Exact Relativistic Antigravity Propulsion

Transcription

1 Eact Relativistic Antigavity Populsion Fanklin S. Fele Physics Division, Stamak, Inc., P. O. Bo 771, San Diego, CA 9198 (858) , Astact. The Schwazschild solution is used to find the eact elativistic motion of a payload in the gavitational field of a 1/ mass moving with constant velocity. At adial appoach o ecession speeds faste than 3 times the speed of light, even a small mass gavitationally epels a payload. At elativistic speeds, a suitale mass can quickly popel a heavy payload fom est nealy to the speed of light with negligile stesses on the payload. Keywods: Antigavity, Populsion, Payload, Gavitational epulsion, Geneal elativity, Relativistic gavity, Eact solutions, Time-dependent fields. PACS: 4..J, 4.5. g, 4.5. g. INTRODUCTION This pape calculates fom the eact Schwazschild solution (Misne, Thone and Wheele, 1973; Ohanian and Ruffini, 1994; Chandasekha, 1983) of Einstein s field equation the elativistically eact motion of a payload in the gavitational field of a souce moving with constant velocity. In the inetial fame of an oseve fa fom the inteaction etween the souce and payload, the payload motion is calculated eactly fo elativistic speeds of oth the souce and payload and fo stong gavitational fields of the souce. This pape pesents the fist elativistically eact solution of the unound oits of test paticles in the time-dependent gavitational field of a moving mass. The elativistically eact ound and unound oits of test paticles in the stong static field of a stationay mass have een thooughly chaacteized, fo eample, in Misne, Thone and Wheele (1973); Ohanian and Ruffini (1994); Chandasekha (1983). Ealie calculations (Misne, Thone and Wheele, 1973; Ohanian and Ruffini, 1994; Baginsky, Caves and Thone, 1977; Hais, 1991; Ciufolini and Wheele, 1995) of the gavitational fields of aitaily moving masses wee done only to fist ode in the atio of souce velocity to the speed of light, c. Even in a weak static field, these ealie calculations have only solved the geodesic equation fo a nonelativistic test paticle in the slow-velocity limit of souce motion. In this slow-velocity limit, the field at a moving test paticle has tems that look like the Loentz field of electomagnetism, called the gavimagnetic o gavitomagnetic field (Ohanian and Ruffini, 1994; Hais, 1991; Ciufolini and Wheele, 1995). Hais (1991) deived the nonelativistic equations of motion of a moving test paticle in a dynamic field, ut only the dynamic field of a slow-velocity souce. An eact solution of the field of a elativistic mass is the Ke solution (Misne, Thone and Wheele, 1973; Ohanian and Ruffini, 1994; Chandasekha, 1983; Ke, 1963; Boye and Lindquist, 1967), which is the eact stationay (time-independent) solution fo a otationally symmetic otating mass. In the stationay gavitational field of a spinning mass, the elativistic unound oits of test paticles have een appoimated in Baaès and Hogan (4). The elativistically eact calculation in this pape shows that a mass adially appoaching o eceding fom a payload 1/ with a elative velocity faste than 3 c gavitationally epels the payload, as seen y a distant inetial oseve. This antigavity is pehaps not so supising when one consides the following: (1) The velocity of a paticle adially incident on a stationay lack hole appoaches zeo as the paticle appoaches the event hoizon, as seen y a distant inetial oseve. Anothe distant inetial oseve would see the same inte- 1374

