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
*m*_{g} = \(\hbar\)/*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,

*g*_{00} = −1 + 2*U*
, *g*_{ij}
= (1 + 2*γ* *M*/*R*) δ_{ij}
+ ... , 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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