Stress-Energy or Energy-Momentum Tensors  

In General > s.a. conservation laws; history of relativistic physics.
* Idea: A tensor field whose components give the energy density, momentum density, pressure and stresses of a field; For example, the total 4-momentum of the field configuration is Pm = ∫Σ Tmn dSn; Usually defined up to the addition of a curl, but it is fixed in general relativity.
$ Canonical energy-momentum tensor: For a matter field with Lagrangian density \(\cal L\),

θmn(φ, ∂φ):= {∂\(\cal L\)/∂(∂mφ)} ∂n φ – δmn\(\cal L\)(φ, ∂φ) .

* Remark: Used in Minkowski spacetime; For some fields, like the Klein-Gordon field, it agrees with the stress-energy tensor, but more generally it may not be symmetric or gauge-invariant, and may not even be conserved in curved spacetime.
$ Stress-energy tensor: For a matter field with action SM,

Tab:= –(αM / 8π) |g|–1/2 δSM / δgab .

* Restrictions: It must be conserved, i.e., satisfy ∇mT mn = 0; It is usually thought to be desirable for the stress-energy tensor to satisfy some positive-energy condition, although we know that all of these conditions can be violated.
@ General references: Goldberg PR(58); Sorkin GRG(77); in Wald 84; Hall & Negm IJTP(86); Magnano & Sokołowski CQG(02)gq/01 [symmetries]; Gamboa Saraví JPA(04)mp/03 [canonical vs metric vs Belinfante]; Forger & Römer AP(04)ht/03 [rev+]; Pons JMP(11) [Belinfante vs Hilbert, and conformal symmetry]; Lehmkuhl BJPS(11) [conceptual]; Voicu IJGMP(16)-a1511 [general geometric approach].
@ And gravity: Leclerc IJMPD(06)gq/05; Dupré a0903 [covariant expression]; Curiel a0908 [non-existence]; Ohanian a1010 [canonical energy-momentum tensor vs gravitational-field source in Einstein's equation]; Chen a1211 [inertial vs gravitational energy-momentum tensors?]; Novello & Bittencourt a1311 [in the Geometric Scalar theory]; Nikolić a1407 [the Einstein tensor as gravitational energy-momentum tensor]; Bamba & Shimizu IJGMP(16)-a1506 [from the Noether theorem]; Padmanabhan GRG(15)-a1506 [momentum density of spacetime]; Shimizu MPLA(16)-a1601 [from Noether's theorem].

Relativistic Particle
* In Minkowski space: For a particle of mass m and velocity v along the x-axis,

Tab(x) = m (1–v2)–1/2 δ(xvt) δ(y) δ(z) sisk ,   si:= δi0 + v δi1 .

@ References: de Souza JPA(97)ht/96 [self-field issues]; Blanchet & Faye JMP(01)gq/00 [regularization in pN expansions of general relativity]; Lechner & Marchetti AP(07) [N charged point-particles].

For Fields > s.a. gravitational energy-momentum; metric matching [matter shell]; Stress; stress-energy pseudotensor.
* Minimally coupled Klein-Gordon field: [@ Wald 84, pp 63+70]

Tab = ∇aφ ∇bφ – \(1\over2\)gab (∇cφ ∇cφ + m2 φ2) .

* Electromagnetic field: [@ Geroch ln(gr), p47; Hawking & Ellis 73, p68; Wald 84, pp64+70]

Tab = \(1\over4\pi\)(Fac Fbc – \(1\over4\)gab Fde Fde) ;

Its components are T 00 = energy density, T 0i = Poynting vector, T ij = Maxwell stress tensor; (Notice that both the Abraham form–kinetic momentum, related to particle properties–and the Minkowski form–canonical momentum, related to wave properties–of the momentum density are correct, but they do not coincide inside a medium).
* Perfect fluid: [@ Geroch ln(gr), p38; Hawking & Ellis 73, 69-70; Wald 84, pp62+69] If ρ = energy density, and p = pressure,

Tab = ρ uaub + p (gab + uaub) = (ρ+p) uaub + p gab .

