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 Φ(r)
= U(r)/r for a test body, then the potential Φ satisfies the Poisson
equation ∇2Φ = 4πGρ,
Φ(r) = −G ∫ dM |r−r'|−1 = −GM/r − G D · r/r3 − \(1\over2\)G Qij ri rj / r5 + ... ,
where D = ∫ dM r'
is the dipole moment of the mass distribution (always zero with respect to the center of mass),
and Qij = ∫
dM [3 ri'rj' −
r' 2 δij]
the quadrupole moment (vanishes for a spherical mass distribution).
* Potential energy: For an
extended body in an external field, U = ∫
ρ(r) Φ(r)
d3r; The self-energy is
Uself = \(1\over2\)∫ ρ(r) Φ(r) d3r = (1/8πG) ∫ (∇Φ)3 d3r + ∫ ρ(r) Φ(r) d3r .
@ References: Deser AJP(05)aug-gq/04 [from field theory]; Counihan EJP(07) [basic principles]; Yurtsever et al a1004 [inverse problem]; in Ohanian & Ruffini 13; in Poisson & Will 14; Pereira a1903 [from its empirical basis to the theory].
Special Topics > s.a. cosmology;
gravitational constant; Newton's
Theorem; teaching [weightlessness, tides].
* Tidal forces: A mass m located
at (x, y, z) with respect to a frame centered at a point a
distance 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];
Farrés et al CMDA(13)-a1208 [high-precision symplectic integrators for the Solar System].
@ Tidal forces:
Masi AJP(07)feb [compressive radial];
Efroimsky & Williams CMDA(09)-a0803 [tidal torques];
> s.a. Love Number.
@ Other situations:
Odagaki & Kawai AJP(98)aug [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];
Ridgely EJP(11) [in material media].
@ Other formulations and issues:
De Pietri et al gq/92,
CQG(95)gq/94,
CQG(95)gq/94 [generalization];
Nardone JPA(98) [regularization];
Natário GRG(06)gq/04 [initial-value form, and warp drive];
Hansen et al PRL(19)-a1807 [action principle, gravitational time dilation];
Banerjee & Mukherjee PRD(18)-a1810 [geometric].
Phenomenology
> s.a. cosmological models [Newtonian cosmology];
phenomenology of gravity; modifications
and tests of newtonian gravity.
@ Effects: Abramowicz et al GRG(97) [curvature of space and perihelion precession];
Ferroglia & Fiolhais AJP(20)dec [tidal locking, pedagogical].
@ Specific objects: Dittrich a1609 [Dirichlet's massive homogeneous ellipsoid].
@ Measurement: Kulikov JMO(06)qp/05 [transparency of cold atoms];
Charrière et al PRA(12) [local g measurements];
Graney PT(12)sep [Giovanni Battista Riccioli];
Harms LRR(15)-a1507 [terrestrial gravity fluctuations];
news sn(19)nov [measuring gravity with trapped atoms];
> s.a. Eötvös Experiment; fifth force.
> Related topics: see critical
collapse; Newton's Theorem [shell theorem]; orbits
in newtonian gravity.
main page
– abbreviations
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send feedback and suggestions to bombelli at olemiss.edu – modified 2 feb 2021