2 action as a lack hole appoaching the paticle, initially at est, and causing the paticle to acceleate away fom the lack hole until the paticle attains a speed nea the hoizon asymptotically appoaching the speed of the lack hole. () Although time-independent, the Ke field ehiits an inetial-fame-dagging effect (Misne, Thone and Wheele, 1973; Ohanian and Ruffini, 1994; Ciufolini and Wheele, 1995) simila to that contiuting to gavitational epulsion at elativistic velocities, namely, a foce in the diection of the moving mass. The inetial-dagging foce can dominate the adial foce of attaction, even nea a lack hole. Inside the so-called static limit suface of a spinning lack hole, an oseve can theoetically halt his descent into the lack hole, ut cannot halt his angula motion induced y inetial-fame dagging (Misne, Thone and Wheele, 1973). (3) Because the affine connection is non-positive-definite, a geneal pediction has een made that geneal elativity could admit a epulsive foce at elativistic speeds (Sachs, 3). But since the epulsive-foce tems ae second-ode and highe in souce velocity, this antigavity at elativistic speeds has not peviously een found. Paticulaly notewothy in the new eact solutions is that aove a citical velocity any mass, no matte how light o how distant, poduces an antigavity field. Though at least twice as stong in the fowad diection of motion, the antigavity field even epels paticles in the ackwad diection. This means that a stationay mass will epel 1/ masses that ae adially eceding fom it at speeds geate than 3 c, with ovious cosmological implications. This pape calculates the eact motion impated to test paticles o payloads y a souce moving at constant velocity. A stong gavitational field is not necessay fo antigavity populsion. Solely fo the pupose of deiving an eact solution, howeve, the souce is consideed to e much moe massive than the payload, so that the enegy and momentum deliveed to the payload have negligile eaction on the souce motion. APPROACH Figue 1 illustates the two-step appoach of this pape to calculating the eact motion of a payload mass m in the field of a souce of mass M and constant velocity câ. Fist, the unique tajectoy in the static Schwazschild field of a stationay mass M is found fo which the peigee, the distance of closest appoach, of the payload is and the asymptotic velocity fa fom the stationay mass is câ. In the static field of M, the tajectoy is time-evesile. Second, the tajectoy is Loentz-tansfomed to a efeence fame moving with constant velocity câ,inwhich the mass M has a constant velocity câ, and the payload is initially at est. The Loentz tansfomation occus etween two inetial oseves fa fom the inteaction, in asymptotically flat spacetime. The eact equations of motion of a allistic payload in spheical coodinates in the static Schwazschild field of the mass M ae (Misne, Thone and Wheele, 1973; Ohanian and Ruffini, 1994; Chandasekha, 1983): (a) Schwazschild Solution in Static Field of M. () Solution Loentz-Tansfomed to Initial Rest Fame of Payload Fa fom M. Pimes Denote Loentz-Tansfomed Quantities. FIGURE 1. Two-Step Eact Solution of Antigavity Populsion of Payload Mass m y Relativistic Mass M. 1375

3 dt / d /, (1) ( d/ d) ( c L / ) c, () d d L / /, (3) whee is the pope time, c is the constant total specific enegy, L is the constant specific angula momentum, and ( ) 1 GM / c is the g component of the Schwazschild metic. By sustitution of Eq. (1), the equations of motion in coodinate time t ecome: 1 / 3 / L c, (4) L / c, (5) whee / c and / c ae the and components of the nomalized payload velocity â() a distant inetial oseve. An ovedot indicates a deivative with espect to t. measued y As shown in Fig. 1(a), if the payload at peigee,, has a speed c, then fom Eqs. (4) and (5): L, (6) c /, (7) /( ) whee ( ). And as shown in Fig. 1(a), if the payload has a speed c fa fom the mass M, whee 1 and f, then fom Eq. (4): 1/(1 ), (8) The payload speeds fa fom the mass M and at peigee ae elated though Eqs. (7) and (8) y: (1 ). (9) Since the payload is moving in the static gavitational potential of the mass M in this efeence fame, the payload speed given y Eqs. (4) and (5) is a function of only: ( ) [1 ( / ) ( L/ c) (1 )]. (1) 1/ And () is found y integating the eact oital equation in a Schwazschild field (Chandasekha, 1983): whee 1/. The adial acceleation of the payload: ( d / d) GM( / c / L ) ( c / L), (11) 3 GM 3 L 3GM c 3, (1) c 1376

