The Gravitational Potential for a Moving Observer, Mercury s Perihelion, Photon Deflection and Time Delay of a Solar Grazing Photon

Transcription

1 Albuquerque, NM 0 POCEEDINGS of the NPA 457 The Gravitational Potential for a Moving Observer, Merury s Perihelion, Photon Defletion and Time Delay of a Solar Grazing Photon Curtis E. enshaw Tele-Consultants, In., 4080 MGinnis Ferry oad, Suite 90, Alpharetta, GA renshaw@telein.om Utilizing the priniple of equivalene and the radiation ontinuum model of EM radiation, it is demonstrated that Newton's gravitational potential applies only for stati or slowly moving objets. The addition of veloity dependent terms, derivable from the priniple of equivalene and the equivalene of matter and energy, produes the full dynami gravitational potential. This dynami potential is applied to the problem of Merury's orbit and photon defletion, fully aounting both, and providing a result idential in form and value to that obtained utilizing the urved spae-time of GT.. Introdution It has been shown in a previous paper that under a Galilean- Newtonian treatment of light, EM radiation may be treated as emanating from a soure at all veloities from 0 to some upper value C, whih is greater than, and may be infinite. In this model, a photon may be viewed as an expanding spherial volume, with the leading edge of that volume expanding at a veloity of C. This is in ontrast to the speial relativisti model of a photon expanding in a spherial shell at a veloity of, but with that value dependent on the veloity of the observer, and not measured with respet to the soure. As a result of modeling the photon as an expanding spherial volume, there an be found a omponent of this extended photon with any veloity one hooses between 0 and C. With respet to the soure, all veloity omponents of the emitted ontinuum photon are at the same frequeny, thus the wavelength inreases proportionally with veloity. Any observer will be suseptible to that omponent with a veloity of relative to the observer. For example, an observer moving away from the soure with a veloity of v would be suseptible to that omponent whih leaves the soure with a veloity of + v with respet to the soure, and thus has a veloity of with respet to the observer. This model has been shown to allow a Galilean invariane of Maxwell's equations [4], and to support all experimentally verified Doppler shift results. Newton's gravitational potential was developed by studying stationary or slowly moving objets ( v ). As suh, we an onsider this to be the stati gravitational potential. In what follows, we will show that there is also a veloity dependent term, derivable from the CM model of light. One the full dynami gravitational potential is developed, we will apply it to an analysis of Merury's anomalous perihelion advane and the defletion of a solar grazing photon, and show that the form and results of the solutions are idential to those derived utilizing GT.. The Gravitational ed-shift For an observer moving diretly away from the soure, with inreasing distane represented as a positive veloity, the observer will be sensitive to that omponent of light whih leaves the soure at a veloity of + v. In what follows, we will refer to the veloity of light measured with respet to the soure as, while the veloity of light observed is always. The red-shifted frequeny due to suh motion is equal to the produt of the soure frequeny times the ratio of, the observed veloity, to the initial veloity omponent of light emitted with respet to the soure, or: v. () v Although it an be demonstrated that any given omponent of light inreases its veloity as it leaves a gravitational field [], the light also loses energy, as measured by a reeiver (atom or lok, for example) loated outside the field, aounting for the gravitational red-shift experiened by the observer. The Pound- ebka-snider experiment demonstrated that we an ounter the effets of the gravitational red-shift by moving the observer toward the soure at a ertain veloity [5]. The veloity required is suh that the inrease in energy due to the motion indued Doppler blue-shift exatly ounters the loss in measured energy due to the gravitational red-shift. If the height of the Pound- ebka test apparatus is h, then the veloity required will be given by: gh v, () where g is the aeleration due to gravity near the surfae of the earth. From this relation, we an express the formula for the gravitational red-shift in the form of Eq. (). If we onsider an observer stationary in the gravitational field, the veloity omponent with respet to the soure is, and the shifted frequeny is given by replaing v in Eq. () with the expression of ():. (3) gh gh

