Equivalence Principle |

**In General, Versions** > s.a. affine connections; mass;
Reference Frames [acceleration and gravity]; quantum equivalence principle.

* __Idea__: All bodies fall with the same
acceleration in a gravitational field; The force of gravity can be made to disappear
locally by going to a suitable reference frame; It motivated the development of general
relativity and is naturally implemented in geometrical theories of gravity, although
alternatives are possible.

* __History__: The heuristic principle
was introduced by Einstein in 1907 as a primary motivation for general relativity,
and formulated more precisely during his time in Prague in 1911-1912.

$ __Weak (Galileo)__: All (pointlike,
neutral) test bodies fall in the same way in a (possibly strong) gravitational
field; Gravity is like an inertial force.

$ __Weak (Newton)__: For (possibly extended)
slowly-moving bodies in weak fields, inertial and gravitational masses are proportional,
independently of composition/form.

$ __Weak Equivalence Principle II__: All
small bodies, including rotating ones, fall in the same way in a (possibly strong)
gravitational field.

* __Relationships__: When all assumptions
are satisfied, the two above versions are equivalent.

$ __Modern versions__: The only long-range
field with gravitational-strength couplings to matter is a massless spin-2 field, the
graviton; The PPN *γ* parameter is the same for all types of matter.

$ __Einstein equivalence principle__:
In a freely falling reference frame, gravity disappears locally.

* __Remark__: This principle concerns the
passive gravitational mass \(m_{\rm pass}^~\), but \(m_{\rm act}^~\) must be equal to
\(m_{\rm pass}^~\) in order for momentum to be conserved and Newton's third law to be
valid, so an exterior gravitational field is independent of what type of matter produces
it; This is more than just a statement on the gravitational effects felt by matter.

* __Strong (idea)__: All (small) objects are
equally affected by gravity in every respect; A stationary observer in a gravitational potential
*V* is indistinguishable from one moving with acceleration −∇*V* and
no gravitational field; All gravitational effects can be locally transformed away and no local
measurement can detect a gravitational field; Requires that matter be coupled to gravity
only through *g*_{ab} and
Γ_{ab}^{c},
not the curvature.

* __And general relativity__:
The weak equivalence principle is built into the theory (in fact, it is one of the three pillars
that support all metric theories of gravity), as one can see using differential geometry and the
connection to relate local Minkowski spaces; In fact, a number of features of general relativity
such as gravitational redshift, light deflection and the fact that space must be curved (and thus
the tensorial nature of the gravitational field) can be deduced from it; The strong equivalence
principle is not built in, and there are situations where it is not satisfied.

@ __Strong version__: Bertotti & Grishchuk CQG(90);
in Ohanian & Ruffini 94 [good];
Aldrovandi et al FP(03)gq/02 [with torsion].

**Violations**
> s.a. geodesics [quantum corrections]; tests
of the equivalence principle; modified lorentz symmetry.

* __Of wep__: May occur if there
are *s* = 0 and 1 particles with gravitational strength couplings
[@ Maddox Nat(91)mar],
such as those predicted by some unified theories like string theory; The best known
consequences are variation of "constants'', non-universality of free fall, and
relative drift of atomic clocks; May also induce neutrino oscillations without
the need for a neutrino mass (& P Halprin).

* __Of sep__: There are at least two
local effects (using infinitesimal-size objects) that can detect gravitational fields,
the tidal distorsion of an object, and the precession of a spinning non-spherical
gyroscope; A gravitational field implies an unambiguous, non-zero \(R^a_{\,\,bcd}\);
The strong equivalence principle fails even in Newtonian gravity; It is violated
in QED in curved spacetime, with "faster than light" photons (> see
causality violations), and by metric-affine theories
that predict vacuum birefringence (> see phenomenology).

@ __Of wep__:
Will PRL(89) [in non-symmetric gravity];
Göklü & Lämmerzahl CQG(08)-a0801 [from metric fluctuations];
Gasperini a2101-ch [gravity at finite temperature].

@ __From string dilaton__: Damour gq/97-proc,
gq/97-proc;
[Landau et al ap/03-wd].

@ __Classical charged particles__: Goto et al CQG(10)-a1007 [and radiation reaction];
Toth a1404.

@ __And cosmology__: Hui et al PRD(09)-a0905 [from modified gravity];
Hees et al a1504-proc [some cosmological consequences].

