Curved spacetime Gravitational Radiation
in the Einstein, Brans-Dicke and Rosen bi-metric theories
Curved spacetime

Joseph H. Taylor Jr. and Russell A. Hulse shared the Nobel Prize in Physics in 1993 "for the discovery of a new type of pulsar [PSR 1913+16], a discovery that has opened up new possibilities for the study of gravitation."   The presentation speech states, in part

One of the most fascinating predictions of relativity theory is that massive objects in vehement motion emit a new kind of radiation, known as gravitational radiation.   This phenomenon is also described as a wave motion, as ripples in the curvature of space-time, and we speak of "gravitational waves."

No one has yet succeeded in recording a gravitational wave in a terrestrial or extraterrestrial receiver, but the Hulse-Taylor pulsar has convinced us that this type of radiation actually exists.   This is because the orbiting period of the pulsar around its companion gradually diminishes with time extremely little, but in exactly the way the general theory of relativity predicts, as a result of the emission of gravitational waves.

Download (72 pages, 242 Kb) a copy, in Adobe Acrobat (PDF) format, of a paper by Warren F. Davis, Ph.D., in which the equations of gravitational radiation are derived to quadruple order in the Brans-Dicke scalar-tensor and Rosen bi-metric theories of gravity, and the results compared with the predictions of general relativity and applied to observations of the binary pulsar PSR 1913+16.

Gravitational Radiation in the Brans-Dicke and Rosen bi-metric Theories of Gravity with a Comparison with General Relativity


General relationships are developed for gravitational radiation in the weak field approximation in the Brans-Dicke scalar-tensor theory and the Rosen bi-metric theory.   Both periodic and aperiodic systems are considered, with results for the former being of quadrupole order.   The specific cases of a binary orbital system and a system of colliding particles are treated.   An attempt to test the validity of the Brans-Dicke, Rosen, and general relativistic theories is made by applying results to the observed binary pulsar PSR 1913+16.   Based on the 1979 Ph.D. thesis of Warren F. Davis.


In recent months there has been an upsurge of interest in the theoretical prediction of gravitational radiation in general relativity following observations by Taylor et al. (1) on the binary pulsar PSR 1913+16. (2,3)   They have found a systematic decrease of the orbital period of the system that is consistent with energy loss due to gravitational radiation as predicted by Einstein's general theory of relativity.   The compact nature of the participating objects is such as to rule out convincingly significant contributions from other mechanisms such as tidal interaction.   These observations represent the first tests of general relativity outside the solar system and also constitute the first convincing experimental evidence, though indirect, for gravitational waves.

These observations simultaneously raise the question as to whether it might not also be possible to discriminate, on the basis of the same data, between general relativity and other competing theories of gravity whose predictions within the solar system are indistinguishable from Einstein's theory.   Here we consider gravitational radiation in the two theories that currently represent viable alternatives: the Brans-Dicke scalar-tensor theory (4,5) and the Rosen bi-metric theory (6,7).

There are principally two distinct methods whereby gravitational radiation can be estimated in the theories.   First, the EIH (Einstein, Infeld, Hoffmann) method (8) consists in solving the equations of motion in a power series in a suitable parameter such as v/c .   The method is recursive and in principle converges to the exact solution.   Gravitational radiation can be estimated by using the motion so derived to deduce the rate of change of total energy and by assuming that any decrease that is not accounted for by other means goes off as gravitational radiation.   The second method, the weak field approximation (9), consists in linearizing the field equations by approximating to the case in which gravitational effects are everywhere small.   The field equations then reduce to linear wave equations from which radiation can be deduced directly.   Results are not in principle exact, being only as good as the validity of the weak field approximation itself.

In the EIH method, radiation is inferred to the order of recursion to which the theorist is willing or capable of going, with exact results possible in principle in the limit.   In the weak field approximation, radiation is seen directly and in many cases can be computed exactly within the linearized theory, but may or may not be exact in terms of the correct nonlinear theory.   So far it has not been possible to find a satisfactory theoretical bridge between the predictions of the two methods, and the problem remains open.   This situation has resulted in an ongoing controversy as to the correct radiation rate predicted by general relativity (10).   It is possible that predictions made in the competing theories may establish bounds on the radiation rate which will be useful in clarifying the relationship between the EIH and weak field methods.

The general theory of relativity predicts radiation that, in the lowest order, is proportional to the third derivative of the quadrupole moment of the mass-energy distribution.   This prediction follows in either of the two methods.   It is a consequence of the conservation equations that the first derivative of the monopole moment and the second derivative of the dipole moment are zero, so that radiation is first seen in the quadrupole term.   In the alternative theories the situation is different.

