Strong Equivalence Principle: Nordtvedt effect

3.1 Strong Equivalence Principle: Nordtvedt effect

The possibility of direct tests of the SEP through Lunar Laser Ranging (LLR) experiments was first pointed out by Nordtvedt [103 ]. As the masses of Earth and the Moon contain different fractional contributions from self-gravitation, a violation of the SEP would cause them to fall differently in the Sun’s gravitational field. This would result in a “polarization” of the orbit in the direction of the Sun. LLR tests have set a limit of $|η| < 0.001$ (see, e.g., [47 , 150 ]), where $η$ is a combination of PPN parameters:

The strong-field formalism instead uses the parameter $Δi$ [43 ], which for object “i” may be written as

( mgrav ) -------- = 1 + Δi minertial i ( ) ( ) Egrav ′ Egrav 2 = 1 + η mc2 i + η mc2 i + .... (13 )

Pulsar–white-dwarf systems then constrain $Δnet = Δpulsar − Δcompanion$ [43

]. If the SEP is violated, the equations of motion for such a system will contain an extra acceleration $Δnetg$ , where $g$ is the gravitational field of the Galaxy. As the pulsar and the white dwarf fall differently in this field, this $Δnetg$ term will influence the evolution of the orbit of the system. For low-eccentricity orbits, by far the largest effect will be a long-term forcing of the eccentricity toward alignment with the projection of $g$ onto the orbital plane of the system. Thus, the time evolution of the eccentricity vector will not only depend on the usual GR-predicted relativistic advance of periastron ( $˙ω$ ), but will also include a constant term. Damour and Schäfer [43

] write the time-dependent eccentricity vector as

where $e (t) R$ is the $˙ω$ -induced rotating eccentricity vector, and $e F$ is the forced component. In terms of $Δnet$ , the magnitude of $eF$ may be written as [43

, 145

]

where $g⊥$ is the projection of the gravitational field onto the orbital plane, and $a = x∕sin i$ is the semi-major axis of the orbit. For small-eccentricity systems, this reduces to

where $M$ is the total mass of the system, and, in GR, $F = 1$ and $G$ is Newton’s constant.

Clearly, the primary criterion for selecting pulsars to test the SEP is for the orbital system to have a large value of $Pb2∕e$ , greater than or equal to 10⁷ days² [145 ]. However, as pointed out by Damour and Schäfer [43 ] and Wex [145 ], two age-related restrictions are also needed. First of all, the pulsar must be sufficiently old that the $˙ω$ -induced rotation of $e$ has completed many turns and $eR(t)$ can be assumed to be randomly oriented. This requires that the characteristic age $τc$ be $≫ 2 π∕˙ω$ , and thus young pulsars cannot be used. Secondly, $ω˙$ itself must be larger than the rate of Galactic rotation, so that the projection of $g$ onto the orbit can be assumed to be constant. According to Wex [145 ], this holds true for pulsars with orbital periods of less than about 1000 days.

Figure 4: “Polarization” of a nearly circular binary orbit under the influence of a forcing vector $g$ , showing the relation between the forced eccentricity $eF$ , the eccentricity evolving under the general-relativistic advance of periastron $eR(t)$ , and the angle $𝜃$ . (After [145

].)

Converting Equation (16 ) to a limit on $Δnet$ requires some statistical arguments to deal with the unknowns in the problem. First is the actual component of the observed eccentricity vector (or upper limit) along a given direction. Damour and Schäfer [43 ] assume the worst case of possible cancellation between the two components of $e$ , namely that $|e | ≃ |e | F R$ . With an angle $𝜃$ between $g ⊥$ and $e R$ (see Figure 4 ), they write $|eF | ≤ e∕ (2 sin (𝜃∕2))$ . Wex [145 , 146 ] corrects this slightly and uses the inequality

where $e = |e|$ . In both cases, $𝜃$ is assumed to have a uniform probability distribution between 0 and $2π$ .

Next comes the task of estimating the projection of $g$ onto the orbital plane. The projection can be written as

where $i$ is the inclination angle of the orbital plane relative to the line of sight, $Ω$ is the line of nodes, and $λ$ is the angle between the line of sight to the pulsar and $g$ [43

]. The values of $λ$ and $|g |$ can be determined from models of the Galactic potential (see, e.g., [84

, 2

]). The inclination angle $i$ can be estimated if even crude estimates of the neutron star and companion masses are available, from statistics of NS masses (see, e.g., [136

]) and/or a relation between the size of the orbit and the WD companion mass (see, e.g., [112 ]). However, the angle $Ω$ is also usually unknown and also must be assumed to be uniformly distributed between 0 and $2π$ .

Damour and Schäfer [43 ] use the PSR B1953+29 system and integrate over the angles $𝜃$ and $Ω$ to determine a 90% confidence upper limit of $Δnet < 1.1 × 10−2$ . Wex [145 ] uses an ensemble of pulsars, calculating for each system the probability (fractional area in $𝜃$ – $Ω$ space) that $Δnet$ is less than a given value, and then deriving a cumulative probability for each value of $Δnet$ . In this way he derives $−3 Δnet < 5 × 10$ at 95% confidence. However, this method may be vulnerable to selection effects; perhaps the observed systems are not representative of the true population. Wex [146 ] later overcomes this problem by inverting the question. Given a value of $Δ net$ , an upper limit on $|𝜃|$ is obtained from Equation (17 ). A Monte Carlo simulation of the expected pulsar population (assuming a range of masses based on evolutionary models and a random orientation of $Ω$ ) then yields a certain fraction of the population that agree with this limit on $|𝜃 |$ . The collection of pulsars ultimately gives a limit of $Δnet < 9 × 10− 3$ at 95% confidence. This is slightly weaker than Wex’s previous limit but derived in a more rigorous manner.

Prospects for improving the limits come from the discovery of new suitable pulsars, and from better limits on eccentricity from long-term timing of the current set of pulsars. In principle, measurement of the full orbital orientation (i.e., $Ω$ and $i$ ) for certain systems could reduce the dependence on statistical arguments. However, the possibility of cancellation between $|eF|$ and $|e | R$ will always remain. Thus, even though the required angles have in fact been measured for the millisecond pulsar J0437–4715 [139 ], its comparatively large observed eccentricity of $∼$ 2 × 10^–5 and short orbital period mean it will not significantly affect the current limits.