Reversed Gravitational Acceleration for High-speed Particles

Transcription

1 1 Revesed Gavitational Acceleation fo High-speed Paticles Hans C. Ohanian Depatment of Physics Univesity of Vemont, Bulington, VT , USA Novembe 14, 011 Abstact. Examination of the fee-fall motion of paticles of extemely high-speed in the Schwazschild geomety eveals that the gavitational acceleation of such paticles is evesed when measued in Schwazschild coodinates. High-speed paticles deceleate when moving adially downwad, and they acceleate when moving upwad. The onset of this abnomal behavio occus at a speed of 1/ 3 times the local value of the speed of light. Howeve, the gavitational foce always emains attactive. PACS numbes: q, 04.0.Cv, g Einstein fist discoveed the gavitational time dilation in 1911, on the basis of his equivalence pinciple. Fom the time dilation, he immediately deduced the slow-down and deflection of light in a gavitational field. 1 His 1911 esult fo the eduction of the speed of light was in eo by a facto of, but he coected this a few yeas late, in his theoy of Geneal Relativity. Howeve, Einstein and his followes failed to notice a cuious consequence of the slowdown of the speed of light: high-speed paticles enteing a gavitational field also slow down, and thei gavitational acceleation is evesed, that is, the paticles deceleate when moving adially downwad in the gavitational field of a massive body, and they acceleate when moving upwad. It is easy to undestand why this must happen in the case of an ultaelativistic paticle of initial speed v 1. When this paticle descends in the gavitational field of a massive body, it 1 A. Einstein, Annalen d. Physik 35, 898 (1911). The case of infall of high-speed paticles seems not to have been teated in the available liteatue. C. W. Misne, K. S. Thone, and J. A. Wheele, Gavitation (Feeman, San Fancisco, 1973 ) teat adial infall in detail, but only fo the case of paticles that stat fom est at a given initial adius R (pp ).

2 must obey the speed limit set by the deceasing speed of light, and this compels it to deceleate in the same way as light. Of couse, a low-speed paticle enteing this gavitational field will acceleate in the nomal way. It is then obvious that thee is some citical speed v 1that seves as a citeion fo acceleation vs. deceleation: paticles of speed smalle than acceleate, but paticles of speed lage than vcit deceleate. Hee I will show that, in the Schwazschild geomety, the citical speed fo evesal of the adial acceleation is vcit 1/ 3 times the local, slowed speed of light. This elationship is independent of position and independent of the value of the mass. In this context, the deceleating speed is, of couse, a coodinate speed, as is the slowed speed of light. That is, the speed is the atio d/dt of the changes in Schwazschild adial and time coodinates. But this coodinate speed is not devoid of physical significance. The coodinate speed d/dt can be eadily detemined by means of measuements with instuments placed in the fa-away egion, fo instance, by ada-anging, with the instantaneous distance calculated accoding to the time-delay and the known fomula fo the coodinate speed of light. Fo a ada pulse taveling adially, emitted at 0 and eflected by the paticle at, the ound tip tavel time t is given by 3 cit vcit o d t 1 GM / 0 / GM 1 / GM 1 0 t ( 0 ) 4GM ln (1) fom which the coodinate can be evaluated immediately. Thus, this coodinate has the status of a measuable quantity, and so does the speed d/dt calculated fom its ate of change. Fo a paticle falling adially, the equation of geodesic motion in the Schwazschild geomety 4 educes to a simple expession, eminiscent of the Newtonian equation: d d GM () Accodingly, fo adial infall, the pope speed d / d always inceases. But the coodinate speed d / dt diffes fom this pope speed by a facto d / dt which is smalle than 1 and 3 W. Rindle, Essential Relativity (Spinge-Velag, New Yok, 1977), p See, e.g., H. C. Ohanian and R. Ruffini, Gavitation and Spacetime (W. W. Noton & Co, New Yok, 1994), p. 401.

3 3 deceasing. Fo a high-speed paticle, the decease of this facto can ovewhelm the incease of d / d and lead to a decease of d / dt, that is, a deceleation. The left side the equation of motion is d d dt d dt d d d dt d d d dt d ddt dt d d With the substitution dt 1 (1 / ) ( / ) / (1 / ) d GM d dt GM (3) this becomes d GM d GM GM d (1 GM / ) v dt (1 GM / ) v v whee v d / dt. Accodingly, the equation of motion becomes GM d GM GM v v GM (1 GM / ) v dt (1 GM / ) The citical speed vcit is detemined by the condition d / dt 0 which implies GM GM 1 4 GM v v (1 GM / ) v (1 GM / ) (4) This is a quadatic equation fo v, with the solution v cit 1 GM 1 3 (5) Since (1 GM / ) is the local, slowed speed of light, this says that the citical speed is 1/ 3 times the local speed of light. Figue 1 shows plots of the speed v d / dt of feely-falling paticles as a function of the adial coodinate. [The speed is detemined by the fist integal of the equation of motion (),

