Updated to take into account early universe cosmology and Penrose s cyclic conformal cosmology

Transcription

1 Bounds upon raviton Mass usin the difference between raviton propaation speed and HFW transit speed to observe post-newtonian corrections to ravitational potential fields Updated to take into account early universe cosmoloy and Penrose s cyclic conformal cosmoloy Andrew Beckwith Chonqin University Department of Physics; abeckwith@uh.edu Chonqin, PRC, The author presents a post-newtonian approximation based upon an earlier arument in a paper by Clifford Will as to Yukawa revisions of ravitational potentials, in part initiated by ravitons havin explicit mass dependence in their Compton wave lenth. Prior work with Clifford Will s idea was stymied by the application to binary stars and other such astrophysical objects, with non-useful frequencies toppin off near 100 Hertz, thereby renderin Yukawa modifications of ravity due to ravitons effectively an experimental curiosity which was not testable with any known physics equipment. This work improves on those results. Key words: raviton mass, Yukawa potential, Post Newtonian Approximation

2 1. Introduction Post-Newtonian approximations to eneral Relativity have iven physicists a view as to how and why inflationary dynamics can be measured via deviation from simple ravitational potentials. One of the simplest deviations from the Newtonian inverse power law is a Yukawa potential modification of ravitational potentials. It is apparent that a raviton s mass (assumin it is massive) would factor directly into the Yukawa exponential term modification of ravity. This present paper tries to indicate how a smart experimentalist could use a suitably confiured ravitational wave detector as a way to obtain more realistic upper bounds for the mass of a raviton, and explores how to use this idea as a template to investiate modifications of ravity alon the lines of a Yukawa potential modification as iven by Clifford Will. Appendix A summarizes why we think ravitons should be massive, i.e. havin a small rest mass. We will show how our findins dovetail with a larer than Planck lenth radius of the initial universe, and its connections with both cyclic conformal cosmoloy (Penrose) and recycled information (from previous cycles) into a new universe. Presumably the information transferred via massive ravitons will be responsible for settin Planck s constant at a particular value at the onset of a new universe. Secondly, this paper will address an issue of reat import to the development of experimental ravity. Namely, if an upper mass to the raviton mass is identified; can an accelerator physicist use the theoretical construction Eric Davis posited in his book in the section Producin ravitons via Quantization of the coupled Maxwell-Einstein fields for obtainin an experimental bound to the raviton mass, to refine our understandin of raviton Synchrotron radiation. A brief review of Chen, Chen, and Noble s application of the ersenshtein effect will be made, to potentially improve their statistical estimates of the rane of raviton production.. ivin an upper bound to the mass of a raviton. The easiest way to ascertain the mass of a raviton is to investiate if or not there is a sliht difference in the speed of raviton particle propaation and that of HFW in transit from a source to the detector. Visser s (1998) mass of a raviton paper presents a theory which passes the equivalence test, but which has possible problem with dependin upon a non-dynamical backround metric. Note that ravitons are assumed by both Visser, and also in Clifford Will s write up of experimental R, to have mass This document also accepts the view that there is a small raviton mass, which the author 60 has estimated to be on the order of 10 kilorams. This is small enouh so the followin approximation is valid. Here, v is the speed of raviton propaation, is the Compton wavelenth of a raviton with h m c, and 10 f 10 Hertz in line with L. rischuck s treatment of relic HFWs. In addition, the hih value of relic HFWs leads to naturally fulfillin hf m c so that 1 v 1 c c f (1)

3 But equation (1) above is an approximation of a much more eneral result which may be rendered as v c 1 m c E () The terms m and E refers to the raviton rest mass and enery, respectively. Now Physics researchers can ascertain what E is, with experimental data from a ravitational wave detector, and the next question needs to be addressed, relatin to Visser s model. Namely; if D is the distance between a detector and the source of a HFW/ raviton emitter source The above formula depends upon, Mpc t 1 v c 510 () D 1sec t ta Z te with where ta and te are the differences in arrival time and emission time of the two sinals (HFW and raviton propaation), respectively, and Z is the redshift of the source. Specifically, the situation for HFW is that for early universe conditions, that Z 1100, in fact for very early universe conditions in the first few milliseconds after the bi ban, 5 that Z ~ 10. This is an enormous number. The first question which needs to be asked is, if the Visser non-dynamical backround metric is correct, for early universe conditions so as to avoid the problem of the limit of small raviton mass does not coincide with pure R, and the predicted perihelion advance, for example, violates experiment. A way forward would be to confiure data sets so in the case of early universe conditions that one is examinin appropriate Z 1100 but with extremely small te times, which would reflect upon eneration of HFW before the electro weak transition, and after the INITIAL onset of inflation. I.e. a ravitational wave detector system should be employed as to pin point experimental conditions so to hih accuracy, the followin is an adequate presentation of the difference in times, t. I.e. t t 1 Z t t a t a (4) a The closer the emission times for production of the HFW and ravitons are to the time of the initial nucleation of vacuum enery of the bi ban, the closer we can be to experimentally usin equation (4) above as to ive experimental criteria for statin to very hih accuracy the followin. e 17 00Mpc t a 1 v c 510 (5) D 1sec More exactly this will lead to the followin relationship which will be used to ascertain a value for the mass of a raviton. By necessity, this will push the speed of raviton