@ __Other situations__: Ellis gq/03 [leptons];
Ellis et al IJMPA(04)gq/03 [from spacetime foam];
Barrow & Scherrer PRD(04)ap [fermions vs bosons];
Hehl & Obukhov GRG(08)-a0705 [and electromagnetic coupling, axion and dilaton];
Bertolami et al PLB(07),
Le Delliou et al AIP(07)-a0709 [dark energy–dark matter interaction in A586];
Carroll et al PRL(09)-a0807 [dark-matter-induced];
Damour & Donoghue PRD(10)-a1007 [through dilaton-like scalar field];
Minazzoli PRD(18)-a1811 [matter with unconventional coupling to geometry];
Blasone et al a1812 [scalar-tensor gravity at finite temperature];
> s.a. Chameleon Field; fifth force;
scalar-tensor gravity.

**References** > s.a. Internal Relativity;
variation of constants.

@ __General__: in Dicke 64;
Klein Sci(71)jan;
Hughes CP(93) [experimental basis and consequences];
Iliev JGP(98)gq;
Camacho MPLA(99)gq [continuous quantum measurement];
Rohrlich FP(00) [critique];
Damour CRAS-gq/01 [rev];
Ghins & Budden SHPMP(01) [conceptual];
Nordtvedt gq/02 [consequences of incorporating special relativity];
Drake AJP(06)jan [and special / general relativity transition];
Fabbri in(12)-a0905 [and the geometrization of gravity];
Damour CQG(12)-a1202 [theoretical aspects];
Nobili et al AJP(13)jul [universality of free fall and gravitational redshift];
Di Casola et al AJP(15)jan-a1310 [precise formulation of the various versions, and relationships];
Brown & Read AJP(16)feb-a1512 [misconceptions];
Kapotis & Kalkanis TPT(16)oct [in class].

@ __History__: Rabinowitz IEEE(90)phy/07 [falling bodies];
Schücking & Surowitz gq/07,
Weinstein a1208 [Einstein 1907];
Janssen SHPMP(12);
> s.a. history of relativistic gravity.

@ __Geometric formulation__:
Coleman & Schmidt JMP(95);
Iliev JPA(96)gq,
JPA(97)gq;
Wesson GRG(03) [5D, weak];
Iliev gq/06-proc [and geodesic deviation].

@ __Criticisms__: Logunov et al SPU(96);
Ginzburg & Froshenko SPU(95),
SPU(96) [reply].

@ __In deformed theories__: Tkachuk PRA(12)-a1301 [and GUP, minimal length, deformed Poisson brackets];
Ghosh CQG(14)-a1303 [and GUP];
Gnatenko & Tkachuk PLA(17)-a1701 [non-commutative theories].

@ __In other theories__: Olmo PRL(07)gq/06 [in *f*(*R*) gravity theories];
Kraiselburd & Vucetich IJMPE(11)-a0902 [Bekenstein's theory];
Deruelle GRG(11)-a1104 [Nordström's scalar theory];
Sheikh-Jabbari IJMPD(11) [Lovelock and other higher-order theories];
Puetzfeld & Obukhov PRD(15)-a1505 [scalar-tensor gravity];
> s.a. gravitational energy-momentum; kaluza-klein
phenomenology; modified gravity ["ultra-strong" version].

@ __For Casimir energy__: Fulling et al PRD(07)ht;
Milton et al JPA(07)-a0705,
JPA(08)-a0710-proc,
a0810-conf;
Shajesh et al JPA(08)-a0711-proc;
Milton et al PRD(14)-a1401.

@ __And electromagnetism__: Özer gq/99;
Trzetrzelewski EPL(18)-a1504 [Lorentz force and geodesics];
Ni IJMPD(16) [and phenomenology, cosmology];
> s.a. electromagnetism in curved spacetime.

@ __Generalized__: Lyre IJMPD(00)gq [for gauge charges];
Chiao gq/02/PRL [extended, and Kramers-Kronig relations];
Mensky PLA(04) [from energy-momentum conservation];
Wiltshire PRD(08)-a0809 [cosmological];
Kopeikin a1311 [in FLRW cosmology];
Di Casola et al PRD(14)-a1401 [for self-gravitating bodies, and purely metric theories of gravity];
Hetzroni FP(20)-a2001 [in abstract spaces];
> s.a. cosmological constant problem; physical
constants ["*c* equivalence principle"].

@ __Related topics__:
't Hooft JGP(84) [and black-hole radiation];
Kreinovich & Zapatrin gq/97 [operational];
Carlip AJP(98)may-gq/99 [and kinetic energy];
Rohrlich PRD(01) [despite self-interaction];
Rodrigues & Sharif FP(01)mp/03 [and local Lorentz invariance];
Maluf et al CQG(07)-a0704 [tetrads and energy in freely falling frames];
Hui & Nicolis PRL(11)-a1009 [for scalar forces];
Hohensee et al PRL(13)-a1308 [for bound kinetic energy].

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