Classical Relativistic Particles| Classical Relativistic Particles |
In General > s.a. particles; particle
models / geodesics; self-force;
test-body motion.
$ Action, reparametrization-invariant:
Two versions, which give the trajectory only as a locus of points, are
S1[x] = m0c ∫ ds |gab x·a x·b|1/2 = ∫ dt (pa x·a − \(1\over2\)g p2) or S2[x] = \(1\over2\)∫ dt (u−1 gab x·a x·b − um2) ;
Here, S1 (Jacobi-type) is proportional
to the world-line length, g(t) is a 1D metric/Lagrange multiplier,
as in 1D gravity with scalars, and u(t) is an additional variable.
$ Action, proper-time-gauge:
S[x] = \(1\over2\)m0 ∫ dt gab x·a x·b .
* Equation of motion: If a particle is coupled to a field, its equation of motion is just the expression of momentum exchange between particle and field, so it can be obtained from ∇a Ttotab = 0; For a scalar particle in electromagnetic field and linearized gravity, respectively,
dpm/dτ = q Fmn un = (q/m) Fmn pn , dpm/dτ = (κ/2) m hab,m uaub .
* Issue: The descriptions
using t and the proper time τ are not equivalent
[@ Kalman PR(61);
Sonego PRA(91)].
* Issue: If a point particle
is included among the field sources, the treatment cannot be made fully consistent.
@ General references: Walstad AJP(18)oct-a1512 [momentum and kinetic energy, thought experiment].
@ Lagrangian formulation: Potgieter AJP(83)jan,
comment Berger AJP(84)may;
Desloge & Eriksen AJP(85)jan.
@ In curved spacetime: Muñoz IJTP(77) [weak-field approximation, Lorentz-force form];
Modanese JMP(92) [fluctuating gravitational field];
Piechocki CQG(03)gq/02 [de Sitter, different topologies];
Bini et al CQG(03)gq/02 [in gravitational wave collision];
Barrabès & Hogan CQG(04)gq/03 [deflection];
Chicone & Mashhoon CQG(05)gq/04 [in Fermi coordinates];
Fukumoto et al PTP(06)gq [finite-size, fast-moving];
in Franklin 10;
Sardanashviky IJGMP(10) [in terms of jets of one-dimensional submanifolds];
Arraut et al CEJP(11)-a1005 [static spherically-symmetric metrics];
Corichi IJMPD(15)-a1207 [stationary black-hole background, energy];
> s.a. kerr spacetimes; kerr-newman
solutions; scattering.
@ Interacting: Bergmann & Komar GRG(82);
Tretyak & Nazarenko CondMP(00)ht;
Damour et al PLB(01)gq [3PN];
Lompay ht/05;
Tarasov AP(10) [non-Hamiltonian, subject to a general force];
Alesci & Arzano PLB(11)-a1108 [coupled to 3D Einstein gravity];
Novello & Bittencourt GRG(13)-a1201 [accelerated motions as geodesics in dragged metrics].
@ Geometric: Balachandran et al JMP(84);
Freidel et al PRD(07)
[Dirac observables and effective non-commutative geometry];
Chanda & Guha IJGMP(18)-a1706.
@ Related topics: Gill et al IJTP(93),
IJTP(98) [proper-time formulation];
Parrott gq/02 [Rohrlich equation];
Uggerhøj RMP(05) [in strong crystalline fields];
Russo & Townsend JPA(09) [jerk, snap, and higher-order derivatives];
Prosekin et al PRD(15)-a1506 [propagation, transition from ballistic to diffusion regime];
> s.a. computational physics areas.
> Other topics: see
gravitomagnetism; mass;
non-commutative physics;
radiation; scalar field theories;
specific heat; statistical-mechanical systems.
Variations and Generalizations > s.a. dissipative system;
quantum particles [including superparticle]; spinning particles.
$ Charged particles:
The length-of-worldline action becomes
S[x] = ∫ ds [m0c |gab x·a x·b|1/2 + q Aa x·a] .
* Modified Lagrangian:
It can depend on the curvature of the worldline (rigidity) and its torsion;
> s.a. higher-order lagrangians.
