The Ultimate Limits of the Relativistic Rocket Equation The Planck Photon Rocket

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1 The Ultimate Limit of the Relativiti Roket Equation The Plank Photon Roket Een Gaade Haug Nowegian Univeity of Life Siene Januay 7, 7 UNRMED SERVIES TEHNIL INFORMTION GENY: Thi infomation mut not be ditibuted o leaked to hotile extateetial life o oganization uoting thee. If thi infomation i leaked, then hotile alien ould be hee befoe we know it! Peole, entitie, and leak-oganization even ueted fo miondut will be oeuted and otentially ued a oket fuel! Fo unlaified doumentation ee <htt:// btat In thi ae we look at the ultimate limit of a hoton oulion oket. The maximum veloity fo a hoton oulion oket i jut below the eed of light and i a funtion of the edued omton wavelength of the heaviet ubatomi atile in the oket. We ae baially ombining the elativiti oket equation with Haug new inight in the maximum veloity fo anything with et ma; ee [,, 3]. n inteeting new finding i that in ode to aeleate any ub-atomi fundamental atile to it maximum veloity, the atile oket baially need two Plank mae of initial load. Thi might ound illogial until one undetand that ubatomi atile with di eent mae have di eent maximum veloitie. Thi an be genealized to lage oket and give u the maximum theoetial veloity of a fully-e ient and ideal oket. Futhe, no additional fuel i needed to aeleate a Plank ma atile to it maximum veloity; thi alo might ound abud, but it ha a vey imle and logial olution that i exlained in thi ae. Key wod: Relativiti oket equation, hoton oulion, oket load, maximum eed oket, Plank ma, Plank length, edued omton wavelength, eleton. Intodution Haug [3] ha eently intodued a new maximum veloity fo ubatomi atile (anything with ma) that i jut below the eed of light given by v max = whee i the edued omton wavelength of the atile we ae tying to aeleate and l i the Plank length, [4]. Thi maximum veloity ut an ue bounday ondition on the kineti enegy, the mentum, and the elativiti ma, a well a on the elativiti Dole hift in elation to ubatomi eenhaugma.om. Thank to Vitoia Tee fo heling me edit thi manuit. lo thank to lan Lewi, Daniel Du y, aue and vt fo ueful ti on how to do high eiion alulation. ()

2 atile. Baially, no fundamental atile an attain a elativiti ma highe than the Plank ma, and the hotet edued omton wavelength we an obeve fom length ontation i the Plank length. In addition, the maximum fequeny i limited to the Plank fequeny, the Plank atile ma i invaiant, and o i the Plank length (when elated to the edued omton wavelength). Hee we will ombine thi equation with the elativiti oket equation in ode to ae how muh fuel would be needed to aeleate an ideal atile oket to it maximum veloity. We will alo extend thi onet to look at the ultimate veloity limit fo a maooi oket taveling unde ideal ondition (in a vauum). The Limit of the Photon Roket The keet [5] elativiti oket equation i given by and olved with eet to veloity we have + v m = m v I v = tanh ln! I m () (3) whee I SP i the eifi imule, whih i a meaue of the e ieny of a oket, m i the final et ma of the oket (ayload), and m i the initial et ma of the oket (ayload lu fuel). We will aume that the intenal e ieny of the oket dive i eent, that i I SP =. Thiibaially equivalent to a oket diven by hoton oulion, o a o-alled hoton oket, ee [,, ]. Next we ae inteeted in etimating the aunt of fuel needed to aeleate a ubatomi atile (uing a hoton oulion atile engine) to the Haug maximum veloity, and we get + vmax m = m v max m = m B + q m = m + q! (4) when >>l q, a i the ae fo any obeved fundamental atile, we an aoximate with a eie l exanion: l, and we get m m + + m m m m 4 l (5) Sine we aume that >>l, then thi an be futhe aoximated quite well by See alo [6], [7], [8] and[9].

3 3 m 4 m m m =m l (6) Thi mean that in ode to aeleate any atile (an eleton, fo examle) to it maximum veloity we need a atile oket with two Plank mae of fuel, m kg. The veloity of the eleton will then be m v max = tanh ln = m e m m e m m e (7) The Eintein elativiti ma of the eleton i then equal to the Plank ma. Thi i the ame maximum veloity a given by [, ]. Thee alulation equie vey high eiion and wee alulated in Mathematia. 3 Remakably, the onet of two Plank mae being ued a fuel to eah the maximum veloity fo a ubatomi atile hold fo any atile. Natually, thi an only wok beaue the maximum veloity of heavie atile i lowe than that of lighte atile. The equation 6 above i only a good aoximation a long a >>l, whih i the ae fo all obeved ubatomi atile o fa. In the eial ae, we initially have a ayload equal to the Plank ma atile with m = m we mut have = l, o we need to ue the equation a it wa befoe we ued the eie aoximation exanion q m = m + q + m = m B m = m = m (8) In othe wod, a we aeleate the Plank ma atile to it maximum veloity we will need no exta ma a fuel. t fit, thi may eem abud, a we will alway need ome enegy fo the aeleation. Howeve, the olution i imle; a [] ha hown, the Plank ma atile mut alway be at et when obeved fom any efeene fame, the Plank ma atile and the Plank length ae emakably invaiant entitie. The maximum veloity of a Plank ma atile i l v max = = l = (9) The Plank ma atile i the vey tuning oint of light. What i the veloity of light at the eie intant when it hange dietion? oding to Haug, at thi vey intant it will be at et. In the vey next intant, the Plank atile will be diolved into enegy. 3 Maximum Veloity of Roket Shi The maximum veloity of any oite objet (even a nuleu) i likely to be limited by the fundamental atile with the hotet edued omton wavelength it i ontuted fom. In othe wod, the eed See [3] and[]. We ued eveal di eent et-u in Mathematia; hee i one of them: N[Sqt[ (6699 ^( 4))^/( ^( 9))^], 5], whee 6699 ^( 4) i the Plank length and ^( 9) i the edued omton wavelength of the eleton. n altenative way to wite it i: N[Sqt[ (SetPeiion[.6699 ^( 35))^, 5]/(SetPeiion[ ^( 3))^, 5]], 5].

