n ( ∂L ) σeab := (− )iδee ∇ei−1 ...∇e1 ----------m-----c...- Δacbe......ed...fih+..1....fng...∇fi+1 ...∇fnΦN hg.(.....2..2 ) n=1 i=1 i ∂ (∇e1 ...∇en ΦN d...) i+1 n
(The terminology will be justified in Section 2.2.) Here ac1...cpg1...gq Δ bd1...dqh1...hp 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: xK Φc... = ∇K Φc...− ∇aKb Δac...g...Φh... d... d... bd...h... g.... (For the general idea behind the derivation of Tab and Eq. (2.2View Equation), see, e.g., Section 3 of [240Jump 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 one-parameter family of diffeomorphisms ϕt, one has

(ϕ∗Lm )(gab,ΦN ,∇e ΦN ,...) − Lm (ϕ∗gab,ϕ∗ΦN ,ϕ ∗∇eΦN ,...) = ∇eBe , t t t t t

for some one-parameter family of vector fields e e ab B t = B t(g ,ΦN ,...). (Lm is called diffeomorphism invariant if ∇eBet = 0, e.g., when Lm is a scalar.) Let Ka be any smooth vector field on M. Then, calculating the divergence ∇a (LmKa ) to determine the rate of change of the action functional I m along the integral curves of Ka, by a tedious but straightforward computation, one can derive the Noether identity: N b... a... 1 ab e E a...xK ΦN b...+ 2TabxKg + ∇eC [K ] = 0, where xK denotes the Lie derivative along Ka, and Ce [K ], the Noether current, is given explicitly by

Ce[K ] = B˙e + 𝜃eaK + (σe[ab] + σa[be] + σb[ae])∇ K . (2.3 ) a a b
Here ˙Be is the derivative of Bet with respect to t at t = 0, which may depend on Ka and its derivatives, and𝜃ab, the canonical energy-momentum tensor, is defined by
∑k ∑n ( ) 𝜃a := − L δa − (− )iδa∇ ...∇ -------∂Lm---------- ∇ ∇ ...∇ Φ c.... (2.4 ) b m b ei ei−1 e1 ∂(∇e1 ...∇en ΦN cd......) b ei+1 en Nd... n=1 i=1
Note that, apart from the term ˙Be, the current Ce [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 K a and its first derivative, respectively, in Ce[K ]. (For the original introduction of these concepts, see [73Jump To The Next Citation Point, 74Jump To The Next Citation Point, 438Jump To The Next Citation Point]. If the torsion c Θ ab is not vanishing, then in the Noether identity there is a further term, 1 ab c 2S cxK Θ ab, where the dynamic spin tensor ab S c is defined by ∘ --- |g|Sabc := 2δIm∕δΘcab, and the Noether current has a slightly different structure [259Jump To The Next Citation Point, 260Jump 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, e B t may be chosen to be an arbitrary nonzero (but divergence free) vector field, even for diffeomorphism-invariant Lagrangians. Thus, to be more precise, if B˙e = 0, then we call the specific combination (2.3View Equation) the canonical Noether current. Other choices for the Noether current may contain higher derivatives of Ka, as well (see, e.g., [304Jump To The Next Citation Point]), but there is a specific one containing Ka algebraically (see 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 (e.g., if the matter is a Maxwell or Yang–Mills field), then in general it is not gauge invariant, even if the Lagrangian is. On the other hand, T ab 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 [73Jump To The Next Citation Point, 74Jump To The Next Citation Point, 438Jump To The Next Citation Point, 259Jump To The Next Citation Point, 260Jump To The Next Citation Point] that

  1. ab ∇aT = 0,
  2. Tab = 𝜃ab + ∇ (σc[ab] + σa[bc] + σb[ac]) c,
  3. a ab a[cb] c[ab] b[ac] C [K ] = T Kb + ∇c((σ − σ − σ )Kb ), where the second term on the right is an identically-conserved (i.e., divergence-free) current, and
  4. Ca [K ] is conserved if Ka is a Killing vector.

Hence, T abKb 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 [536Jump To The Next Citation Point, 287Jump To The Next Citation Point, 191Jump To The Next Citation Point].)

The interpretation of the conserved currents, a C [K ] and ab T Kb, depends on the nature of the Killing vector, a K. In Minkowski spacetime the ten-dimensional Lie algebra K of the Killing vectors is well known to split into the semidirect sum of a four-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 gabEaaEbb = ηab := 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 {Eaa} (i.e., for which the one-form basis dual to {Eaa} has the form 𝜗aa = ∇axa-), the general form of the Killing vectors (or rather one-forms) is Ka = Ta-𝜗aa + Mab-(xa-𝜗ba − xb𝜗aa) for some constants T a and M = − M ab ba. Then, the corresponding canonical Noether current is e e ea ea b eb a e[ab] C [K ] = E e(𝜃 Ta-− (𝜃 x − 𝜃 x − 2σ )Mab-), and the coefficients of the translation and the boost-rotation parameters Ta- and Mab- are interpreted as the density of the energy-momentum and of 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 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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