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
.
![]() |
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 |