A non-zero implies that the velocity of a binary pulsar system (relative to a “universal” background reference frame given by the Cosmic Microwave Background, or CMB) will affect its orbital evolution. In a manner similar to the effects of a non-zero , the time evolution of the eccentricity will depend on both and a term that tries to force the semi-major axis of the orbit to align with the projection of the system velocity onto the orbital plane.
The analysis proceeds in a similar fashion to that for , except that the magnitude of is now written as [37, 19]
The figure of merit for systems used to test is . As for the test, the systems must be old, so that , and must be larger than the rate of Galactic rotation. Examples of suitable systems are PSR J2317+1439 [28, 19] with a last published value of in 1996 , and PSR J1012+5307, with . This latter system is especially valuable because observations of its white-dwarf component yield a radial velocity measurement , eliminating the need to find a lower limit on an unknown quantity. The analysis of Wex  yields a limit of . This is comparable in magnitude to the weak-field results from lunar laser ranging, but incorporates strong field effects as well.
Tests of can be derived from both binary and single pulsars, using slightly different techniques. A non-zero , which implies both a violation of local Lorentz invariance and non-conservation of momentum, will cause a rotating body to experience a self-acceleration in a direction orthogonal to both its spin and its absolute velocity :
An ensemble of single (isolated) pulsars can be used to set a limit on in the following manner. For any given pulsar, it is likely that some fraction of the self-acceleration will be directed along the line of sight to the Earth. Such an acceleration will contribute to the observed period derivative via the Doppler effect, by an amount assumes random orientations of both the pulsar spin axes and velocities, and finds that, on average, , independent of the pulse period. The sign of the contribution to , however, may be positive or negative for any individual pulsar; thus, if there were a large contribution on average, one would expect to observe pulsars with both positive and negative total period derivatives. Young pulsars in the field of the Galaxy (pulsars in globular clusters suffer from unknown accelerations from the cluster gravitational potential and do not count toward this analysis) all show positive period derivatives, typically around 10–14 s/s. Thus, the maximum possible contribution from must also be considered to be of this size, and the limit is given by .
Bell  applies this test to a set of millisecond pulsars; these have much smaller period derivatives, on the order of 10–20 s/s. Here, it is also necessary to account for the “Shklovskii effect”  in which a similar Doppler-shift addition to the period derivative results from the transverse motion of the pulsar on the sky:[131, 32]. Nevertheless, once this correction (which is always positive) is applied to the observed period derivatives for isolated millisecond pulsars, a limit on on the order of 10–15 results [17, 20].
In the case of a binary-pulsar–white-dwarf system, both bodies experience a self-acceleration. The combined accelerations affect both the velocity of the centre of mass of the system (an effect which may not be readily observable) and the relative motion of the two bodies . The relative-motion effects break down into a term involving the coupling of the spins to the absolute motion of the centre of mass, and a second term which couples the spins to the orbital velocities of the stars. The second term induces only a very small, unobservable correction to and . The first term, however, can lead to a significant test of . Both the compactness and the spin of the pulsar will completely dominate those of the white dwarf, making the net acceleration of the two bodies effectively), the spin axis of the pulsar may be assumed to be aligned with the orbital angular momentum of the system, hence the net effect of the acceleration will be to induce a polarization of the eccentricity vector within the orbital plane. The forced eccentricity term may be written as
The figure of merit for systems used to test is . The additional requirements of and being larger than the rate of Galactic rotation also hold. The 95% confidence limit derived by Wex  for an ensemble of binary pulsars is , much more stringent than for the single-pulsar case.
Another PPN parameter that predicts the non-conservation of momentum is . It will contribute, along with , to an acceleration of the centre of mass of a binary system [148, 149]° since the pulsar’s discovery. In this case, the projection of along the line of sight to the Earth will change considerably over the long term, producing an effective second derivative of the pulse period. This is given by [148, 149] × 10–30 s–1 [132, 148]. This leads to an upper limit on of 4 × 10–5 . As is orders of magnitude smaller than this (see Section 3.2.2), this can be interpreted as a limit on alone. Although PSR B1913+16 is of course still observed, the infrequent campaign nature of the observations makes it difficult to set a much better limit on (J. Taylor, private communication, as cited in ). The other well-studied double-neutron-star binary, PSR B1534+12, yields a weaker test due to its orbital parameters and very similar component masses. A complication for this test is that an observed could also be interpreted as timing noise (sometimes seen in recycled pulsars ) or else a manifestation of profile changes due to geodetic precession [80, 76].
© Max Planck Society and the author(s)