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*_{pass},
but *m*_{act} must be equal to *m*_{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].

@ __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]; > 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 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 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 a1504 [Lorentz force and geodesics]; Ni IJMPD(16) [and phenomenology, cosmology].

@ __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]; > s.a. physical constants ["*c* equivalence principle"]; Di Casola et al PRD(14)-a1401 [for self-gravitating bodies, and purely metric theories of gravity].

@ __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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