Modifications of Newtonian Gravity  

In General > s.a. newtonian gravity / covariant quantum gravity; graviton.
* Motivation: 1740s, Clairaut experiments with adding a 1/r 3 term to the force to explain quantitatively the Moon's apsidal motion; 1905, The realization that Newtonian gravitation was incompatible with special relativity; Poincaré made the first attempt to modify the theory; 2001, Interest has increased recently because of proposed higher-dimensional models with 4D spacetime as brane, and to some extent because of possible alternative to dark matter in explanations of galactic dynamics.
* Radial dependence: The only corrections to the potential consistent with relativistic field theory are Yukawa-type,

U(r) = −(G mm'/r) (1 + α exp{−r/λ}) ,

where the range λ is related to the graviton mass by mg = \(\hbar\)/λc.
@ General articles: Shelupsky AJP(85)dec [and light bending]; Cook CP(87); Milgrom Rech(88)feb; Parker & Zumberge Nat(89)nov; Onofrio MPLA(98) [motivation and tests]; Mostepanenko proc(04)gq/03 [Casimir force]; Cannella PhD-a1103 [effective field theory approach and tests of gravity]; Hansen et al IJMPD(19)-a1904-GRF [non-relativistic extension].
@ Early work on non-Newtonian effects: Heaviside 1894 [gravitomagnetism]; Forward PIRE(61); in Bodenmann PT(10)jan [Clairaut].
@ Phenomenology: Marinoni & Piazza ap/03/ApJL [GraS]; Dvali ht/04-conf [infrared]; > s.a. MOND; Yilmaz Theory.
> Phenomenology: see newtonian orbits; phenomenology of gravity [PN approximation]; Sagnac Effect; tests of general relativity; tests of newtonian gravity.

Sources of Corrections > s.a. brane phenomenology; fifth force; higher-dimensional gravity; modified gravity [non-local theory].
* Astrophysics: The Newtonian description of stars does not take into account all pressure effects and therefore can not be used in strong field regimes.
@ On curved manifolds: Abramowicz et al GRG(14)-a1303 [perihelion advance and light bending]; Vigneron PRD-a2010 [1+3 formulation].
@ General relativistic: Iorio GJI(06)gq [on Earth, and possible measurement]; Brannen IJMPD(09) [and graviton interactions]; Van den Bleeken CQG(17)-a1703 [twistless torsion from large-c expansion, for strong gravitational time dilation]; Bailey & Havert PRD(17)-a1706 [velocity-dependent inverse cubic force correction]; Hansen et al a1905-MG15 [non-relativistic expansion of the Einstein-Hilbert Lagrangian]; Accettulli et al a1911 [no corrections from quadratic curvature terms].
@ Quantum mechanical: Donoghue PRL(94); Kazakov CQG(01); Kirilin & Khriplovich JETP(02)gq; Kirillov & Turaev G&C(03)ht/01 [spacetime foam]; Ward IJMPA(05) [resummed quantum gravity]; Bailey & Kostelecký PRD(06)gq [with Lorentz symmetry violation]; Akhundov & Shiekh EJTP(08)-gq/06 [rev]; Faller PRD(08)-a0708 [from effective quantum gravity and scalar QED]; Caldwell & Grin PRL(08) [modifications below 0.05 mm and cosmology]; Modesto & Randono a1003 [and modifications to the entropy-area relation]; Wu et al PLB(16)-a1607 [gravitational interaction between polarizable objects].
@ Other sources: Antoniadis hp/99-conf; Zinoviev a1201 [causality, and planet motion]; Floratos & Leontaris JCAP(12)-a1202 [non-trivial spatial topology]; Gorbatsievich & Schmutzer IJMPE(12)-a1203 [projective unified field theory]; Diósi EPJwc(14)-a1406 [delay time]; Almeida et al JPCS(18)-a1802 [Kaluza-Klein gravitons]; > s.a.  quantum-gravity effects; string phenomenology.

Parametrized Post-Newtonian (PPN) Formalism > s.a. gravitational phenomenology; tests of general relativity.
* Idea: A framework for comparing general relativity with other metric theories, using 10 parameters which can be experimentally determined; It includes the PN approximation of (nearly) every metric theory, but only treats conservative aspects and does not include radiation; Some parameter-value ranges correspond to Lorentz-symmetry violating theories and overlap with the Standard Model Extension framework.
* Rem: Some modified-gravity theories, such as massive scalar-tensor theories, have no PPN expansion.
* Interpretation: For an isolated, spherical object, if β represents the non-linearity in gravitational attraction, and γ the amount of curvature produced by a body,

g00 = −1 + 2U ,   gij = (1 + 2γ M/R) δij + ... ,   where   U = M/R − β (M/R)2 + ...

* Status, parameter values: 2004, From Cassini data, β−1 = (1.2 ± 1.1) × 10−4; 2006, Bounds on β and γ are 10−4–10−5, various missions to improve limits by several orders of magnitude (LATOR, ASTROD, BepiColombo, GAIA) under consideration; 2006, on galactic scales, γ = 0.98 ± 0.07 (68% cl); 2010, The data that currently provide the tightest constrains on the parameterized post-Einsteinian framework come from the orbital period decay of binary pulsars; 2016, Using a 0.4-ns giant pulse from the Crab pulsar, \(\Delta\gamma < 8\times 10^{-16}\).
@ General references: in Eddington 22; Robertson in(62); Schiff SIAM(62), in(67); Nordtvedt PR(68), PR(68), PR(69); Will ApJ(71); Will & Nordtvedt ApJ(72); in Misner et al 73, Ch39; Rendall PRS(92); Kopeikin & Vlasov PRP(04) [reference frames, scalar-tensor theory]; Blanchet et al PRD(05)gq [structure, near zone]; Allemandi et al GRG(05)gq [alternative theories]; Hwang et al a1012 [cosmologically-motivated].
@ Specific theories: Alexander & Yunes PRD(07), PRL(07)ht [for Chern-Simons gravity]; Hohmann et al PRD(16)-a1607 [multiscalar-tensor gravity, parameter γ]; > s.a. bimetric gravity; brans-dicke theory; gravity; higher-order theories; lorentz-symmetry phenomenology; newton-cartan theory; theories of gravity [C-theory].
@ Constraints on values: Arminjon in(03)gq [new method]; Bolton et al PRD(06)ap [γ on galactic scales]; Yunes & Hughes PRD(10)-a1007 [constraints from binary pulsars]; Yang & Zhang a1608 [Crab pulsar]; Rivera-Tapia et al a2101 [using quantum metrology, Hong-Ou-Mandel effect].
@ Effects: Rama & Ghosh PLB(96), same as PLB(96) [values for repulsive gravity]; Schröder & Treder FP(02) [Earth-Moon system]; Arminjon PRD(05)gq, in(06)gq/05 [corrections to equations of motion and new self-acceleration].
@ Variations: Xu & Ma PLB(07) [for 5D gravity, and tests of Kaluza-Klein theory]; Clifton PRD(08)-a0801 [for fourth-order theories].
@ Experiments: Gai et al EA(12)-a1203 [GAME mission]; Verma et al ApJ-a1707 [determining β using asteroid radar astronomy].
> Related topics: see cosmological models [post-Friedmann framework]; Frame Theory; motion of test particles and orbits in newtonian gravity and relativistic gravity; self-force; sources of gravitational waves; dark matter.

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