The first binary pulsar used to test GR in this way was PSR B1913+16 discovered by Hulse & Taylor in 1974 . Measurements of three PK parameters (, and ) were obtained from long-term timing observations at Arecibo [359, 360, 392]. The measurement of orbital decay, which corresponds to a shrinkage of about per orbit, is seen most dramatically as the gradually increasing shift in orbital phase for periastron passages with respect to a non-decaying orbit shown in Figure 26. This figure includes recent Arecibo data taken since the upgrade of the telescope in the mid 1990s.
The measurement of orbital decay in the B1913+16 system, obtained from observations spanning a 30-yr baseline , is within 0.2% of the GR prediction and provided the first indirect evidence for the existence of gravitational waves. Hulse and Taylor were awarded the 1993 Nobel Physics prize [371, 153, 356] in recognition of their discovery of this remarkable laboratory for testing GR. A similar, though less precise, test of GR from this combination of PK parameters has recently been performed in the double neutron star binary PSR B2127+11C in the globular cluster M15 . For this system, the measurement uncertainties permit a test of GR to 3% precision which is unlikely to improve due to various kinematic contaminations including the acceleration of the binary system in the cluster’s gravitational potential.
Five PK parameters have been measured for the double pulsar discussed in detail below and for the double neutron star system PSR B1534+12  where the test of GR comes from measurements of , and . In this system, the agreement between theory and observation is within 0.7% . This test will improve in the future as the timing baseline extends and a more significant measurement of can be made.
Currently the best binary pulsar system for testing GR in the strong-field regime is the double pulsar J0737–3039. In this system, where two independent pulsar clocks can be timed, five PK parameters of the 22.7-ms pulsar “A” have been measured as well as two additional constraints from the measured mass function and projected semi-major axis of the 2.7-s pulsar “B”. A useful means of summarising the limits so far is Figure 27 which shows the allowed regions of parameter space in terms of the masses of the two pulsars. The shaded regions are excluded by the requirement that . Further constraints are shown as pairs of lines enclosing permitted regions as predicted by GR. The measurement  of gives the total system mass .
The measurement of the projected semi-major axes of both orbits gives the mass ratio . The mass ratio measurement is unique to the double pulsar system and rests on the basic assumption that momentum is conserved. This constraint should apply to any reasonable theory of gravity. The intersection between the lines for and yield the masses of A and B as and . From these values, using Equations (13–16) the expected values of , , and may be calculated and compared with the observed values. These four tests of GR all agree with the theory to within the uncertainties. Currently the tightest constraint is the Shapiro delay parameter where the observed value is in agreement with GR at the 0.05% level .
Another unique feature of the double pulsar system is the interactions between the two pulsars’ radiation beams. Specifically, the signal from A is eclipsed for 30 s each orbit by the magnetosphere of B [242, 187, 261] and the radio pulses from B are modulated by the relativistic wind from A during two phases of the orbit . These provide unique insights into plasma physics  and, as shown shown in Figure 27, a measurement of relativistic spin–orbit coupling in the binary system and, hence, another constraint on GR. By careful modeling of the change in eclipse profiles of A over a four-year baseline, Breton et al.  have been able to fit a remarkably simple model 3 and determine the precession of B’s spin axis about the orbital angular momentum vector. This remarkable measurement agrees, within the 13% measurement uncertainty, to the GR prediction.
It took only two years for the double pulsar system to surpass the tests of GR possible from three decades of monitoring PSR B1913+16 and over a decade of timing PSR B1534+12. On-going precision timing measurements of the double pulsar system should soon provide even more stringent and new tests of GR. Crucial to these measurements will be the timing of the 2.7-s pulsar B, where the observed profile is significantly affected by A’s relativistic wind [242, 259]. A careful decoupling of these profile variations is required to accurately measure TOAs for this pulsar and determine the extent to which the theory-independent mass ratio can be measured. This task is compounded by the fact that pulsar B’s signal is now  significantly weaker compared to the discovery observations five years ago . This appears to be a direct result of its beam precessing out of our line of sight.
The relativistic effects observed in the double pulsar system are so large that corrections to higher post-Newtonian order may soon need to be considered. For example, may be measured precisely enough to require terms of second post-Newtonian order to be included in the computations . In addition, in contrast to Newtonian physics, GR predicts that the spins of the neutron stars affect their orbital motion via spin-orbit coupling. This effect would most clearly be visible as a contribution to the observed in a secular  and periodic fashion . For the J0737–3039 system, the expected contribution is about an order of magnitude larger than for PSR B1913+16 . As the exact value depends on the pulsars’ moment of inertia, a potential measurement of this effect would allow the moment of inertia of a neutron star to be determined for the first time . Such a measurement would be invaluable for studies of the neutron star equation of state and our understanding of matter at extreme pressure and densities .
Finally, in the neutron star–white dwarf binary J1141–6545, the measurement of , , and permit a 6% test of GR . Despite this being nominally only a relatively weak constraint for GR, the significantly different self gravities of the neutron star and white dwarf permits constraints on two coupling parameters between matter and the scalar gravitational field. These parameters are non-existent in GR, which describes gravity in terms of a tensor field, but are predicted by some alternative theories of gravity which consider tensor and scalar field components. The constraints provided by PSR J1141–6545 for the strongly non-linear coupling parameter are currently the most stringent to date and are set to improve dramatically over the next five years of timing observations.
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