Plank Quantization of Newton and Einstein Gavitation Espen Gaade Haug Nowegian Univesity of Life Sienes Mah 0, 06 Abstat In this pape we ewite the gavitational onstant based on its elationship with the Plank length and, based on this, we ewite the Plank mass in a slightly di eent fom (that gives exatly the same value). In this way we ae able to uantize a seies of end esults in Newton and Einstein s gavitation theoies. The fomulas will still give exatly the same values as befoe, but eveything elated to gavity will then ome in uanta. Numeially this only has impliations at the uantum sale; fo mao objets the disete steps ae so tiny that they ae lose to impossible to notie. Hopefully this an give additional insight into how well o not so well (ad ho) uantized Newton and Einstein s gavitation ae potentially linked with the uantum wold. Key wods: Quantized gavitation, gavitational onstant, esape veloity, gavitational time dilation, Shwazshild adius, Plank length, bending of light, Plank mass, Plank length. Foundation We suggest that the gavitational onstant should be witten as a funtion of Plank s edued onstant G p = @ () whee is the edued Plank s onstant and is the well tested ound-tip speed of light. We ould all this Plank s fom of the gavitational onstant. The paamete @ is an unknown onstant that is alibated so that G p mathes ou best estimate (measuement) fo the gavitational onstant. As shown by Haug (06), the Plank fom of the gavitational onstant enables us to ewite the Plank length as and the Plank mass as l p = m p = Gp = s s = G @ p s @ = @ () = Using the gavitational onstant in the Plank fom, as well as the ewitten Plank units, we ae easily able to modify a seies of end esults fom Newton and Einstein s gavitational theoies to ontain uantization as well. Newton Univesal Gavitational Foe The Newton gavitational foe is given by F G = G p m m (4) e-mail espenhaug@ma.om. Thanks to Vitoia Tees fo helping me edit this manusipt. In vesion 5 a mathematial typo in the gavitational aeleation and Newtons vesion of Kelles thid law was fixed. If you find this pape of inteest you will possibly also find my eent pape in the Relativity and Cosmology setion The Collapse of the Shwazshild Radius: The End of Blak Holes of inteest. ()
Using the gavitational onstant of the fom G p = @ ewite the Newton gavitational foe fo two Plank masses as and the Plank mass of m p = we an In the speial ase whee = @ we get F GP = G p m pm p F GP = @ = (5) be F Gp = (6) @ @ It seems fom this that gavity ould be intepeted as hits pe seond. Fo lage masses the fom will F Gp = G p N m pn m p F Gp = G p N N m p F Gp = N N (7) whee N is the numbe of Plank masses in objet one and N is the numbe of Plank masses in objet two. In the ase when the two masses ae of eual size we have F GP = N (8) Esape Veloity at the Quantum Sale The taditional esape veloity is given by GM v e = (9) whee G is the taditional gavitational onstant and M is the mass of the objet we ae tying to esape fom, and is the adius of that objet. In othe wods, we stand at the sufae of the objet, fo example a hydogen atom o a planet. Based on the gavitational onstant witten in the Plank fom we an find the esape veloity at Plank sale; see also the Appendix fo a deivation fom sath. It must be GpNm p v e,p = s N @ v e,p = v e,p = v e,p = whee N is the numbe of Plank masses in the planet o mass in uestion. A patiulaly inteesting ase is when we only have one Plank mass N =and =@ (this is atually the Shwazshild adius of a Plank mass objet). This gives us (0) v e,p = @ @ v e,p = () as the esape veloity fo a patile with Plank mass with adius @ is. Next we will see if the fomula above an also be used to alulate the esape veloity of Eath. The Eath s mass is 5.97 0 4 kg. We must onvet this to the numbe of Plank masses. The Plank mass is