* Imperfect fluid: [@ Coley PLA(89), > s.a. fluids] If ζ ≥ 0 is the coefficient of bulk viscosity, θ the expansion, σ the shear, η ≥ 0 the coefficient of shear viscosity, and q the heat conduction vector (q · u = 0),

Tab = ρ uaub + (p–2ζθ) (gab+uaub) – 2η σab qaub + uaqb .

@ Electromagnetic: Accioly AJP(97)sep [from field equations]; Carminati & Zakhary CQG(99) [+ fluid, Segre classification]; Gamboa Saraví JPA(02)mp; Montesinos & Flores RMF(06)mp [+ Yang-Mills + Proca, from Noether's theorem]; Navarro & Sancho AIP(09)-a1101 [characterization]; > s.a. Momentum; self-force.
@ Electromagnetic, in media: Antoci & Mihich NCB(97)gq [Abraham tensor]; Pfeifer et al RMP(07) [electromagnetic wave in a dielectric]; Ravndal a0805; Barnett PRL(10) + news pw(10)mar [resolution of Abraham-Minkowski dilemma]; Philbin PRA(11)-a1008 [dispersive media]; Jiménez et al EJP(11) [magnetic media]; Ramos et al PLA(11) [and dielectric "Einstein box" thought experiment]; Medina & Stephany a1703 [resolution of the Abraham-Minkowski controversy]; > s.a. electromagnetism in matter [Abraham and Minkowski tensors].
@ Gauge theories: Sardanashvily ht/02-conf; Deser ht/04-conf [higher-spin gauge fields]; Blaschke et al a1605-NPB.
@ Spinors: Carlson et al PRL(03) [massless spin-1/2 around static black hole]; Zhang CTP(05)mp/04 [equivalence of Belinfante's and metric Tab].
@ Conservation laws: Sardanashvily gq/94 [Hamiltonian]; Mensky PLA(04) [covariant]; Deser FP(05) [and equations of motion]; Koivisto CQG(06) [in modified gravity]; Obukhov & Rubilar PRD(06)gq [3-form, in tetrad gravity]; Mann et al CQG(08)-a0804 [for asymptotically flat spacetimes].
@ In quantum field theory, renormalized: Cannella & Sturani GRG(10)-a0808 [via effective field theory]; Barceló et al PRD(12)-a1112 [equivalence of two procedures]; > s.a. quantum field theory in curved backgrounds.
@ Other fields and topics: Deser & Jackiw IJMPB(96)ht/95 [2D scalar field, and conformal anomaly]; Muñoz AJP(96)sep [and Poincaré invariance]; Percus JMP(96) [non-local]; Magnano & Sokołowski GRG(98)gq [from field equations]; Gerhold et al ht/00 [scalar in non-commutative geometry]; Saharian PRD(04)ht/03 [boundary terms]; Giulini a1502-in [for extended objects, in special relativity]; Mukherjee et al a1609 [for gravitationally-coupled theories]; > s.a. momentum [pressure contribution, for fluids].

In Quantum Theories > s.a. quantum field theory effects; semiclassical general relativity.
* Applications: In quantum field theory in curved spacetimes, the vacuum expectation value of the stress-energy tensor is important in order to assess the importance of back-reaction effects, and as a better probe of the physical situation than a particle count.
* Properties: This expectation value can be conserved even with particle creation, if we violate the dominant energy condition.
@ General references: Hawking CMP(70); Zel'dovich & Pitaevski CMP(71); Roman PRD(86) [and weak energy condition]; Moretti CMP(03)gq/01.
@ Fluctuations: Borgman & Ford PRD(04)gq [with compact extra dimensions]; Ford & Wu AIP(08)-a0710 [physical effects].


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