2 458 enshaw: The Gravitational Potential for a Moving Observer Vol. 9 As a hek on the validity of the form of Eq. (3), we an derive the standard formula for the gravitational red-shift as follows, where gh, g GM r, and h. gh, gh or gh GMh GM. (4) 3. The Gravitational ed-shift for a Moving Observer If we now onsider an observer moving away from the soure with a veloity v, the observer is sensitive to light leaving the soure at a veloity of v (Fig. ). For the moving observer arrying a reeiver with him, the height, h, traversed by the light, depited in Fig., is derived by the following expressions: h h h hvt v t, or t, v or hvt h. (5) motion has no effet on the measured gravitational red-shift or on the effetive gravitational potential for suh motion. Next we onsider an observer arrying a reeiver and moving transversely to the gravitational field. As an be seen in Fig., the effetive height, h, does not hange. Sine we are interested in the omponent of light that strikes the reeiver with apparent perpendiular inidene, the aberrated veloity of light with respet to the soure,, is given by: v. (7) Fig.. adial motion and gravitational Doppler In what follows, we will use g for the aeleration due to gravity as experiened by moving observers, and verify whether, for eah type of motion (radial and transverse), g = g. Using the proper values for the veloity of light with respet to the soure,, and the effetive value for h in Eq. (3) yields: gh v gh( v ) v v. gh v gh v Thus, the total red-shift in reeived energy is equal to the expeted motion-indued Doppler shift (given by the first term in the last expression) times the expeted gravitational red-shift (given by the seond term). From this we an dedue that radial (6) Fig.. Transverse motion and gravitational Doppler The reason we are interested in light that appears to be perpendiularly inident to the reeiver is that we are going to develop the equivalent gravitational relation, and we are interested in that fore that appears in the referene frame of the test partile to be along the line joining the gravitational soure and the test partile. In other words, all gravitational fore is direted along this line. The gravitational fore vetor always appears radial, with no transverse omponents. Thus, the total equation for the hange in reeived energy is given by: gh gh v v gh v gh v v v (8). In this ase, the hange in energy is equal to the produt of the transverse Doppler shift for apparent perpendiular inidene, given by the first term, times the gravitational redshift, given by the seond term. However, if the above equation were orret, unlike the ase of radial motion, the gravitational red-shift in energy observed in the moving system would be different from that measured at the same point in the field by a stationary observer. Imagine two observers in free-fall in empty spae, far removed from any gravitational fields. Allow one observer to have an initial veloity, v, with respet to the other observer. Further allow the moving observer to pass very near the stationary observer, at the same instant that eah observes the light from a distant soure. If eah observer reords the

3 Albuquerque, NM 0 POCEEDINGS of the NPA 459 frequeny of light reeived from a distant soure, then, after aounting for the Doppler shift experiened by the moving observer, they will eah obtain the same frequeny for that light. Now, the priniple of equivalene tells us that the same experiment performed by these same two observers in free-fall in a strong gravitational field should obtain the same results as they did in free-fall far removed from any gravitational fields. Thus, the observed shift in frequeny due to the gravitational field alone (after baking out motion indued Doppler effets) should be the same for both observers. In order for this to be the ase, Eqs. (3) and the right hand side of Eq. (8) must produe the same result--eah observer will measure the same value for the gravitational red-shift. From this, the following relation beomes immediately apparent: v v g g, or g g. (9) If we now replae this transverse motion v with r, and append the dynami adjustment of Eq. (9) to Newton's stati potential, we arrive at the dynami gravitational potential for a moving observer: V Gm m r r. (0) One might argue that the priniple of equivalene doesn't atually go as far as to imply that a free fall observer with an initial veloity should see the same gravitational red-shift at a partiular point in spae as does a stationary observer, so we will present a gedanken experiment that illustrates that if suh is not the ase, a ontradition will our. It has been demonstrated experimentally that loks in a gravitational field slow down aording to a relation idential in form (though inverted) to that for the gravitational red-shift. In other words, the amount of slowing depends only on the strength of, and thus the position in, any partiular gravitational field. It has also been demonstrated that a lok plaed in motion slows down aording to