Tests of GR – Equivalence Principle Violations

3 Tests of GR – Equivalence Principle Violations

Equivalence principles are fundamental to gravitational theory; for full descriptions, see, e.g., [95 ] or [149 ]. 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 [150 ], 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 [36 , 39 ] 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

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 [150

]. References: $γPPN$ , solar system: [53 ]; $β$ , solar system: [116

]; $ξ$ , solar system: [104

]; $α1$ , solar system: [96 ], pulsar: [146

]; $α2$ , solar system: [104 , 149

]; $α 3$ , solar system: [149

], pulsar: [146

]; $ζ 2$ , pulsar: [148

]; $ζ 3$ , solar system: [16 , 149

]; $η$ , $Δ net$ , solar system: [47

], pulsar: [146

]; $2 (αc1 − α0)$ , pulsar: [8

]; $˙ G ∕G$ , solar system: [47

, 113

, 62

], pulsar: [135

Parameter	Physical meaning	Solar-system test	Limit	Pulsar test	Limit
$γPPN$	Space curvature produced by unit rest mass	VLBI, light deflection; measures $\|γPPN − 1\|$	3 × 10^–4
$β$	Non-linearity in superposition law for gravity	Perihelion shift of Mercury; measures $\|β − 1\|$	3 × 10^–3
$ξ$	Preferred-location effects	Solar alignment with ecliptic	4 × 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	4 × 10^–7
$α 3$	Preferred-frame effects and non-conservation of momentum	Perihelion shift of Earth and Mercury	2 × 10^–7	Ensemble of binary pulsars	1.5 × 10^–19
$ζ 1$	Non-conservation of momentum	Combined PPN limits	2 × 10^–2
$ζ2$	Non-conservation of momentum			Limit on $¨ P$ for PSR B1913+16	4 × 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	9 × 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	6 × 10^–12 yr^–1	Changes in Chandrasekhar mass	4.8 × 10^–12 yr^–1