Test-Body
Orbits in Curved Spacetime| Test-Body
Orbits in Curved Spacetime |
In General > s.a. force; particle
models [including torsion]; orbits of gravitating
bodies; self-force [radiation reaction].
* Idea: Pointlike test
particles (neutral, spinless) that do not radiate follow geodesics; 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 nonexistence 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 & Plebanski 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)
[unfamiliar trajectories]; Nandi et al EJP(06)gq/05 [from
modified Newtonian equations].
@ 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].
@ Spinning particles: Thorne & Hartle PRD(85) [corrections]; Hestenes
IJTP(86) [spinor approach]; Karpov JETP(03)gq;
> s.a. Weyl Solutions.
@ In modified theories: Mannheim GRG(93)
[and dynamical mass]; Dereli & Tucker gq/01 [in
Brans-Dicke]; > s.a. Birkhoff's Theorem, inertia.
@ 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.
In Black Hole Backgrounds > s.a. black
hole phenomenology; chaos in gravitation;
geodesics.
* In Schwarzschild: Test bodies can follow stable circular orbits
at a radial distance r depending on their angular momentum, but always
greater than 6R, where R is the Schwarzschild radius.
* Precession: Non-circular
orbits in Schwarzschild do not close; They are only approximately ellipses;
With a quadrupole moment, the main contributions are 
quadr =
6
(GM)Qm3/L4,
and gr
= 6(GM)2m2/L2c2,
both positive.
@ Particles in Schwarzschild: in Weinberg 72; Wald 84, pp 140ff; Do-Nhat PLA(98);
Dean AJP(99)
[phase plane analysis]; Mitra gq/99 [radial];
Kerner et al CGQ(01)gq [nearly
circular]; Ajith et al PRD(05)gq/04 [post-Newtonian
approximants]; Boccaletti et al re GRG(05)
[Beltrami's method]; Hall a0807 [exact results].
@ Particles in Kerr: Hartl PRD(03)gq/02,
PRD(03)gq [spinning,
(no) chaos]; Teo GRG(03)
[photons, closed]; Boccaletti et al re GRG(05)
[Beltrami's method]; Rosquist et al a0710 [meaning of Carter's constant]; > s.a. kerr
spacetime.
@ Precession: in Weinberg 72; Stump
AJP(88), comment
Doggett
AJP(91); Scales & Cornelius
JURP(93).
@ Epicyclic oscillations: Abramowicz & Kluzniak GRG(03)gq/02;
Biesiada GRG(03)gq [vs Newtonian].
> Other spacetimes: see gravitational
waves; schwarzschild; schwarzschild-de
sitter.
> Online resources:
see John Walker's page and
applet.
Other Effects and Backgrounds > s.a. classical
particle models.
* Reverse centrifugal force:
The critical distance, within which the effect is seen is (3/2) rS;
> 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)-a0708 [intuitive
approach].
@ Antigravity: Felber gq/05 [repulsive
weak field at high v], gq/05 [propulsion].
@ Periastron precession: Bini et al GRG(05)
[Weyl solutions]; > s.a. newtonian
orbits.
@ 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].
In Cosmological Backgrounds > s.a. chaos
in gravitation.
* 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
27 jul 2008