a well-defined formula dependent only on the veloity of the lok relative to its rest frame. Now, suppose we have three, idential, synhronous loks, far removed from a gravitational field. We hold one of these loks in spae, stationary with respet to a distant, gravitating body. We lower another of the loks into the field and leave it there, at rest with respet to the stationary spae lok, though far removed from it and deep within the field. The lok in the field will tik more slowly due to its presene in the field, aording to the strength of the field at that point. Next we aelerate the third lok outside the field to some known initial veloity with respet to the stationary loks outside and within the field. One this veloity is reahed, we allow this lok to pass very near the stationary lok outside the field and note that this lok is running slow due to its veloity as measured by that lok, and we measure the rate of slowing. We allow the moving lok to ontinue on with the same veloity until it passes the stationary lok inside the field. We note again that this lok is running slow due to its veloity with respet to that lok, even though both loks have also slowed due to their presene in the gravitational field, and we measure the rate of slowing. We ompare the rate of the moving lok as determined by eah of the stationary loks. Sine the moving lok was at the same point in or out of the gravitational field as were the stationary loks at the time eah measured the moving lok s rate, eah of these loks should have determined the same rate of motion indued slowing for that lok, sine the gravitational slowing would be the same for the moving and the stationary lok. However, if the gravitational red-shift for the observer with an initial veloity is different than that for the stationary observer, then the gravitational effet on the moving lok will also not be to the same degree as that on the stationary lok. This ontradits the original assumption and analysis whih showed that the slowing of a lok due to a gravitational field is due only to the strength of that field and the distane from the soure of that field, not the veloity through that field as well. Therefore, the gravitational red-shift as measured by a moving observer at a ertain point in the field must be the same as that measured by a stationary observer, and we see that Newton's stati gravitational potential must be modified to reflet the effets of motion through a gravitational field as is done in Eq. (0). Sine we wish to onsider the ase of planetary orbits and solar grazing photons, we obtain the following relations regarding angular momentum, l: l mr r, or r l mr, or r l m r. () Substituting Eq. () into (0) yields (ignoring terms of higher order than / ) a useful form of the dynami gravitational potential for a moving observer: Gmm l V. () r mr We an now write diretly the fore due to the potential of Eq. (), where Gm m has been replaed by k [3]: F d dv dv dt dr dr k 3l r m r. (3) Utilizing a hange of variables to u = /r, and applying the LaGrangian treatment of motion to Eq. (3) yields: u u km 3ku. (4) l m This is, of ourse, the exat form derived under general relativity. 4. The Perihelion Shift of Merury Eq. (4) has the form of an elliptial orbit plus a small perturbation term. Thus we an hoose a solution of the form: km u0 os. (5) l Eq. (5) shows that with eah orbit of the planet, the perihelion will advane by an amount given by:

4 460 enshaw: The Gravitational Potential for a Moving Observer Vol. 9 k Gm 3 6 l a. (6) Eq. (6) is of ourse the exat equation derived in relativity theory, and results in an anomalous perihelion advane for Merury of 43 per entury, as onfirmed by observation. 5. The Defletion of a Solar Grazing Photon Beginning with the equation of motion derived from the dynami gravitational potential, and onsidering a photon approahing from, we see that l, and we an neglet the first term on the right-hand side of Eq. (4). Thus, we an assume a homogenous solution of the form: u h os, (7) and obtain: x k GM. (8) y m r x Sun y Fig 3. Defletion of a solar grazing photon. In Eq. (8), is the tangent of the angle of defletion on one side of the sun, as seen in Fig. 3. Sine the value of is very small, the tangent of the angle is almost idential to the angle itself, expressed in radians. The total defletion of the photon then, in radians, is, or: 4 total defletion GM. (9) Eq. (9) is the exat value obtained in general relativity theory and results in.75 of defletion as onfirmed by observation. 