Newtonian
Gravitation| Newtonian
Gravitation |
Theory > s.a. equivalence principle; models
of spacetime structure.
* Idea: Bodies interact
through a gravitational force Fg = G mg m'g/r2
acting at a distance, and accelerations are proportional
to forces with proportionality constant equal to the inertial mass, a = F/mi.
* Masses: The force is
proportional to the gravitational masses, but as Newton knew mg 
mi,
which makes gravitation similar to "inertial" forces,
in that the acceleration is independent of each body's individual characteristics.
* Potential: If we define 
(r)
= U(r)/r for a test body, then the potential satisfies 
2 =
4
G
,
(r)
= –G
dM |r–r'|–1 = –GM/r – G D · r/r3 – 
G Qij ri rj / r5 +
... ,
where D = 
dM r'
is the dipole moment of the mass distribution (always zero wrt the center of
mass), and Qij
= dM [3 ri'rj' – r' 2
ij]
the quadrupole moment (vanishes for a spherical
mass distribution).
* Potential energy: For
an extended body in an external field, U =
(r) (r)
d3r; The self-energy
is
Uself = (r)
(r)
d3r = (1/8G)
()3 d3r +
(r) (r)
d3r .
@ References: in Ohanian & Ruffini 94; Deser AJP(05)gq/04 [from field theory]; Counihan EJP(07) [basic principles].
Special Topics > s.a. cosmology; gravitational
constant; Newton's
Theorem; teaching [weightlessness,
tides].
* Tidal force: A mass m located
at (x, y, z) with respect to a frame centered
at a point a distance r0 away
from a center of attraction of mass M, feels a tidal force
Fx = –x (GMm/r03) , Fy = –y (GMm/r03) , Fz = 2z (GMm/r03) ;
In general relativity the expression is more complicated, uses the equation
for geodesic deviation.
* Other formulations:
A (slightly generalized) geometric version is the Newton-Cartan
theory.
@ N-body problem: Volchan a0803 [Sundman-Weierstrass theorem of total collapse].
@ Other situations: Odagaki & Kawai AJP(98)
[many-particle
statistics]; Beig & Schmidt PRS(03)gq/02 [self-gravitating
extended bodies]; Teixeira phy/03 [infinite
straight line of mass]; Buchert PLA(06)gq/05 [self-gravitating
dust]; Masi AJP(07)
[compressive radial tidal forces].
@ Other formulations and issues: De Pietri et al gq/92,
CQG(95)gq/94,
CQG(95)gq/94 [generalization];
Nardone JPA(98)
[regularization]; Natario GRG(06)gq/04 [initial
value form, and warp drive].
Phenomenology > s.a. cosmological
models; phenomenology
of gravity; modifications and tests
of newtonian gravity.
@ Effects: Abramowicz et al GRG(97) [curvature of space and perihelion
precession].
@ Measurement: Kulikov qp/05 [transparency
of cold atoms]; > s.a. Eötvös Experiment; fifth
force.
> Other: see critical
collapse; orbits
in newtonian gravity.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
13 jun 2008