4 propaation very close to the speed of liht. In this, we are assumin an enormous value for D 17 00Mpc t a v c (6) D 1sec This equation (6) relationship should be placed into h / m c, with a way to relate this above value of v c 1 m c E, with an estimated value of E as an averae value from field theory calculations, as well as to make the followin arument riorous, namely Mpc ta D 1sec mc 1 E A suitable numerical treatment of this above equation, with data sets could lead to a rane of bounds for m, as a refinement of the result iven by Clifford Will for raviton Compton wavelenth bounded behavior assumin that h is the Planck s constant. (7) for a lower bound to the raviton mass, h 10 m c km 00 km 00 D Mpc D Mpc 100Hz f 100Hz f 1/ 1 ft a 1/ 1/ 1 ft 1/ (8) The above equation (8) ives an upper bound to the mass 1/ 1 D 100Hz 1 m as iven by c m km (9) h Mpc f f ta Needless to say, an estimation of the bound for the raviton mass m, and the resultin Compton wavelenth would be important to et values of the followin formula for experimental validation V M r exp r ravity 1/ (10) r Clifford Will ave for values of frequency f 100 Hertz enormous values for the Compton wavelenth, i.e. values like km. Such enormous values for the Compton wavelenth make experimental tests of equation (10) practically infeasible.

5 Values of 10 5 centimeters or less for very hih HFW data makes investiation of equation (10) above far more tractable.. Application to ravitational Synchrotron radiation, in accelerator physics Eric Davis, quotin Pisen Chen s article written in 1994 estimates that a typical 6 storae rin for an accelerator will be able to ive approximately ravitons per second. Eric uses Chen's article about Photon conversion into ravitons, to suest a way we can use accelerators as a somewhat focused raviton source. Quotin Pisen Chen s 1994 article, the followin for raviton emission values for a circular accelerator system, with m the mass of a raviton, and M P bein the Planck mass. N as mentioned below is the number of particles in a rin for an accelerator system, and nb is an accelerator physics parameter for bunches of particles, which for the LHC is set by Pisen 11 Chen to the value of 800, and N for the LHC is about 10. And, for the LHC Pisen Chen sets at , with m 400. Here, m ~ mraviton acts as a mass chare. 4 m c NSR ~ 5.6 nb N (11) M P The immediate consequence of the prior discussion would be to obtain a more realistic 11 set of bounds for the raviton mass, which could considerably refine the estimate of 10 ravitons produced per year at the LHC, with realistically 65 x seconds = seconds in a year, leadin to ravitons produced per second. Refinin an actual permitted value of bounds for the accepted raviton mass, m, as iven above, while keepin M P ~ ev/c would allow for a more precise value for ravitons per second which sinificantly enhances the chance of actual detection, since riht now for the LHC there is too much eneral uncertainty about where to place a detector for actually capturin / detectin a raviton. Note that Eric Davis explicitly uses 6 Pisen Chen s calculations of ravitons produced to make a feasibility arument as to non zero raviton mass. 4. Conclusion, falsifiable tests for the raviton are closer than the physics community thinks The physics community now has an opportunity to experimentally infer the existence of ravitons as a knowable and verifiable experimental datum with the onset of the LHC as an operatin system. Even if the LHC is not used, Pisen Chen s parameterization of inputs from the table riht after his equation (8) as inputs into equation (11) above will permit the physics community to make proress toward the detection of ravitons for, say the Brookhaven laboratory site circular rin accelerator system. See Appendix B for that table. Tony Rothman s statement about needin a detector the size of Jupiter to obtain a sinle experimentally falsifiable set of procedures is defensible only if the wave-particle duality induces so much uncertainty as to the mass of the purported raviton, that worst case model buildin and extraordinarily robust parameters for a Rothman style raviton detector have to be put in place.