6. The Time Delay of a Solar Grazing Photon We have seen in previous papers [,], that moving loks and loks in a gravitational field slow down, not due to the effets of speial relativity nor to the spae-time urvature of general relativity, but due only to the priniple of equivalene and the onservation of energy. However, some might argue that there has been a further test of the effet of gravity on time, namely the measurement of the time-delay of a round-trip, solar-grazing radar beaon performed by Shapiro in the 960 s. In this test, Shapiro bouned a radar pulse off Mars at superior onjuntion (a feat in itself for the time), and ompared the measured round-trip travel time of this pulse with the expeted round-trip time of a signal traveling at for the entire trip, as determined from highly aurate planetary ephemerides. Shapiro had predited this time-delay long before being tehnologially able to make suh a measurement. While general relativity an be used to orretly obtain the magnitude of this delay, it is not the only explanation. As we did earlier with the analysis of loks in motion and of loks in a gravitational field, this paper will derive the same result without invoking the spae-time urvature of general relativity. 6.. Photon Veloity in a Gravitational Field We begin by onsidering the Priniple of Equivalene. eall that, by this priniple, if one is in free-fall (not being aelerated by a roket or held in plae by a floor), one annot tell the differene between floating freely in spae or falling toward a gravitating body suh as the sun. No experiment one might perform ould provide any knowledge about whih state one is in, as the results of all experiments would be idential in both ases. This is also true of a photon in free-fall. Now, a photon, upon entering a gravitational field, or well, aquires exess energy, ompared with the energy of the surrounding gravitational field, muh the way a ball dropped from a tower gains energy as it heads toward the ground. We an express the energy of this photon as: h E, or E m. (0) Thus, an inrease in energy an be viewed as an inrease in effetive mass, or, onversely, as a shortening or bunhing up of the wavelength omponents. A visual representation of this is when smoothly flowing traffi suddenly omes upon slowing due to rubberneking a stalled vehile aross the road. The traffi slows, and the ars whih had a omfortable, even spaing now beome bunhed up as they pass through this area. Upon leaving the ongestion, the ars one again resume their original, spread out onfiguration and speed. Now, due to the priniple of equivalene and its presene in the gravitational field, the photon s veloity must slow down proportional to its dereased wavelength. For the photon, it travels in its frame of referene a distane l in a time given by l. If l dereases by to l, then the photon s veloity will slow to as well, so that it now travels the distane l in a time given by ( l ) ( ), or l, as before. If we let the inreased energy of the photon in the gravitational well (as ompared with the energy of the surrounding field) be represented by E, then we an rewrite Eq. (0): h h E. () From Eq. (), we an again derive the new speed of this photon, realizing again that, due to the priniple of equivalene, the photon s frequeny must remain unhanged:. () Thus, the photon must slow down inversely proportional to the inrease in energy it would otherwise gain during its fall through the gravitational field. We ould also have simply stated that, sine the veloity of a photon is equal to its frequeny times its wavelength, the new veloity must be given by:. (3)

5 Albuquerque, NM 0 POCEEDINGS of the NPA 46 Either approah yields the same value for the new veloity of the photon. We must now turn our attention to deriving the value of the fator. If a hydrogen atom is arried into a gravitational well, it will generate or absorb energy at a lower frequeny than it would outside the field. Thus, if we build a lok based on the frequeny absorbed or emitted by this atom, it will run more slowly than an idential lok not inside the field. The rate of slowing of this lok will be the inverse of the equation used to derive the gravitational blue-shift for a photon falling into a gravitational well. If represents the frequeny generated by the lok inside the gravitational field, and the frequeny generated far removed from that field, we have the following equation for the lok slowing: GM. (4) If this lok measured the frequeny of a photon generated far outside the field, the measured frequeny would be blue shifted by the inverse of the above fator. This is graphi evidene that the frequeny of the photon in free-fall has not hanged. We know that the lok in the gravitational field runs slowly by the relation in Eq. (4), beause this has been tested many times with atual esium loks. We also know that the blue-shift of a photon in free fall as measured by these same loks is exatly the inverse of Eq. (4), as the Pound-ebka- Snyder experiment (960, for example) has shown. Now, if the photon itself were atually blue-shifted, the measured shift would be twie that given by the inverse of (4). First, the photon would be blue shifted