4 4 limit of a oket i limited by the heaviet ubatomi fundamental atile it i built fom. When thi atile eahe it maximum veloity that i given by v max = it will fit tun into a Plank ma atile and then will likely but into enegy. If thi tye of fundamental atile i a ignifiant at of the maooi objet (aehi) we ae taveling in, then the whole hi i likely to be detoyed at the ment we eah thi veloity. If the oton wa a fundamental atile, then the maximum veloity of a oket taveling unde ideal ondition (in a vauum) would likely be 4 v = () P = () Fo omaion, at the Lage Hadon ollide in 8, the team talked about the oibility of aeleating oton to the eed of % of the eed of light [4]. When the Lage Hadon ollide went full foe in 5, they ineaed the maximum eed lightly above thi (likely to aound % of the eed of light). In eality, if a oton onit of a eie of othe ubatomi atile, then the eed limit given above fo a oton will not be vey auate. ltenatively, we ould have looked at the edued omton wavelength of the quak that the tandad del laim make u the oton. 4 Summay and onluion The maximum aunt fuel needed fo any fully-e ient atile oket i equal to two Plank mae. Thi aunt of fuel will bing any ubatomi atile u to it maximum veloity. t thi maximum veloity the ubatomi atile will itelf tun into a Plank ma atile and likely will exlode into enegy. Inteetingly, we need no fuel to aeleate a fundamental atile that ha a et-ma equal to Plank ma u to it maximum veloity. Thi i beaue the maximum veloity of a Plank ma atile i zeo a obeved fom any efeene fame. Howeve, the Plank ma atile an only be at et fo an intant. The Plank ma atile an be een a the vey tuning oint of two light atile; it exit when two light atile ollide 5. Haug newly-intodued maximum ma veloity equation eem to be fully onitent with aliation to the elativiti oket equation and it give an itant new inight into the ultimate limit of fully-e ient atile oket. endix Thi how a lightly di eent and lightly e omlex way to deive the ame eult. In the ae of a hoton oket, when ombined with Haug maximum veloity fo ubatomi atile, we have 4 Hee auming l = and P = See [] foadetaileddiuionandeentationofanatomimatiledel.

5 5 atanh v max = tanh ln v max = tanh ln vmax = ln vmax e atanh( ) = m m = m e m = m e m m m = m e atanh m = m e Futhe, when >>l we an ue a eie aoximation, m m e ln B m m e m m v u m m t 4 m vmax atanh( ) l atanhb ln + B q ln + l B + l l l l! Futhe, when >>l, then then thi an be vey well-aoximated by l,thigive () (3) m 4 m m m =m l (4) Thi i the ame eult a we obtained in the main at of the ae uing a lightly eaie deivation. Refeene [] E. G. Haug. The Plank ma atile finally dioveed! Good bye to the oint atile hyothei! htt://vixa.og/ab/67.496, 6. [] E. G. Haug. new olution to Eintein elativiti ma hallenge baed on maximum fequeny. htt://vixa.og/ab/69.83, 6. [3] E. G. Haug. The gavitational ontant and the Plank unit. a imlifiation of the quantum ealm. Phyi Eay Vol 9, No 4, 6. [4] M. Plank. The Theoy of Radiation. Dove 959 tanlation, 96.

6 6 [5] J. keet. Zu theoie de aketen. Helvetia Phyia ta, 9:3 59, 946. [6] W. L. Bade. Relativiti oket theoy. meian Jounal of Phyi, 3(), 953. [7] K. B. Pomeanz. The elativiti oket. meian Jounal of Phyi, 565(34), 966. [8] G. Vuletti. Maximum teminal veloity of elativiti oket. ta tonautia, 85():8 9, 985. [9]. F. ntia. The elativiti oket. Euoean Jounal of Phyi, 3:65 63, 9. [] G. G. Zelkin. hoton oket. Pioda (Natue), : 9,96. [] V. Snilga. Thee will be no hoton oket. Bulletin of the tonomial Intitute of zeholovakia, 7:3 33, 96. [] P. Buev. On the mehani of hoton oket. Bulletin of the tonomial Intitute of zeholovakia, 5:79 8,964. [3]. Eintein. Zu elektodynamik bewegte köe. nnalen de Phyik, (7), 95. [4] G. Bumfiel. LH by the numbe. New: Biefing, Natue, Publihed online 9 Setembe, 8.