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 mgmi, 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 |rr'|–1 = –GM/rG 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.

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 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]; 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 a1807 [action principle, gravitational time dilation].

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].
@ 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]; > s.a. Eötvös Experiment; fifth force.
> Related topics: see critical collapse; Newton's Theorem [shell theorem]; orbits in newtonian gravity.

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