4 indicates epulsion y the stationay mass M wheneve: 3 L GM 1 GM 3 c. (13) Equation (13) is the eact elativistic stong-field condition fo antigavity epulsion of a payload to e measued y a distant inetial oseve. A payload fa fom the stationay mass M, fo oth and GM / c,isseen 1/ to e epelled y M wheneve 3/ o 3. Net, we tansfom to the inetial efeence fame shown in Fig. 1(), in which the mass M moves in the diection at constant speed c,andthemassm is initially at est at () and (). (Since only unound oits ae consideed hee, can always e chosen lage enough that is negligile.) In the est fame of M, the and y components of â wee: cos sin, (14) sin cos. (15) y In the Loentz-tansfomed fame, in which M moves in the diection at constant speed c, the components of the eact payload velocity câ and acceleation cd â/dt ae given y (Jackson, 196; 1975): ( )/(1 ), (16) ( / )/(1 ), (17) y y d dt, (18) (1 ) 3 d y (1 ) y y dt (1 ) 3. (19) RESULTS Equations (4), (5), and (14) (17) completely define the eact payload velocity measued y a distant inetial oseve, to whom the mass M appeas to e moving in the diection at constant speed c. Two cases of special inteest fo thei simplicity ae iefly analyzed hee: (1) The eact payload velocity with puely adial motion ( L ) in the stong field of a lack hole; and () the appoimate velocity with aitay unounded motion in a weak field. Fo puely adial motion, Eqs. (4), (14), and (16) give the eact speed of a payload in the field of a moving lack hole as: ( / ), () 1 ( / ) 3 1/ 3 1/ whee 1 BH /,and BH is the adius of the lack hole. Figue shows two fames fom the animated eact solutions of Eq. (), which may e viewed o downloaded ( Download Souce ) at Fele (5a). The fist fame in Fig. shows the initial configuation of a lack hole diectly appoaching an initially stationay payload at.9c. The second fame shows the positions of the lack hole and the payload afte the payload has een acceleated to.6c y gavitational epulsion, as seen y a distant inetial oseve in the initial est fame of the payload. 1377

5 FIGURE. Two Fames fom Animated Eact Solutions of Eq. () Showing Payload (Diamond) Repelled y Black Hole (Cicle). FIGURE 3. Eact Payload Speed vs. Nomalized Distance to Black Hole fo Constant Black-Hole Appoach Speeds Indicated. Figue 3 shows the adial velocity of the payload fo seveal values of the lack hole speed,. (In Fig. 3, all 4 payloads stat fom est at 1 BH.) Figue 3 illustates seveal inteesting esults concening the osevations of a distant inetial oseve witnessing a lack hole diectly appoaching a payload. At any closing speed, the final payload speed is always the speed of the lack hole, appoached as the payload appoaches BH. At a closing speed 1/ faste than 3 c, the payload will e continuously epelled y the lack hole at any distance. At slowe closing speeds, the payload will always e seen to e epelled y the stong field of the lack hole at least within a adius 3 BH. Equation (18) shows that Eq. (13) is the eact elativistic stong-field condition fo antigavity epulsion of a payload measued y a distant oseve in the est fame of the souce o the initial est fame of the payload. That 1/ is, eithe oseve will see the payload epelled when (1 /3 ). Fo any unounded motion of a payload aout a stationay mass, a nomalized gavitational potential is defined as 1/ V() ( )/(1 ),whee ( ) (1 ). This eact potential is the payload total specific enegy, less the specific est plus kinetic enegies, nomalized to times the classical potential at peigee. In the weak-field 3 appoimation, this nomalized potential, V ( / R)(13 / R ), plotted against R / in Fig. 4, is independent of field stength and of angula momentum. The dashed cuve is the nomalized citical adius, 1/ (1 1/3 ), eyond which the potential is epulsive ( dv / d ). Rc By comining Eqs. (4), (1), and (18) with the FitzGeald-Loentz contaction (Jackson, 196; 1975), s /, whee s ct, the weak field of mass M diectly appoaching a payload, as measued y a distant inetial oseve in the initial est fame of the payload (when it is fa fom M), is found to e: cd/ dt (1 3 ) GM / s. (1) 1378