6 A suitably confiured detector can help with bracketin a rane of masses for the raviton, as a physical entity subject to measurements, without needin to be so massive. Such an effort requires obtainin riorous verification of the approximation used to the effect that t ta 1 Z te ta ta is a defensible approximation. Furthermore, havin realistic estimates for distance D as inputs into equation (9) above is essential. The expected pay offs of makin such an investment would be to determine the rane of validity of equation (10), i.e. to what deree is ravitation as a force is amendable to post Newtonian approximations. The author asserts that equation (10) can only be realistically be tested and vetted for sub atomic systems, and that with the massive Compton wavelenth specified by Clifford Will cannot be done with low frequency ravitational waves. Furthermore, a realistic boundin of the raviton mass would permit a far more precise calibration of equation (11) as iven by Pisen Chen in his 1994 article. We refer the reader to Appendix C for how we expect that Eq. (11) and Appendix A, Eq. (A4) may be combined to yield experimentally falsifiable tests for a massive raviton. Note that Appendix D ives the details of how Yurov links both initial inflation and the speed up of acceleration of the universe a billion years ao. In addition a brief mention of Padmanabhan s contruction of the present era infaton is mentioned. Notice that the inflaton for today is iven by Padmanabhan, which Beckwith views as important to understand what Yurov meant by linkin initial inflation with the expansion of the universe speedin up today, i.e. Yurov s second inflation. Also, if ravitons are linkable to DE, as stated by Beckwith s Journal of cosmoloy publication (011), in the present era, and if ravitons are super partnered with ravitinos as stated by Beckwith (01), Kauffman s non zero initial radius becomes essential, and we also have a description of how the fine structure constant over time, as iven by Appendix C, Eq. (C), could be invariant. That in turn would lead to Planck s constant bein invariant from cosmoloical cycle to cycle without invokin the Anthropic principle. As well as also keepin track of how the followin would be possible at the electro weak era. Note, from Valev, m raviton RELATIVIST IC raviton m raviton h c ev / c 8 meters (1) Extendin M. Marklund et al. and Valev, some ravitons may become larer, i.e. due to red shiftin as iven by Clifford Will. 4 raviton 10 meters or larer (1) c m raviton

7 The author maintains that the information necessary for both Eq. (1) and Eq. (1) plus space-time eometry, as well as DM and DE linked as iven by Beckwith(01) states require the finite initial radius result (Kauffman) with initially in the electro weak era, tiny HFW as iven by Eq.(1) super partnered with ravitinos. We hope for experimental confirmation, as soon as possible. Appendix A ives a crucial lower bound for the initial radius of the universe. Appendix C makes the case for a non zero raviton mass super partered to ( via SUSY) to ravitinos. The non zero ravitino mass leads credence to a non zero initial enery necessary for a the initial cosmoloical radii, R, bein non zero, and Appendix D makes the case for linkin non zero raviton mass as linked to the expansion of the universe, via a mechanism linkin initial inflation with the speed up of inflation as stated by Yurov, which the author wishes to improve upon. Appendix A: Indirect support for a massive raviton We follow the recent work of Steven Kenneth Kauffmann, which sets an upper bound to concentrations of enery, in terms of how he formulated the followin equation put in below as Eq. (A1). Equation (A1) specifies an inter-relationship between an initial radius R for an expandin universe, and a ravitationally based enery expression we will call T r which lead to a lower bound to the radius of the universe at the start of the Universe s initial expansion, with manipulations. The term T r is defined via Eq.(A) afterwards. We start off with Kauffmann s 4 c R T r r d r (A1) r R 4 c Kauffmann calls a Planck force which is relevant due to the fact we will employ Eq. (A1) at the initial instant of the universe, in the Planckian reime of space-time. Also, we make full use of settin for small r, the followin: T r r T 0 r const ~ V( r) ~ mraviton ninitial entropy c (A) I.e. what we are doin is to make the expression in the interand proportional to information leaked by a past universe into our present universe, with N style quantum infinite statistics use of n ~ S Initial entropy raviton count entropy (A) Then Eq. (A1) will lead to

