Stress-Energy or Energy-Momentum Tensors

In General > s.a. conservation laws.
* 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 P^m = _Sigma T^mn dS_n; 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 ,

^m_n(,):= {/(_m)} _n – ^m_n (,) .

* Remark: Used in Minkowski spacetime; For some fields, like Klein-Gordon, 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 S_M,

T_ab:= –(_M / 8) |g|^–1/2 S_M / g^ab .

* Restrictions: Must be conserved, i.e., satisfy _mT ^mn = 0; Usually thought to be desirable that it satisfy some positive energy condition, although we know that all of them can be violated.
@ References: Goldberg PR(58); Sorkin GRG(77); in Wald 84; Hall & Negm IJTP(86); Magnano & Sokolowski CQG(02)gq/01 [symmetries]; Gamboa Saraví JPA(04)mp/03 [canonical vs metric vs Belinfante]; Forger & Römer AP(04)ht/03 [rev+]; Leclerc IJMPD(06)gq/05 [and gravity].

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

T_ab(x) = m (1–v²)^–1/2 (x–vt) (y) (z) sⁱs^k ,   sⁱ:= ⁱ₀ + v ⁱ₁ .

@ 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]

T_ab = _a b – g_ab (^c _c + m^{2 2}) .

* Electromagnetic field: [@ Geroch on general relativity, p47; Hawking & Ellis 73, p68; Wald 84, pp64+70].

T_ab = (1/4) (F_ac F_b^c – g_ab F_de F^de) .

* Perfect fluid: [@ Geroch on general relativity, p38; Hawking & Ellis 73, pp69-70; Wald 84, pp62+69] If = energy density, and p = pressure,

T_ab = u_au_b + p (g_ab + u_au_b) = (+p) u_au_b + p g_ab .

* Imperfect fluid: [@ Coley PLA(89)] If 0 is the coefficient of bulk viscosity, the expansion, the shear, 0 the coefficient of shear viscosity, and q the heat conduction vector (q · u = 0),

T_ab = u_au_b + (p–2) (g_ab+u_au_b) – 2 _ab – q_au_b + u_aq_b .

@ Electromagnetic: Antoci & Mihich NCB(97)gq [in matter, Abraham tensor]; Accioly AJP(97) [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]; Pfeifer et al RMP(07) [electromagnetic wave in a dielectric medium]; Ravndal a0805 [in matter]; > s.a. Momentum, self-force.
@ 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 T_ab].
@ 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 a0804 [for asymptotically flat spacetimes].
@ Other fields and topics: Deser & Jackiw ht/95 [2D scalar field, and conformal anomaly]; Muñoz AJP(96) [and Poincaré invariance]; Percus JMP(96) [non-local]; Magnano & Sokolowski GRG(98)gq [from field equations]; Gerhold et al ht/00 [scalar in non-commutative geometry]; Sardanashvily ht/02-in [gauge theory]; Saharian PRD(04)ht/03 [boundary terms]; Deser ht/04-in [higher-spin gauge fields]; Vishwakarma a0705 [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 a0710-in [physical effects].

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