6 FIGURE 4. Nomalized Potential of Weak Gavitational Field of Stationay Mass vs. Nomalized Distance fo Values of Indicated. Aove Dashed Cuve ( / Rc ), Potential is Repulsive ( dv / d ). This same esult can e deived fom the geodesic equation. On the ais, the weak-field metic of mass M moving with constant speed c along the ais is lineaized as g =diag(1, 1, 1, 1) h. The nonzeo components of h in Catesian coodinates, ( ct,, y, z ),ae h h11 (1 ), h1 h1 4,and h h33 /, whee the dimensionless potential, GM / c s, satisfies the hamonic gauge condition, / t c /. Fom the geodesic equation, the equation of motion fo the payload in the weak field of 4 M is d s/ d ( c / ) d / ds. The fist integals of the motion ae: d / d c (13 ), () dt/ d 1 (1 ). (3) In the weak-field appoimation, tems of ode ae neglected, so that d/ d d/ dt, and the acceleation of the payload fom Eq. () is d / dt (1 3 ) GM / s, in ageement with Eq. (1). The sepaation { R} et of the payload and a diectly appoaching mass M at the etaded time t{ R} et / c { R} s/(1 ). In tems of this etaded sepaation, the acceleation of the payload fom Eq. (1) is: et is cd/ dt (1 ) (1 3 ) GM /{ R }, (4) 5 et in ageement with the weak etaded field found y Liénad-Wiechet methods (Fele, 5). In the weak-field appoimation, the field on a stationay test paticle is the same as the field on a payload moving feely along a geodesic, as long as the payload stats fom est. That is, the gavitomagnetic tems in the geodesic equation ae of the same ode as tems that ae neglected in the weak-field appoimation. Fom Eq. (1), the minimum speed eached y the payload in Fig. 1(a), coesponding to the maimum deceleation of the payload y the weak static field of M, is: min [1 ( GM / c )/ R c ], (5) 1379

7 at the nomalized citical adius, R c. Fom Eqs. (16) and (5), in the Loentz-tansfomed fame of Fig. 1(), the maimum speed deliveed to a payload, initially at est, is: ( GM / c ) ( 1/3 ). (6) ma This field of a adially appoaching mass is geate than that of a adially eceding mass y a facto of aout (1 ), accoding to Eq. (18). CONCLUSION The weak-field condition used to deive Eq. (6) is ( GM / c ) 1. Theefoe, we find that the maimum speed that can e deliveed to a payload, initially at est, y the weak field of a much heavie mass moving at 1/ constant speed 3 is ma 1/(3 ). As was seen in Fig., the maimum speed that can e deliveed to a payload, initially at est, y the stong field of a lack hole diectly incident on it at any speed is ma. Whethe the payload is acceleated y a stong o a weak field, the payload tavels along a geodesic. The only stesses on the payload, theefoe, ae the esult of tidal foces in the acceleated fame of the payload. These stesses can e aanged y choice of the tajectoy to e kept within acceptale limits. Geate pactical polems fo gavitational populsion ae finding a suitale and accessile dive mass at elativistic velocities, and maneuveing the payload in and out of the dive tajectoy. The seeming scacity of suitale elativistic dives in ou galactic neighohood may well e due to dag y just the sot of gavitational epulsion analyzed in this pape. The analysis found that at adial appoach o ecession speeds faste than 3-1/ times the speed of light, any mass gavitationally epels a payload at any distance. The fowad antigavity field of a suitaly heavy and fast mass might e used to popel a payload fom est to elativistic speeds. c diag G g h NOMENCLATURE = distance (m) in coodinate fom cente of souce to closest point on payload tajectoy = speed of light (m/s) = diagonal tenso with elements given y the aguments of diag = gavitational constant (m 3 /s kg) = metic tenso = lineaized pat of metic tenso (weak-field metic tenso minus the Loentz metic) L = constant specific angula momentum (m /s) of payload in est fame of souce m = est mass (kg) of payload M = est mass (kg) of gavitational souce = Schwazschild adial coodinate (m) = adial component of payload velocity (m/s) measued y distant inetial oseve in est fame of souce BH = adius (m) of a Schwazschild lack hole = coodinate (m) Loentz-tansfomed to distant inetial oseve in initial est fame of payload = coodinate (m) at initial time t, when payload is at est R c = citical adius eyond which potential is epulsive, nomalized to { R } et = etaded sepaation (m) of the payload fom a moving souce s = ange (m) equal to the Loentz-tansfomed quantity ct t = Schwazschild time coodinate (s) t =timet (s) Loentz-tansfomed to distant inetial oseve in initial est fame of payload V() = nomalized gavitational potential of a stationay mass = Catesian Schwazschild coodinate (m) in diection opposite â 138