The essential feature of the Brans-Dicke theory is that the gravitational "constant" G is in fact not a constant but is determined by the totality of matter in the universe through an auxiliary scalar field equation.   G expresses the ability of mass-energy to interact gravitationally.   The non-universality of G means that the effective interaction strength of a quantity of mass-energy is determined by the local value of the scalar G field.   To make an analogy with electromagnetic theory, the effective gravitational "charge-to-mass" ratio is not a constant.   For the same reason that the existence of differing charge-to-mass ratios among the particles produces non-vanishing electric dipole moments, the variation of G in the Brans-Dicke theory introduces dipole terms in the EIH gravitational radiation equations.

Will (11) has computed the radiation due to these terms in the Brans-Dicke theory for both the scalar and the tensor field.   Eardley (12) has investigated the effect of these terms on PSR 1913+16.   The extent of the dipole effect depends on the difference of the self-gravitational binding energy per unit mass for the two bodies and is thus dependent also on the internal structure of the objects.   When the objects are in circular orbits, the time variation of the scalar field at each object due to the motion of the other is zero and the dipole contributions consequently drop out.   Under these circumstances the dominant surviving terms are of quadrupole order.

Here we develop expressions for quadrupole gravitational radiation in the Brans-Dicke theory using the weak field technique and apply these results, which are applicable in general, to the specific example of PSR 1913+16, though its orbit is eccentric.   We also use the weak field approach to compute the power spectral density to all multipole orders associated with a system of colliding particles.   This result is useful as a basis for the study of gravitational radiation emitted by a hot gas in the Brans-Dicke theory.

Dipole radiation also appears in the application of the EIH method to the bi-metric theory of Rosen.   In Rosen's theory there are two metrics: one that describes purely gravitational effects, as is general relativity, and a second that accounts for inertial effects independent of gravity.   While the gravitational constant does not vary with space-time position as in the Brans-Dicke theory, the energy-momentum tensor in the field equations is scaled by the square-root of the ratio of the determinants of the two metrics.   Thus the effective mass-energy is dependent on the local field, and dipole radiation follows as a consequence as in the Brans-Dicke theory.   Again the dipole radiation is a function of the internal structure of the participating bodies and is zero in certain circumstances such as the case of circular orbits.   For this reason, it is again of some interest to compute the radiation rate due to the quadrupole term.

Will and Eardley (13) have computed the dipole radiation rate in the bi-metric theory and have found that it carries negative energy.   That is, the dipole term acts to increase, rather than decrease, the energy of the system.   Rosen (14) has argued that it is possible to assume a time-symmetric solution, rather than the retarded solution used by Will and Eardley, with the consequence that there is predicted no energy gain or loss due to radiation of any order.

We investigate here quadrupole gravitational radiation in Rosen's bi-metric theory using the conventional retarded solution in the weak field approximation.   As in the Brans-Dicke theory, we also investigate the power spectral density associated with a system of colliding particles.

  1. J. H. Taylor, L. A. Fowler and P. M. McCulloch, Nature, 227, 437 (1979).   Return from footnote.
  2. This paper is written from the perspective of 1979.   Return from footnote.
  3. Joseph H. Taylor, Jr., and Russell A. Hulse received the Nobel Prize in Physics in 1993 "for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation."   Return from footnote.
  4. C. Brans and R. H. Dicke, Phys, Rev., 124, 925 (1961).   Return from footnote.
  5. R. H. Dicke, Phys. Rev., 125, 2163 (1962).   Return from footnote.
  6. N. Rosen, J. Gen. Rel. and Grav., 4, 435 (1974).   Return from footnote.
  7. N. Rosen, Ann. Phys., 84, 455 (1974).   Return from footnote.
  8. A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math., 39, 65 (1938).   Return from footnote.
  9. For example: S. Weinberg, Gravitation and Cosmology (John Wiley & Sons, Inc., New York, 1972), Chapter 10.   Return from footnote.
  10. J. Ehlers, A. Rosenblum, J. N. Goldberg, P. Havas, Ap. J., 208, L77 (1976).   Return from footnote.
  11. C. M. Will, Ap. J., 214, 826 (1977).   Return from footnote.
  12. D. M. Eardley, Ap. J., 196, L59 (1975).   Return from footnote.
  13. C. M. Will and D. M. Eardley, Ap. J., 212, L91 (1977).   Return from footnote.
  14. N. Rosen, Ap. J., 221, 284 (1978).   Return from footnote.

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