m p = 8.765 0 5.97 0 The Eath s mass in tems of the numbes of Plank masses must be 4.7488 0..765 0 8 Futhe the adius of the Eath is 6 7 000 metes. We an now just plug this into the Plank sale esape veloity: v e,p =.7488 0.6687 0 v e,p = 99 79 458 5 85.7 metes/seond 67000 whih is eual to 40,69 km/h, the well-known esape veloity fom the Eaths gavitational field. We think ou new way of looking at gavity ould have onseuenes fo the undestanding of gavity. Gavitation must ome in disete steps and the esape veloity must also ome in disete steps fo a given adius; this is beause the amount of matte likely omes in disete steps. 4 Obital Speed The obital speed is given by GM v o We an ewite this in the fom of the Plank gavitational onstant and the Plank mass as () This an also be witten as v o,p GpNm p s @ N v o,p N@ v o,p. () v o,p p ve N@ = (4) 5 Gavitational Aeleation The gavitational aeleation field in moden physis is given by This an be ewitten in uantized fom as g = GM (5) g = GpM g = @ N g = N@ (6)
4 6 Gavitational Paamete The standad gavitational paamete is given by This an be ewitten in uantized fom as µ = GM (7) µ p = G pm µ p = G pnm p 7 Keple s Thid Law of Motion µ p = @ N µ p = N@ (8) The Newton mehanis vesion of Keple s thid law of motion fo a iula obit is given by a = 4 G(M s + m) Whee M s is the mass of the Sun, m the mass of the planet, P is the peiod, and a is the semi-majo axis. This an be e-witten as (9) a = a = a = 4 G p(n m p + N m p) @ 4 N + N 4 @ (N + N ) (0) whee N is the numbe of Plank masses in the mass of the Sun M s and N is the numbe of Plank mass of the planet m. IntheasetheplanetsmassismuhsmallethantheSunsmass,weanusethe following appoximation a 4 @ N whee N is now the numbe of Plank masses in the Sun. () 8 Gavitational Time Dilation at Plank Sale Einstein s gavitational time dilation is given by t 0 = t f GM ve = t f () whee v e is the taditional esape veloity. We an ewite this in the fom of uantized esape veloity (deived above). v u t Let s see if we an alulate the time dilation at, fo example, the sufae of the Eath fom Plank sale gavitational time dilation. The Eath s mass is 5.97 0 4 kg. And again, the Eath s mass in ve,p N @ ()
5 5.97 0 tems of the Plank mass must be 4.7488 0. Futhe, the adius of the Eath is.765 0 8 6 7 000 metes. We an now just plug this into the uantized gavitational time dilation.7488 0.6687 0 5 t f 0.999999999095 67000 That is fo evey seond that goes by in oute spae (a lok fa away fom the massive objet), 0.99999999909500 seonds goes by on the sufae of the Eath. That is fo evey yea in in oute spae (vey fa fom the Eath), thee ae about milliseonds left to eah an Eath yea. This is natually the same as we would get with Einstein s fomula. Still, the new way of witing the fomula gives additional insight. Ciula obits gavitational time dilation The time dilation fo a lok at iula obit is given by GM ve t 0 = t f = (4) whee v e is the taditional esape veloity. We an ewite this in the fom of uantized esape veloity (deived above). v u t ve,p N@ N @ (5) 9 The Shwazshild Radius The Shwazshild adius of a mass M is given by s = GM (6) Rewitten into the uantum ealm as desibed in this atile it must be s = GpM s = GpNmp Fo a lok at the Shwazshild adius we get a time dilation of s = @ N s = (7) =0. (8) At the Shwazshild adius, time stands still. Fo a adius shote than that the gavitational time dilation euation above beaks down. At obital adius lage then s Exept if we assume the @ epesents the adius of an indivisible patile. Thus if we move away fom the point patile onept, this would simply mean that we ould not go below the Plank sale Shwazshild Radius.