@ Charged particles: Hyman AJP(97)mar,
Muñoz AJP(97)may [with constant fields];
Horwitz ht/98 [Lorentz force equation from Stückelberg mechanics];
Aldaya et al JPA(02) [group cohomology];
Carvalho et al IJMPA(04)gq [with defect distribution];
Timoumi RPMP(04) [closed trajectories];
Marmo & Tulczyjew RPMP(06) [Poincaré-covariant, and T reflection];
Rohrlich 07;
Chin JMP(09)-a0809 [constant field];
Boozer JPA(08) [advanced effects];
Arrayás & Trueba JPA(10) [in a knotted electromagnetic field];
Kar & Rajeev AP(11)-a1010 [with magnetic moment, and radiation reaction];
Gallo & Moreschi PRD(12)-a1112 [without explicitly divergent quantities];
Torromé IJGMP-a1207 [pointlike];
Franklin & LaMont BJP(14)-a1310
[two oppositely charged particles in 1D, with retarded potentials and no radiation reaction];
Azzurli & Lechner AJP(14)aug [massless];
Nekouee et al a1711 [in deformed phase space, Lagrangian];
Kiessling & Tahvildar-Zadeh IJMPD(19)-a1906-MG15;
> s.a. electromagnetism; modified electromagnetism
[non-linear]; self-force [including preacceleration].
@ Charged particles, radiation:
Lidsky TMP(05) [radiating];
Franklin & Griffiths AJP(14)aug [particle in hyperbolic motion];
Garfinkle a1901 [uniform acceleration].
@ Charged particles, curved spacetime: Frolov & Shoom PRD(10) [near a weakly magnetized Schwarzschild black hole];
Shiose et al PRD(14)-a1409 [around a weakly-magnetized rotating black hole];
Noble & Jentschura PRA(16)-a1603 [in Reissner-Nordström spacetime].
@ With torsion:
Barros et al CQG(05) [and rigidity];
Gaitan et al a0912-conf;
> s.a. torsion in physics.
@ In gravity's rainbow spacetime: Ling et al MPLA(07)gq/06;
Garattini & Mandanici PRD(12)-a1109.
@ In generalized backgrounds:
Johnson & Hu PRD(02)
[modified Abraham-Lorentz-Dirac equation in a quantum field];
Cabo-Bizet & Cabo Montes de Oca PLA(06) [stochastic medium];
Punzi et al PRD(09)-a0901 [in area-metric manifolds, Finsler norm];
Stern PLA(11) [on quantum spacetime];
Calcagni PRD(13)-a1306 [multiscale spacetimes];
> s.a. kaluza-klein phenomenology; non-commutative
geometry [Snyder spacetime]; types of field theories [double field theory].
@ Frenet-Serret formalism: Arreaga et al CQG(01)ht [Frenet-Serret curvature action];
Bini et al CQG(06)-a1408 [for null world lines].
@ Other generalizations: Pavón JMP(01)qp [stochastic dynamics];
Sonego & Pin JMP(09)-a0812 [anisotropic];
Markov et al JPG(10)-a1008 [with color charge];
Lahiri & Lee a1011 [with non-Abelian charge];
Tarakanov a1010-conf
[internal degrees of freedom, potential depending on v and a];
Arzano & Kowalski-Glikman CQG(11) [with de Sitter momentum space];
Amelino-Camelia et al PRD(12)-a1206
[in de Sitter spacetime, with deformed Lorentz symmetry and relative locality];
Romero & Vergara MPLA(15)-a1501
[Lifshitz field theories, Snyder non-commutative spacetime and momentum-dependent metric];
Muñoz-Díaz & Alonso-Blanco a1907 [composite systems, description];
> s.a. classical systems [two particles].
Coupled to Gravity
> s.a. 3D gravity; dynamics of gravitating
particles; particle models [point particles]; spinning particles;
test-body motion.
@ Non-spinning: Kaniel & Itin gq/01;
Blanchet & Faye JMP(01)gq/00 [regularization].
@ Charged: Rosen AP(62) [field of particle in motion];
Khriplovich & Pomeransky SHEP(99)gq/98-conf;
Khriplovich ht/00-proc [spinning];
Das et al mp/05 [review];
Gorbatenko TMP(05)
[Einstein-Infeld-Hoffmann, order (v/c)3].
@ With torsion: Fiziev & Kleinert gq/96 [action];
Kleinert & Pelster GRG(99)gq/96 [autoparallels];
Barros e Sá gq/97;
Geyer et al IJMPA(00)ht/99;
Pezzaglia gq/99/IJTP;
Arroyo et al GRG(04)ht/03;
Barros et al IJMPA(04).
> Related topics: see multipoles
[extended objects]; orbits of gravitating bodies; quantum
particles; radiation.
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