Test-Body Motion in Curved Spacetime |

**In General**
> s.a. classical particles [including torsion];
force; geodesics; motion
of gravitating bodies; self-force [radiation reaction].

* __Idea__: Pointlike test particles
(neutral, spinless) that do not radiate follow timelike geodesics (geodesic
hypothesis/principle); Charged ones feel, in addition, a Lorentz force; For spinning
bodies, their angular momentum is Fermi transported along the geodesic.

* __Historical issue__: General
relativity is the only known theory in which the equation of motion for a particle
in the field can be obtained from the field equations; This result was obtained
in 1938 by Einstein, Infeld and Hoffmann – it is an important indication of
the self-consistency of the theory, and it requires that the field equations be
non-linear; The proof uses conservation of the stress-energy tensor applied to dust;
This overrides the difficulty of non-existence of solutions of Einstein's equation
with support on a timelike line; Note that one needs for the energy density of
matter to be non-negative.

* __Corrections__: The first corrections to
geodesic motion were obtained by Einstein, Infeld & Hoffmann (Lorentz-Droste equations).

* __Other modifications__: In a theory with
torsion, one needs to distinguish between extremal lines and autoparallels.

@ __General references__:
Brillouin JPR(23)phy/00;
Einstein & Grommer SPAW(27);
Einstein et al AM(38);
Fock JPUSSR(39);
Einstein & Infeld AM(40),
CJM(49);
Infeld & Schild RMP(49);
in Infeld RMP(57);
Infeld & Plebański 60;
Dixon PTRS(74);
Geroch & Jang JMP(75);
Nevin GRG(95) [Infeld-Schild theorem],
CQG(99);
Fernández-Jambrina & Hoenselaers JMP(01) [higher-order corrections];
Boyer AJP(04)aug [unfamiliar trajectories];
Nandi et al EJP(06)gq/05 [from modified Newtonian equations];
Barone et al JGP(11);
Weatherall FP(12)-a1106 [role of energy conditions];
Weatherall FP(12) [energy conditions and the Geroch-Jang theorem];
Tamir SHPMP(12) [the geodesic principle and universality];
Yang CMP(14) [rigorous derivation];
Savickas IJMPD(14) [in terms of Newton's laws within curved geometries];
Weatherall a1810
[new approach based on "tracking"].

@ __And gravitoelectromagnetism__:
Jantzen et al in(87)gq/97.

@ __Charged particles__: Howard et al PRL(00)
+ pn(00)mar;
Balakin et al CQG(00)gq [in pp-waves and Reissner-Nordström spacetime].

@ __Spinning particles__:
Thorne & Hartle PRD(85) [corrections];
Hestenes IJTP(86) [spinor approach];
Karpov JETP(03)gq;
Hojman & Asenjo a1810 [non-geodesic circular motion];
Keresztes & Mikóczi a1907 [in rotating black-hole spacetimes];
> s.a. Weyl Solutions.

@ __Particles with internal structure__:
Sławianowski & Gołubowska RPMP(10) [curved spaces, analysis based on non-holonomic frames].

@ __In other theories__:
Mannheim GRG(93) [and dynamical mass];
Dereli & Tucker gq/01 [Brans-Dicke theory];
Puetzfeld & Obukhov PRD(08) [with non-minimal matter-curvature coupling];
Gralla PRD(10)-a1002;
Weatherall JMP(11)-a1010,
SHPMP(11)-a1106 [in Newton-Cartan theory];
Puetzfeld & Obukhov PRD(13)-a1303,
PLA(13) [with general non-minimal coupling];
Kahil a1502 [in bimetric gravity];
> s.a. Birkhoff's Theorem;
hořava-lifshitz gravity; inertia.

@ __In higher dimensions__: Dadhich et al PRD(13)-a1308
[Lovelock gravity, but not general relativity, has bound orbits around a static source].

@ __Modifications__: Almeida gq/01 [optical approach];
Accioly & Ragusa CQG(02) [semiclassical];
Dvali et al PRD(03)hp/02 [and Lunar ranging];
Guéron & Letelier ap/03 [interacting geodesics];
Arminjon PRD(05)gq;
> s.a. motion in newtonian gravity [pseudo-Newtonian limit].

**In Black Hole Backgrounds**
> s.a. black-hole phenomenology; Carter Constant;
chaos in gravitation; geodesics.

@ __General references__: Palit et al IJTP(09)-a0808 [stability of circular orbits, phase-space method];
Tejeda & Rosswog a1402
[spherically symmetric spacetimes, generalized Newtonian description];
Bhattacharya et al PRD(15)-a1412 [around static black holes, in Einstein and Lovelock gravity].

@ __Precession__: in Weinberg 72;
Stump AJP(88)dec,
comment Doggett AJP(91)sep;
Scales & Cornelius JURP(93).

@ __Epicyclic oscillations__: Abramowicz & Kluzniak GRG(03)gq/02;
Biesiada GRG(03)gq [vs Newtonian].

> __Other spacetimes__: see
gravitational waves; particles in kerr spacetimes;
schwarzschild solution; schwarzschild-de sitter
solution.

> __Online resources__:
see John Walker's page and applet.

**In Cosmological Backgrounds** > s.a. chaotic motion.

* __Issues__: Do orbits of planets expand because of the global universal
expansion? If so, at what rate? Are there any other local effects of the global expansion?

@ __With a cosmological constant__: Kerr et al CQG(03)gq [clocks and orbit precession];
> s.a. cosmological constant.

@ __And cosmological expansion__: Cooperstock et al ApJ(98)ap;
Baker ap/99-wd,
ap/00/PRD;
Bolen et al CQG(01)gq/00 [precession];
Licht gq/01 [no effect on Pioneer];
> s.a. anomalous acceleration;
relativistic cosmology [local effects].

**Other Effects and Backgrounds** > s.a. classical
particle models; kaluza-klein phenomenology.

* __Reverse centrifugal force__:
The critical distance, within which the effect is seen is (3/2) *r*_{S};
> s.a. force.

@ __Reverse centrifugal force__: Allen Nat(90)oct;
Abramowicz MNRAS(92), SA(93)mar;
Bini et al IJMPD(97)gq/01,
IJMPD(97)gq/01;
Prasanna MPLA(02) [ADM view];
Jonsson AJP(06)oct-a0708 [intuitive approach];
Dalakishvili a1110 [from the perspective of rigidly rotating observer].

@ __Antigravity__: Felber gq/05 [repulsive
weak field at high *v*], AIP(06)gq/05 [propulsion].

@ __Periastron / perihelion precession__: Bini et al GRG(05) [Weyl solutions];
Poveda & Marín a1802 [higher-order corrections];
> s.a. newtonian orbits; schwarzschild-de sitter
spacetime.

@ __Resonances__: Abramowicz et al CQG(02)gq;
Rebusco AN(05)ap [compact objects].

@ __Swimming in curved spacetime__: Blau PT(03)jun;
Guerón et al PRD(06)gq/05;
> s.a. Extended Objects.

@ __Related topics__: Rothman et al CQG(01)gq/00 [quantized holonomy];
Bini et al CQG(03) [in colliding plane waves];
Iorio NS(10)gq/05 [isotropic mass loss by the central body];
Ilhan MSc(09)-a0911 [equations of motion in terms of multipoles];
Ahmadzadegan et al PRD(14)-a1406 [atoms with internal degrees of freedom as probes];
> s.a. wormhole solutions [Einstein-Rosen bridge].

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send feedback and suggestions to bombelli at olemiss.edu – modified 6 jul 2019