4 4 v0 const. 0 1 d GM GM 1 d 1 v (6) whee v0 is the initial speed of the paticle at lage distance. The combination of Eqs. (6) and (3) then gives the value of d / dt, d [ GM / v / (1 v )](1 GM / ) dt 1 [ GM / v / (1 v )] / (1 GM / ) ] (7) Note that fo an initial speed v0 1/ 3 (not plotted in Fig. 1), the paticle poceeds with constant speed as long as the linea appoximation fo the gavitational field is valid, that is, fo GM / 1 (within this linea egime, the citical speed is simply v 1/ 3 times the standad speed of light). The paticle then deceleates when it entes the nonlinea egime of the Schwazschild geomety. Also note that fo a paticle moving in a tansvese, o tangential, diection, the adial acceleation is always downwad, that is, the acceleation does not evese at high speed. Thus, such a paticle deflects in the nomal way and, fo a paticle of speed v 1, the deflection is the same as fo a light signal. Taken at face value, the discepancy between the signs of the acceleations of low-speed and high-speed paticles is a peplexing violation of the equivalence pinciple. Geneal Relativity attibutes this discepancy to a bad choice of coodinates the coodinates and t do not epesent locally measued distances and times. In local geodesic coodinates, with 0, the acceleations of all paticles ae zeo, and the discepancy disappeas. In 1911, Einstein would not have known about this way of avoiding the violation of the equivalence pinciple. If he had noticed that the slowed speed of light equies a slowed speed fo ultaelativisic paticles, he would have been in a quanday. But he didn t notice, and neithe did anybody else (until seveal yeas late; see Coection attached at end of this pape). cit

5 5 d/dt /GM Fig. 1 Speed of a feely-falling paticle vs. adial coodinate in units of GM. Uppe cuve: light signal. Middle cuve: paticle of initial speed moe than v cit ; the paticle deceleates monotonically as it falls downwad fom a lage initial distance. Lowe cuve: paticle of initial speed less than v cit ; the paticle acceleates until its speed eaches the citical value at a adial coodinate of 3.1 units, and it then deceleates. The value of v cit at the peak is 1/ 3 times the local, slowed speed of light.

6 6 Coection and Addendum Contay to my assetion that Einstein s followes failed to notice the deceleation of high-speed paticles enteing a gavitational field, thee ae actually a handful of publications that discuss this deceleation, anging fom a 1917 pape by Hilbet 5 to ecent papes by Mashoon 6 and by Felbe 7 (McGude 8 gives a compehensive list of papes befoe 198). I am indebted to B. Mashoon, F. Felbe, and J. Mooe fo binging these publications to my attention. Hilbet, a bette mathematician than physicist, unfotunately misconstued this deceleation as a epulsive gavitational foce ( die Gavitation wikt abstossend ), and some authos imitated this mistake. Hilbet naively assumed that the foce is in the diection of the coodinate acceleation d / dt, wheeas he should have known that the foce is in the diection of the ate of change of momentum. Accoding to Eq. (), the ate of change of the elativistic momentum is d d d p GMm m d d d which is always negative. Theefoe the diection of the foce (and also the diection of the pope, o elativistic, acceleation d / d ) is always downwad, that is, the foce is always attactive. As aleady mentioned on p., fo a high-speed paticle moving downwad, and d / dt d / d can have opposite signs, because d / dt diffes fom d / d by a facto d / dt, and the decease of this facto can ovewhelm the incease of d / d and lead to a decease of d / dt, that is, a deceleation. But the sign of d / dt does not detemine the sign of the foce, which always emains attactive and inceases in magnitude with deceasing, accoding to the invese-squae law (8). Hilbet s epulsive foce is a delusion that ests on bad physics, and Felbe s contention that this epulsive foce can be exploited fo an antigavity spacecaft populsion scheme ests on equally bad physics. Felbe s scheme 7 is meely a elativistic vesion of the familia Newtonian slingshot effect that has been used to boost the teminal speed (and momentum) of seveal spacecaft by using obits that swing the spacecaft aound a moving planet; this involves the gavitational attaction of the planet, not any kind of epulsion. (8) 5 D. Hilbet, Nachichten Königl. Ges. Wiss. Göttingen, 1917, p. 53, available at The evesal of acceleation is mentioned in passing on the last page of this pape. 6 B. Mashoon, Int. J. Mod. Phys. D 14, 05 (005). 7 F. Felbe, AIP Confeence Poceedings 108, 47 (010), also available at axiv: v. 8 C. H. McGube, III, Phys. Rev. D 5, 3191 (198).