3 Tests of GR – Equivalence Principle Violations

Equivalence principles are fundamental to gravitational theory; for full descriptions, see, e.g., [95] or [149Jump To The Next Citation Point]. Newton formulated what may be considered the earliest such principle, now called the “Weak Equivalence Principle” (WEP). It states that in an external gravitational field, objects of different compositions and masses will experience the same acceleration. The Einstein Equivalence Principle (EEP) includes this concept as well as those of Lorentz invariance (non-existence of preferred reference frames) and positional invariance (non-existence of preferred locations) for non-gravitational experiments. This principle leads directly to the conclusion that non-gravitational experiments will have the same outcomes in inertial and in freely-falling reference frames. The Strong Equivalence Principle (SEP) adds Lorentz and positional invariance for gravitational experiments, thus including experiments on objects with strong self-gravitation. As GR incorporates the SEP, and other theories of gravity may violate all or parts of it, it is useful to define a formalism that allows immediate identifications of such violations.

The parametrized post-Newtonian (PPN) formalism was developed [151] to provide a uniform description of the weak-gravitational-field limit, and to facilitate comparisons of rival theories in this limit. This formalism requires 10 parameters (γPPN, β, ξ, α1, α2, α3, ζ1, ζ2, ζ3, and ζ4), which are fully described in the article by Will in this series [150Jump To The Next Citation Point], and whose physical meanings are nicely summarized in Table 2 of that article. (Note that γPPN is not the same as the Post-Keplerian pulsar timing parameter γ.) Damour and Esposito-Farèse [36Jump To The Next Citation Point39] extended this formalism to include strong-field effects for generalized tensor-multiscalar gravitational theories. This allows a better understanding of limits imposed by systems including pulsars and white dwarfs, for which the amounts of self-gravitation are very different. Here, for instance, α1 becomes ′ ˆα1 = α1 + α1(c1 + c2) + ..., where ci describes the “compactness” of mass mi. The compactness can be written

( grav) c = − 2∂-lnmi- ≃ − 2E----- , (11 ) i ∂ ln G mc2 i
where G is Newton’s constant and grav Ei is the gravitational self-energy of mass mi, about –0.2 for a neutron star (NS) and –10–4 for a white dwarf (WD). Pulsar timing has the ability to set limits on ˆα1, which tests for the existence of preferred-frame effects (violations of Lorentz invariance); αˆ 3, which, in addition to testing for preferred-frame effects, also implies non-conservation of momentum if non-zero; and ζ2, which is also a non-conservative parameter. Pulsars can also be used to set limits on other SEP-violation effects that constrain combinations of the PPN parameters: the Nordtvedt (“gravitational Stark”) effect, dipolar gravitational radiation, and variation of Newton’s constant. The current pulsar timing limits on each of these will be discussed in the next sections. Table 1 summarizes the PPN and other testable parameters, giving the best pulsar and solar-system limits.


Table 1: PPN and other testable parameters, with the best solar-system and binary pulsar tests. Physical meanings and most of the solar-system references are taken from the compilations by Will [150Jump To The Next Citation Point]. References: γPPN, solar system: [53]; β, solar system: [116Jump To The Next Citation Point]; ξ, solar system: [104Jump To The Next Citation Point]; α1, solar system: [96], pulsar: [146Jump To The Next Citation Point]; α2, solar system: [104149Jump To The Next Citation Point]; α 3, solar system: [149Jump To The Next Citation Point], pulsar: [146Jump To The Next Citation Point]; ζ 2, pulsar: [148Jump To The Next Citation Point]; ζ 3, solar system: [16149Jump To The Next Citation Point]; η, Δ net, solar system: [47Jump To The Next Citation Point], pulsar: [146Jump To The Next Citation Point]; 2 (αc1 − α0), pulsar: [8Jump To The Next Citation Point]; ˙ G ∕G, solar system: [47Jump To The Next Citation Point113Jump To The Next Citation Point62Jump To The Next Citation Point], pulsar: [135Jump To The Next Citation Point].
Parameter

Physical meaning

Solar-system test

Limit

Pulsar test

Limit
γPPN

Space curvature produced by unit rest mass

VLBI, light deflection; measures |γPPN − 1|

× 10–4

β

Non-linearity in superposition law for gravity

Perihelion shift of Mercury; measures |β − 1|

× 10–3

ξ

Preferred-location effects

Solar alignment with ecliptic

× 10–7

α1

Preferred-frame effects

Lunar laser ranging

10–4

Ensemble of binary pulsars

1.4 × 10–4
α2

Preferred-frame effects

Solar alignment with ecliptic

× 10–7

α 3

Preferred-frame effects and non-conservation of momentum

Perihelion shift of Earth and Mercury

× 10–7

Ensemble of binary pulsars

1.5 × 10–19
ζ 1

Non-conservation of momentum

Combined PPN limits

× 10–2

ζ2

Non-conservation of momentum

Limit on ¨ P for PSR B1913+16

× 10–5
ζ3

Non-conservation of momentum

Lunar acceleration

10–8

ζ 4

Non-conservation of momentum

Not independent

η, Δ net

Gravitational Stark effect

Lunar laser ranging

10–3

Ensemble of binary pulsars

× 10–3
2 (αc1 − α0)

Pulsar coupling to scalar field

Dipolar gravitational radiation for PSR B0655+64

2.7 × 10–4
G˙∕G

Variation of Newton’s constant

Laser ranging to the Moon and Mars

× 10–12 yr–1

Changes in Chandrasekhar mass

4.8 × 10–12 yr–1

 3.1 Strong Equivalence Principle: Nordtvedt effect
 3.2 Preferred-frame effects and non-conservation of momentum
  3.2.1 Limits on αˆ1
  3.2.2 Limits on αˆ3
  3.2.3 Limits on ζ2
 3.3 Strong Equivalence Principle: Dipolar gravitational radiation
 3.4 Preferred-location effects: Variation of Newton’s constant
  3.4.1 Spin tests
  3.4.2 Orbital decay tests
  3.4.3 Changes in the Chandrasekhar mass

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