| Related sites for http://www.davis-inc.com/relativity |
| Gravity_Probe_B Gravity Probe B is the relativity gyroscope experiment being developed by NASA and Stanford University to test two extraordinary, unverified predictions of Albert Einstein's general theory of relati | | Handbook_of_Space_Astronomy_and_Astrophysics_-_Relativity A collection of equations important to both special relativity and general relativity. | | Hyperspace_at_the_University_of_British_Columbia Includes links to online journals, news and preprint archives, as well as lists of faculty at this university. | | Interactive_Experiments_in_Gravity Try an experiment that illustrates the gravitational attraction between two objects or use a Java applet to understand how orbits work in strongly curved space-time. | | Lecture_Notes_on_General_Relativity Sean M. Carroll's lecture notes from a one-semester graduate-level course he taught at MIT in 1996. Notes are in postscript format. | | NOVA_Online/Einstein_Revealed Profile of Albert Einstein, with additional teaching resources, Shockwave demonstrations, and animations of relativity concepts. | | Physics_Bookshelf_-_Relativity A collection of articles about relativity | | Publications_in_Physics_and_Mathematics_by_Walter_Pfeifer A textbook for relativistic quantum mechanics is presented. The Dirac equation is dealt with in detail. | | QMW_Hyperspace A set of hypertext based services for general relativity research provided by the QMW Relativity group. | | Relativity Provides information on the history, experiments and paradoxes of relativity. | | Relativity_bookmarks Rob Salgado's bookmarks | | Ricci A Mathematica package for doing tensor calculations in differential geometry and general relativity. | | Shapiro_Radar_Bounce_Test A two part overview of the Shapiro radar bounce test of general relativity. (The two parts consist of a section for normal people, and one for nerds) | | Solving_Einstein\'s_Equations_in_Three_Dimensions This is a Mathematica notebook showing how to plot light cones in two plus one spacetime dimensions with a point mass. | | The_Special_Theory_of_Relativity Self-tutorial with short essays, questions and answers. | | Why_Time_is_Absolute,_and_Relative,_But_Never_Universal An article by Vincent Sauvé elaborating upon a materialist view of the absolute and relative nature of time. | | Wrong__Some_Scientifically_Inaccurate_Claims_Concerning_Cosmology_and_Relativity Easy to understand 'debunking' of some common misconceptions about cosmology and relativity and why they are wrong. | | Gravity_of_Gravity An experiment at the University of Washington seeks to determine whether the gravitational binding energy of an object generates gravity of its own. (October 26, 1999) | | Derwent_Innovations_Index An international overview of inventions in chemical, electronic and electrical, and mechanical engineering. Over ten million basic inventions from over 40 authorities. Subscription required. | | Lemelson-MIT_Program Celebrating inventors who have turned their ideas into accomplishments. | | Mahi_Consultant_-_The_Art_of_Invention Tips for protecting and marketing an invention. | | Mandy_Haberman The story of the inventor of the Anywayup Cup and her fight against companies that stole her idea. | | National_Inventor_Fraud_Center,_Inc_ Information regarding invention marketing companies, from Neustel Law Offices, LTD. | | Old_Technology Information for collectors and restorers of late 19th and early 20th century technology. | | A_country_that_innovates Discussing inventions and innovations in Finland. (April 1, 2004) | | Biomechanical_Engineering_Division Information on admissions, degree programs, courses, facilities and laboratories. Stanford University, U.S. | | Biomechanics_Classes_on_the_Web A resource for faculty and students who are interested in biomechanics classes. Includes information on animal biomechanics, biomedical engineering, and related subjects, plus textbooks and related l | | Cummings_Engineering_-_Specializing_in_Biomechanics Consulting firm specializing in biomechanics, accident reconstruction, animation and exhibit preparation for plaintiff and defense attorneys including criminal and insurance companies. | | Fossil_Biomechanics Current research into the biomechanics of fossil sea scorpions (eurypterids). | | Human_Biomechanics_Course Lecture and laboratory notes, assignments, interactive simulations, and related links. St. Francis Xavier University, Canada. | | Human_Lab_Sports Distributors of CD based sport biomechanics programs, designed to improve the performance of golfers, runners and cyclists. | | International_College_of_Biomechanics Provides practical training in assessment procedures and treatments. Contains list of courses. New South Wales, Australia. | | Motion_Analysis_Research_and_Rehabilitation_Center Offers services and research to sports, physiotherapy, legal and industrial clients. University College Worcester, U.K. | | Orthopaedic_Biomechanics Detailed information on biomechanics of the muskuloskeletal system relevant to everyday orthopaedic practice. Includes short paper review, bibliography, news and related links. | | AWC_Electronics Resources for basic stamp programming and accessories, tutorials, projects, with a basic stamp FAQ. | | BASIC_Stamp PIC16/17 Microcontroller | | Basic_Stamp_2 Tutorial and Applications. Sadly only for the BASIC stamp 2. | | Basic_Stamp_Microcontroller_Projects Basic Stamp, Pic microcontroller projects, BASIC stamp books | | Eric\'s_PIC_Page Nice reference page. | | HVW_Technologies_Inc_ A distributor of BASIC Stamp, development tools and software. They also have a download section about PICs. |
|
Gravitational radiation in the Einstein, Brans-Dicke and Rosen bi-metric theories
Gravitational Radiation
in the Einstein, Brans-Dicke and Rosen bi-metric theories
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
ABSTRACT
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.
INTRODUCTION
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.
J. H. Taylor, L. A. Fowler and P. M. McCulloch, Nature, 227,
437 (1979). Return from footnote.
This paper is written from the perspective of 1979.
Return from footnote.
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.
C. Brans and R. H. Dicke, Phys, Rev., 124, 925 (1961).
Return from footnote.
R. H. Dicke, Phys. Rev., 125, 2163 (1962).
Return from footnote.
N. Rosen, J. Gen. Rel. and Grav., 4, 435 (1974).
Return from footnote.
N. Rosen, Ann. Phys., 84, 455 (1974).
Return from footnote.
A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math., 39,
65 (1938). Return from footnote.
For example: S. Weinberg, Gravitation and Cosmology (John Wiley & Sons,
Inc., New York, 1972), Chapter 10. Return from
footnote.
J. Ehlers, A. Rosenblum, J. N. Goldberg, P. Havas, Ap. J.,
208, L77 (1976). Return from
footnote.
C. M. Will, Ap. J., 214, 826 (1977).
Return from footnote.
D. M. Eardley, Ap. J., 196, L59 (1975).
Return from footnote.
C. M. Will and D. M. Eardley, Ap. J., 212, L91 (1977).
Return from footnote.
N. Rosen, Ap. J., 221, 284 (1978).
Return from footnote.
If you do not have a copy of
Acrobat Reader, it is available free from
Adobe
Classic Texts on General Relativity
Gravitation
Comprehensive text by Misner, Wheeler and Thorne,
sometimes referred to affectionately as "The Telephone Book," using
geometrodynamic approach. For the advanced reader.
Gravitation and Cosmology
Very well written, clear, classic approach to general relativity and
cosmology by Steven Weinberg. For the advanced reader.
The Riddle of Gravitation
A clear exposition by Peter Bergmann of the ideas
of Einstein's general relativity accessible to the lay reader.
Davis Associates, Inc.
• 43 Holden Road • Newton, MA 02465-1909 U.S.A. • 617-244-1450 • FAX: 617-964-4917
|
|
