],
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 Eqs. (60
). These formulas depend on the
masses of the two neutron stars, on their self-gravitational binding
energy, represented by ``sensitivities'' s 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
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) [147] 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 [153].
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 [42] 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 [44]. 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.
- plane allowed by solar-system,
binary-pulsar, and future gravitational wave observations. A
polytropic equation of state for the neutron stars was assumed.
The shaded region is that allowed by all tests. For positive
values of , solar-system bounds (labelled 1PN) still are
the best. (From [44], ©
1998 by the American Physical Society, reproduced by permission.)
|
The Confrontation between General Relativity and Experiment Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