8 4 c R T r r d r const m raviton ninitialentropy ~ S ravitoncountentropy r R 4 c R const mraviton ninitialentropy ~ S ravitoncountentropy 1 4 c R const mraviton n 5 Here, ninitial entropy ~ S raviton count entropy ~ 10, Initialentropy ~ Sravitoncountentropy mraviton 1 Planck lenth = l Planck = meters ~ 10 6 rams, and (A4) where we set lplanck with R~ l 10 Planck, and 0. Typically R~ l Planck 10 is c about 10 lplanck at the outset, when the universe is the most compact. The value of const is chosen based on common assumptions about contributions from all sources of early universe entropy, and will be more riorously defined in a later paper. Appendix B: raviton paper table from Pisen Chen Storae PEP-11 LEP-1 LEP-II HERA LHC Rins ev N n b lcm m ravitational SR N SR 10 sec Resonant conversion 9 10 Hz c N Re s 10 sec ,

9 Appendix C: How to et stricter bounds to Eq. (A4) above We follow the recent work of Steven Kenneth Kauffmann, which sets an upper bound to concentrations of enery, in terms of how he formulated Eq. (A4) as stated earlier in Appendix A above. The methodoloy for obtainin a specific set of initial conditions to specify a lower non zero bound for obtainin a radii of the universe, initially requires a specific set of initial conditions. The first is to state how a raviton would have non zero mass, initially, as discussed by Beckwith in 011 in the journal of cosmoloy, as well as the assumption that a ravitino would be super partnered with ravitons accordin to a mass conservation expression iven as in the followin section in Beckwith, 01, sent to the Rencontres De Moriond conference. i.e. the section entitled. Formin m / for a ravitino and linkin it to Massive raviton Contributions in electro weak era: For the Machian relationship We assume that the mass between ravitons and ravitinos is super 50 parterned and that the number of ravitons, i.e. about 10 in the electro 8 1 weak era, reflects in about ravitinos, each ravitino with at least 8 10 times the mass of a raviton. Each of the ravitinos is initially up to 1 TeV in mass, and if mass is initially the same as enery, the presence of non zero enery will show up in the formulation of T r of Eq. (A) as havin a non zero value. Since T r has a non zero value, this super partnerin of ravitinos with non zero mass ravityons insures that there is a lower non zero bound for the initial radii, R. We now below document how to obtain a super partner relationship between ravitons and ravitinos which will lead to a non zero T r The idea was to mix results in Salvoy s 198 documet with M N m N 10 m 8 electroweak electro weak / electro weak raviton N m 10 m 88 today raviton raviton Then the electro weak reime would have and entropy value of would be compared with than the raviton, whereas the m/ (C1) 50 Nelectro weak ~10 for ravitons which Ninitial ~10 10 for ravitinos, with m / 10 times larer in mass is a ravitino mass value. With (C) below bein the initial entropy 88 value in electro weak reime which rows to 10 today, i.e. via use of N s re statement of entropy 50 N ~10 (C) electro weak The first and second eqn. above ravitons linked to SUSY ravitinos form our relationship for makin a linkae of massive This would lead to, say m mfermions 0 (C) BOSONS Table 1, mass of different particles and cosmoloical parameters (rounded off) M Planck M TEV ~ M DM ~ M ravitino M DE M raviton 8 10 k ~ TeV 4 10 k = 10 1 ev 10 k M DM

