### 5.4 Binary pulsars and scalar-tensor gravity

Making the usual assumption that both members of the system are neutron stars, and using the
methods summarized in TEGP 10 – 12 [281], one can obtain formulas for the periastron shift, the
gravitational redshift/second-order Doppler shift parameter, and the rate of change of orbital period,
analogous to Equations (78). These formulas depend on the masses of the two neutron stars, on
their self-gravitational binding energy, represented by “sensitivities” and , and on the
Brans–Dicke coupling constant . First, there is a modification of Kepler’s third law, given by
Then, the predictions for , and are
where , and, to first order in , we have
where is defined in Equation (72). The quantities and are defined by
and measure the “sensitivity” of the mass and moment of inertia of each body to
changes in the scalar field (reflected in changes in G) for a fixed baryon number N (see TEGP 11,
12 and 14.6 (c) [281] for further details). The quantity is related to the gravitational
binding energy. These sensitivities will depend on the neutron-star equation of state. Notice how
the violation of SEP in Brans–Dicke theory introduces complex structure-dependent effects in
everything from the Newtonian limit (modification of the effective coupling constant in Kepler’s
third law) to gravitational radiation. In the limit , we recover GR, and all structure
dependence disappears. The first term in (see Equation (84)) is the combined effect of
quadrupole and monopole gravitational radiation, while the second term is the effect of dipole
radiation.
Unfortunately, because of the near equality of the neutron star masses in the binary pulsar, dipole
radiation is suppressed, and the bounds obtained are not competitive with the Cassini bound on [293],
except for those generalized scalar-tensor theories, with [74]. Bounds on the parameters and
from solar system, binary pulsar, and gravitational wave observations (see Sections 5.1 and 6.3) are
found in [74].

Alternatively, a binary pulsar system with dissimilar objects, such as a white dwarf or black hole
companion, would provide potentially more promising tests of dipole radiation. In this regard, the recently
discovered binary pulsar J1141+6545, with an apparent white dwarf companion, may play an important
role. Here one can treat s_{WD} 10^{–4} as negligible. Then, from Equation (84), it is straightforward
to show that, if the timing reaches sufficient accuracy to determine to an accuracy
in agreement with the prediction of GR, then the resulting lower bound on would be

Thus, for s_{NS} 0.2, a 4 percent measurement would already compete with the Cassini bound (for
further details, see [118, 102]).