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];
Lei et al a1903 [new form];
Baker et al NPB-a2011 [derivation methods].
@ 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];
Acquaviva et al CQG(08)-a1802 [square root of the Bel-Robinson tensor];
Curiel SHPMP(19)-a1808 [there can be no such tensor];
Dupré a2012 [Jacobi curvature tensor].
Relativistic Particles
* In Minkowski space: For a
particle of mass m and velocity v along the x-axis,
Tab(x) = m (1−v2)−1/2 δ(x−vt) 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]; Gratus et al a2005 [quadrupole].
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 ua ub + p (gab + uaub) = (ρ+p) ua ub + 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 ua ub + (p − 2ζθ) (gab + ua ub) − 2η σab − qa ub + ua qb .
@ 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 NPB(16)-a1605.
@ 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];
Giulini IJGMP(18)-a1808 [integrability and global conserved quantities].
@ 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 MPLA(18)-a1609 [for gravitationally-coupled theories];
Ilin & Paston a1807 [higher-derivative tensor fields];
> 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];
Wu et al a2104 [spacetime average].
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