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, need 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].

@ __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;
> 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 11
feb 2018