Then, the predictions for , and are
where , and, to first order in , we have
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)  for further details). The quantity is related to the gravitational binding energy. 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 (Eq. (68)) is the effect of quadrupole and monopole gravitational radiation, while the second term is the effect of dipole radiation.
In order to estimate the sensitivities and , one must adopt an equation of state for the neutron stars. It is sufficient to restrict attention to relatively stiff neutron star equations of state in order to guarantee neutron stars of sufficient mass, approximately . The lower limit on required to give consistency among the constraints on , and as in Figure 6 is several hundred . The combination of and give a constraint on the masses that is relatively weakly dependent on , thus the constraint on is dominated by and is directly proportional to the measurement error in ; in order to achieve a constraint comparable to the solar system value of , the error in would have to be reduced by more than a factor of ten.
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. Unfortunately, none has been discovered to date; the dissimilar system B0655+64, with a white dwarf companion is in a highly circular orbit, making measurement of the periastron shift meaningless, and is not as relativistic as 1913+16. From the upper limit on (Table 7), one can infer at best the weak bound .
Damour and Esposito-Farèse  have generalized these results to a broad class of scalar-tensor theories. These theories are characterized by a single function of the scalar field , which mediates the coupling strength of the scalar field. For application to the solar system or to binary systems, one expands this function about a cosmological background field value :
A purely linear coupling function produces Brans-Dicke theory, with . The function acts as a potential function for the scalar field , and, if , during cosmological evolution, the scalar field naturally evolves toward the minimum of the potential, i.e. toward , , or toward a theory close to, though not precisely GR [47, 48]. Bounds on the parameters and from solar-system, binary-pulsar and gravitational wave observations (see Sec. 6.3) are shown in Figure 8 . Negative values of correspond to an unstable scalar potential; in this case, objects such as neutron stars can experience a ``spontaneous scalarization'', whereby the interior values of can take on values very different from the exterior values, through non-linear interactions between strong gravity and the scalar field, dramatically affecting the stars' internal structure and the consequent violations of SEP. On the other hand, is of little practical interest, because, with an unstable potential, cosmological evolution would presumably drive the system away from the peak where , toward parameter values that could easily be excluded by solar system experiments. On the plane shown in Figure 8, the axis corresponds to pure Brans-Dicke theory, while the origin corresponds to pure GR. As discussed above, solar system bounds (labelled ``1PN'' in Figure 8) still beat the binary pulsars. The bounds labelled ``LIGO-VIRGO'' are discussed in Sec. 6.3 .
|The Confrontation between General Relativity and
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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