10 The net up shot of the super partner pairin of ravitons and ravitinos is to obtain a non zero T r due top no zero ravitons leadin up to possibly an upper ravitino mass value with this mass bein initially enery. The minimum enery for T r will then entail the lower bound to the initial radii, R are not zero. Now to the problem How to insure that the raviton has non zero mass. To start, we can first review briefly what was done by Beckwith in 011, in the Journal of Cosmoloy. In this publication, Beckwith outlined how there may be a contribution via a minimally massive raviton as to re acceleration of the universe. Here the value of 6 mraviton ~ 10 rams to et a speed up of acceleration of cosmoloical expansion a billion years ao. This thouh, does not cleave to the essence of the problem, thouh, as seen in rowin neutrino mass theories and cosmoloy, there conceivably could be rowin mass for early universe ravitons. The author thinks not, and appeals to what was done in another publication, a talk in Italy 011 (see paes 6-4) where a variant of Penrose cyclic conformal cosmoloy is employed as a means to recycle prior universe information in order to have a uniform Planck s constant, per cycle via a consistent small raviton mass. Note that recently in La Thuile, Italy (01) in Rencontres De Moriond, Beckwith outlined a relationship between ravitons and ravitinos which implies that at or before the electro weak era, i.e. that due to a revision of SUSY, the up to TeV mass of ravitinos at or before the electro weak era with ravitinos super partnered with ravitons leads to a non zero initial cosmoloical radius. Furthermore, as stated in FXQI, in a discussion by Tom Ray, No sinularity at the Schwarzchild radius not only confirms the quantum nature of the cosmoloical initial condition, it implies non-quantization of classical space-time. For if the quantum field does not collapse, the universal wavefunction, which is continuous (Kauffmann concludes, "... only the universe itself, with its cosmoloical redshift, is actually capable of 'containin' the arbitrarily hih frequencies of a quantum field") is physically real and dark enery isn't. This end result leads to a finite initial radius of space-time, not zero, with enouh information transported from a prior universe to the present so as to preserve the continuity of physical constants from space-time collapse to rebirth, aain and aain. Note that the almost infinite amount of enery within a radii within a Planck value ( but not smaller) is enouh to et ravitinos formed which are super partered via re done SUSY as stated by Beckwith (01) to obtain durin the Electro weak reime massive ravitons obeyin the equation of state, as iven by Maiore m raviton h T 1 vt mraviton T ( C4) Furthermore, we state that both the raviton and ravitino would have to be consistent, with respect to the well known De Brolie matter-wave hypothesis, with the resultant consistency of a constant fine structure constant,which does not vary over time, as iven by Beckwith (010a,b), so that there would be constant fine structure values, as iven by

11 ~ e e c (C5) d hc The idea that Planck s constant would remain consistent is also stated by R. Penrose in his recent book. Accordinly; the author views that a minimum entropy of 10^5 as a transfer mechanism from prior to present space time is important and should be reviewed, as it ives constant values equal to the present value of Planck s constant from cycle to cycle. Cyclic conformal cosmoloy, and tests confirmin its initial CMBR pattern, would do much to add such a plausible explanation showin how consistency in Planck s constant from cycle to cycle can be maintained. Lastly; the author appeals to a calculation iven by iovanni, to the effect that if the Penrose supposition of Final entropy ravitons S FINAL RAVITONS ~ Phase space due to ravitons (C6) is true (as iven by Penrose) then the relationship iven by iovanni in his book for entropy that the present entropy of the universe, i.e. about is obtained by interatin from 10-^ 11 Hertz down to 10^-19 Hertz would do much to arue for a constant raviton mass, as iven above. Furthermore, this could be with work also connected with what the author referred to as first and second inflation, as iven by his work in 010. This would be a project warrantin serious investiation. First inflation is the typical inflation iven in cosmoloy, whereas the nd inflation is the speed up of the universe, as referenced in Beckwith s 011 publications. Appendix D What if an inflaton partly re-emeres in space-time dynamics? At z ~. 4? In this section, the author will ive further elaboration of a suestion by Yurov as to linkin of initial inflation with the speed up of expansion of the universe, commencin up to today. This section will be focused upon sayin somethin about the inflaton which may be basic as to why the universe has a speed up of inflation. We review, in doin so, the work by Padmanabhan. Padmanabhan has written up how the nd Friedman equation which for z ~. 4 may be simplified to read as m H 4 a (D1) would lead to an inflaton value of, when put in, for scale factor behavior as iven by a t t, 1/,0 1, of, for the inflaton and inflation of (Padmanabhan) t dt H 4 (D) Assumin a decline of a t t, 1/,0 1, Eq. (D) yields