8 = coodinate (m) Loentz-tansfomed to distant inetial oseve in initial est fame of payload y = Catesian Schwazschild coodinate (m) in diection nomal to â y = coodinate y (m) Loentz-tansfomed to distant inetial oseve in initial est fame of payload â = payload velocity, nomalized to c, measued y distant inetial oseve in est fame of souce â min = constant velocity of souce, nomalized to c, in initial est fame of payload = payload speed, nomalized to c, at peigee measued y distant inetial oseve in est fame of souce = minimum speed, nomalized to c, attained y payload at citical adius of stationay souce = adial component of â() = deivative of with espect to t (s -1 ) = azimuthal component of â() â = nomalized velocity â Loentz-tansfomed to distant inetial oseve in initial est fame of payload ma = maimum speed, nomalized to c, attained y payload in initial est fame of payload = component of â = y component of â y c = constant total (est + kinetic + potential) specific enegy (J/kg) of payload in est fame of souce = invese of coodinate (m -1 ) = pope time of payload (s) = Schwazschild azimuthal coodinate (ad) = angula velocity (ad/s) of payload measued y distant inetial oseve in est fame of souce = coodinate (ad) Loentz-tansfomed to distant inetial oseve in initial est fame of payload = coodinate (ad) at initial time t, when payload is at est = dimensionless gavitational potential () = g component of the Schwazschild metic = constant value at peigee of g component of the Schwazschild metic () = g component of the metic of a Schwazschild lack hole REFERENCES Baaès, C., and Hogan, P. A., Scatteing of High-Speed Paticles in the Ke Gavitational Field, Phys. Rev. D 7, (4). Boye, R. H., and Lindquist, R. W., Maimal Analytic Etension of the Ke Metic, J. Math. Phys. 8,65 81 (1967). Baginsky, V. B., Caves, C. M., and Thone, K. S., Laoatoy Epeiments to Test Relativistic Gavity, Phys. Rev. D 15, (1977). Chandasekha, S., The Mathematical Theoy of Black Holes, Ofod Univesity Pess, New Yok, Ciufolini,I.,andWheele,J.A.,Gavitation and Inetia, Pinceton Univesity Pess, Pinceton, Fele, F. S., Eact Relativistic Antigavity Populsion, (5a), accessed 7 June 5. Fele,F.S., Weak Antigavity Fields in Geneal Relativity, (5), accessed 7 June 5. Hais, E. G., Analogy Between Geneal Relativity and Electomagnetism fo Slowly Moving Paticles in Weak Gavitational Fields, Am. J. Phys. 59,41 45 (1991). Jackson, J. D., Classical Electodynamics (John Wiley & Sons, NY, 196); Classical Electodynamics, nd Ed., John Wiley & Sons, New Yok, Ke, R. P., Gavitational Field of a Spinning Mass as an Eample of Algeaically Special Metics, Phys. Rev. Lett. 11, (1963). Misne, C. W., Thone, K. S., and Wheele, J.A., Gavitation, W.H. Feeman & Co., San Fancisco, Ohanian, H. and Ruffini, R., Gavitation and Spacetime, nd Ed., W. W. Noton & Co., New Yok, Sachs, M., in Mach s Pinciple and the Oigin of Inetia, edited y M. Sachs and A. R. Roy, C. Roy Keys, Monteal, Canada,

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