by this amount, and this blue shifted frequeny would appear to be shifted even higher when measured with the slow running lok. But the effet appears only one, and it is due only to the lok slowing--the frequeny of the photon, due to the priniple of equivalene, has remained unhanged. So now we have the new veloity of a photon in free-fall through a gravitational field. Combining Eqs. (4) and (3), we get: GM (5) However, Eq. (5) must be modified slightly. The veloity of the photon is onstantly hanging during its trip, in a manner always proportional to /r, r being the distane from the sun at any given point. We therefore replae, the grazing radius of the sun, with r, the instantaneous distane from the sun at any time along the photon s journey: GM r. (6) This is still not the entire story. Eq. (6) represents the veloity of the photon as measured loally by a lok traveling with the photon (in the viinity of the sun). However, we are interested in the value of this veloity as measured using an Earth based lok. Clearly, a veloity measured with a standard lok will be less than the same veloity measured with a gravitationally slowed lok. Taking the Earth based lok to be our standard, we see that the veloity as measured by that lok will be even slower than the veloity as measured by a loal lok arried with the photon. The additional saling of the veloity will be idential to the saling already provided in (6). The effet appears twie. One due to the atual slowing of the photon in its own referene frame measured by its own loks, and again to translate this veloity to the veloity as measured by an Earth based lok. The veloity as measured by an Earth based lok, omitting terms of order higher than, is then given by: GM GM GM e r r r 6.. The Derivation of the Time Delay. (7) From Eq. (7), the following relations an be derived, where T is the total time of the photon s travel from 0 to d, as measured by an Earth based lok: T T dt, (8) 0 ds GM dt ds ds, (9) e GM r r T d GM, (30) r 0 0 T dt ds / ds r rdr, (3) d GM /. (3) r T r rdr In Eq. (3), we an obtain the effetive path length of a photon traveling at for a time T by multiplying through by. Doing this, and breaking the integral into parts, yields the effetive path length of the photon: d T d r / rdr d GM r r r GM r / dr / / ln d GM d ln d d ln GM d d GM d d ln d ln. (33) (34) (35) In Eq. (35), the last approximation holds for d>>. The last term on the right-hand side of Eq. (35) represents the effetive inrease in path length, and is given by:

6 46 enshaw: The Gravitational Potential for a Moving Observer Vol. 9 GM d d ln. (36) For distanes on the order of Earth or Mars, d is approximately 9 km. Thus, for a light signal traveling from Earth to Mars and bak, with Mars at superior onjuntion, the round trip inrease in effetive path length due to the time-delay of the photon is approximately 76 km, resulting in an inrease in round trip travel time of 50 miroseonds (out of a total trip on the order of thirty minutes). This result is equivalent to the results of relativity theory, and is onfirmed (Shapiro, et al) to an auray of 0.%. 7. Conlusion Aeptane of Einstein's seond postulate, ombined with the priniple of equivalene, requires the introdution of spae-time urvature in the presene of massive objets. In this paper it has been demonstrated that a Galilean-Newtonian model of light results in a modifiation to Newton's stati gravitational potential. Applying this modified potential to the ase of Merury's anomalous perihelion advane aounts for the entire 43 per entury, and provides a form equivalent to that of GT. Thus this model will work equally well for any system to whih it is applied, without resorting to the spae-time urvature of GT. The same potential, applied to the problem of a solar grazing photon, gives a solution idential in form and ontent to that of GT as well. Finally, we showed that the same treatment applied to the time delay of light passing massive objets produes results idential to GT. Suh results lead to the onlusion that these effets are not attributable to massive objets urving spae-time, but rather are just a natural onsequene of the Prinipal of Equivalene applied in a Galilean-Newtonian framework. eferenes [ ] Curtis E. enshaw, The Effets of Motion and Gravity on Cloks, IEEE: Aerospae and Eletroni Systems Magazine 0 (0) (995). [ ] Curtis E. enshaw, Moving Cloks, eferene Frames and the Twin Paradox, IEEE: Aerospae and Eletroni Systems Magazine () ( 996). [ 3 ] J.B. Marion, Classial Dynamis of Partiles and Systems, pp (Harourt and Brae Publishers, New York, 970). [ 4 ] Curtis E. enshaw, The adiation Continuum Model of Light and the Galilean Invariane of Maxwell's Equations, Galilean Eletrodynamis 7 () (996). [ 5 ]. V. Pound, G. A. ebka, Jr. and J. L. Snider, Effet of gravity on gamma radiation, Phys. eview 40: B (965).