12 m t ~ t (D) 4 a t t, 1/,0 had time of the value of As the scale factor of 1 rouhly t t, 1/,0 1 1/ t t a have a power law relationship drop below a, the inflaton took Eq. (D) s value which may have been a factor as to the increase in the rate of acceleration, as noted by the q (z) factor, iven in Beckwith s Journal of cosmoloy publication. Note that there have been analytical work projects relatin the inflaton, and its behavior to entropy via notin that inflation stopped when the inflaton field settled down into a lower lower enery state. The way to relate an enery state to the inflaton is, if a t a0t potential enery term of ( Padmanabhan), then in the early universe, one has a 16 V V0 exp t (D4) A situation where both 1 / rows smaller, and, temporarily, t takes on Eq. (D) s value, even if the time value ets lare, and also, if acceleration of the cosmic expansion is taken into account, then there is infusion of enery by an amount dv. The entropy ds dv/t, will lead, if there is an increase in V, as iven by Eq. (D4) a situation where there is an effective increase in entropy. If there is, as will be related to later, in pae eiht, circumstances, where S N number of raviton states ( as stated by Y. N) Linkin the two analytically, partly due to Yurov s suestion about an explicit linkae between initial and final inflation (initial inflation bein what happens riht after Planck scale time, and final inflation bein the speed up of acceleration seen as of the present era) would in the mind of the author, clinch the case for a non zero raviton mass. The Yurov suestion about linkin initial and final inflation ( as stated above) will be worked upon seriously by the author in future publications and is crucial to makin initial and final raviton mass, about the same order of manitude. Acknowledements This work is supported in part by National Nature Science Foundation of China rant No The author thanks Jonathan Dickau for his review and suestions for improvement iven over several months in 01 and 01.

13 Biblioraphy A. Beckwith, Applications of Euclidian Snyder eometry to the Foundations of Space Time Physics, v 6 (newest version) [February 010a] Andrew Beckwith, Applications of Euclidian Snyder eometry to Space Time Physics & Deceleration Parameter ( DE Replacement?) Analysis of Linkae Between 1st, nd Inflation?, [ 010b] A.W. Beckwith, Identifyin a Kaluza Klein Treatment of a raviton Permittin a Deceleration Parameter Q(z) as an Alternative to Standard DE, November, 011a Andrew Beckwith, The Nature of Semi-Classical Nature of ravity Reviewed, and Can We Use a raviton Entanlement Version of the EPR Experiment to Answer if the raviton is Classical or Quantum in Oriin? November 011b Andrew Beckwith, If a Machian Relationship Between ravitons and ravitinos Exists, What Does Such a Relationship Imply as to Scale Factor and Quinessence Evolution and the Evolution of DM?, March 01 Pisen Chen, Resonant Photon-raviton Conversion in E M fields: From Earth to Heaven, SLAC Pub 6666, September (1994) Eric Davis, Producin ravitons via Quantization of the Coupled Maxwell Fields, in Frontiers of Propulsion Science, Proress in Astronautics and Aeronautics Series, Vol. 7, eds. M.. Millis and E. W. Davis, AIAA Press, Reston, VA, (009) Steven Kauffmann, A Self ravitational Upper bound On Localized Enery Includin that of Virtual Particles and Quantum Fields, which Yield a Passable Dark Enery Density Estimate, PUT IN WHICH VERSION OF THE DOCUMENT. I CITE THE EQN ON PAE 1 M. iovannini, A Primer on the Physics of the Cosmic Microwave Backround, World Press Scientific, Sinapore, Republic of Sinapore, 008 M. Maiore, ravitational Waves, Volume 1 : Theory and Experiment, Oxford Univ. Press(008) M. Marklund,. Brodin, and P. Shukla, Phys. Scr. T (1999). Y.J. N, Entropy 10(4), pp (008); Y.J. N and H. van Dam, Found Phys 0, pp (000) Y.J. N and H. van Dam, Phys. Lett. B477, pp49 45 (000) Y.J. N, Quantum Foam and Dark Enery, in the conference International workshop on the Dark Side of the Universe, International Workshop on The Dark Side of the Universe/Talks/Monday/Quntum From And Dark Enery.pdf missin link --

14 T. Padmanabhan, An Invitation to Astrophysics, World Scientific series in Astronomy and Astrophysics, Vol. 8 R. Penrose, Cycles of Time, The Bodley Head, 010, London, UK C.A. Salvoy, Super symmetric Particle physics: A panorama, pp 99-8; from 18 Rencontres De Moriond, Volume, Beyond the Standard Model, edited by J. Tran Thanh Van, Editions Frontieres, B. P IF Sur Yvette France 198 D. Valev, Aerospace Res. Bul. :68-8, 008; ; Matt Visser, Mass of the raviton, arxiv: r-qc/ v Feb 6, 1998 Clifford Will, Boundin the Mass of the raviton usin ravitational-wave observations of inspirallin compact Binaries, arxiv: r-qc/ v1 Sept 4, 1997 Clifford Will, The confrontation between eneral Relativity and Experiment, Livin Rev. of Relativity, 9, (006),, A. Yurov; arxiv: hep-th/0819 v1, 19 Au, 00