6 Mass in Shwazshild mete The Shwazshild mass in tems of metes is given by This an be e-witten as mete = GM (9) mete = GpNmp mete = @ mete = N@ (0) 0 Quantized Gavitational Bending of Light The angle of defletion in Einstein s Geneal elativity theoy is given by This an be ewitten as GR = 4GM GRH = 4GpM GRH = 4GpNmp GRH = 4 @ GRH = 4N @ N () whee N is the numbe of Plank masses making up the mass we ae inteested in. Fom the fomula above, this means that the defletion of angles omes in uanta. Lets also ontol that ou Plank sale defletion ooted in Plank and GR is onsistent fo lage bodies like the Sun, fo example. The sola mass is M s.988 0 0 kg. The Sun s mass in tems of the numbe of Plank masses must be.988 0 0.765 0 8 9.4 0 7. Futhe, the adius of the Sun is s 696 4 000 metes. We an just plug this into the Plank sale defletion: GRH = 4N@ = 4 9.4 07.6687 0 5 696 4 000 8.48 0 06 () 648 000 If we multiply this by we get a bending of light of about.75 aseonds o.75 of a degee. 600 This is the same as has been onfimed by expeiments and helped make Einstein famous, as Newton gavitation supposedly only pedited half of the bending of light. Newton bending of light is given by Newton = GM () See fo example Soaes (009) and Momeni (0) fo deivations of bending of light unde Newton gavitation. Gavitational Redshift The Einstein gavitational edshift is given by lim z() =!+ (4) GM R whee R e is the distane between the ente of the mass of the gavitating body and the point at whih the photon is emitted. This we an ewite as
7 lim z() =!+ lim z() = s!+ lim z() =!+ GM R e @ N @ R e (5) Futhe in the Newtonian limit when R e is su an appoximate the above expession with iently lage ompaed to the Shwazshild adius we GM lim z()!+ R e lim z() N!+ R e Einstein s Field Euation And finally we get to Einstein s field euation. It is given by N@ lim z() (6)!+ R e 8 G R µv gµvr = Tµv (7) 4 I am fa fom an expet on Einstein s field euation, but based on the Plank gavitational onstant given in this pape we an ewite it as 8 Gp R µv gµvr = T 4 µv @ 8 R µv gµvr = T 4 µv 8 @ R µv gµvr = Tµv (8) Bea in mind = h and based on this we an altenatively wite Einstein s field euation as R µv gµvr = 6 @ T µv (9) h The potential intepetation and usefulness of this ewitten vesion of Einstein s field euation we leave up to othe expets to onside. An inteesting uestion is natually whethe o not it is onsistent with some of the deivations given above in this fom. Table Summay The table below summaizes ou ewiting of some gavitational fomulas. The output is still the same, but based on this view of gavity, masses, gavitational time dilation, and even esape veloity all ome in disete steps. 4 Conlusion By making the gavitational onstant a funtion fom of the edued Plank onstant one an easily ewite many of the end esults fom Newton and Einstein s gavitation in uantized fom. Even if this is seen as an ad ho method, it ould still give new insight into what degee uantized Newton s gavitation and Geneal elativity ae onsistent with the uantum ealm.
8 Table : The table shows some of the standad gavitational elationships given by Newton and Einstein and thei expession in uantized fom. Units: Newton and Einstein fom: Quantized-fom: Gavitational onstant G 6.67408 0 G p = @ Newton s gavitational foe F G = G MM Newton s gavitational foe F G = G mpmp @ Keple s thid law Newton s Esape veloity fom any mass v e = Obital veloity fo any mass 4 a = G(M s+m) GM v o GM F G = G p m pm p F G = N N = @ @ = @ @ 4 a = @ (N +N ) v e,p = v o,p N @ N Gavitational aeleation field g = GM g p = N@ Gavitational paamete µ = GM µ p = N@. GM v Gavitational time dilation t 0 = t f = e t o = t f Obital time dilation t 0 = t f GM = v e N@ t o = t f Shwazshild adius s = GM s = Bending of light GR = 4GM GRH = 4N@ Blak holes Possible Depends on uantum intepetation Appendix: Esape veloity Deivation of the esape veloity fom Plank sale E mv GmM E GN m pn m p Nmpv E N N v E N v @ N N @ N N N (40) whee N is the numbe of Plank masses in the smalle mass m (fo example a oket) and N is the numbe of Plank masses in the othe mass. This we have to set to 0 and solve with espet to v to find the esape veloity: N v N N = 0 v = NN N v = N @ v = N @ This is a uantized esape veloity. Sine N anels out we an simply all N fo N and wite the esape veloity as v = N @ whee N is the numbe of Plank masses in the mass we ae tying to esape fom. Refeenes Einstein, A. (96): Näheungsweise Integation de Feldgleihungen de Gavitation, Sitzungsbeihte de Königlih Peussishen Akademie de Wissenshaften Belin. (4) (4)
9 Haug, E. G. (06): The Gavitational Constant and the Plank Units. A Deepe Undestanding of the Quantum Realm, www.vixa.og Mah 06. Momeni, D. (0): Bending of Light a Classial Analysis, Woking pape, pp. 4. Newton, I. (686): Philosophiae Natualis Pinipia Mathematis. London. Plank, M. (90): Uebe das Gesetz de Enegieveteilung im Nomalspetum, Annalen de Physik, 4. Shwazshild, K. (96a): Übe das Gavitationsfeld eine Kugel aus Inkompessible Flussigkeit nah de Einsteinshen Theoie, Sitzungsbeihte de Deutshen Akademie de Wissenshaften zu Belin, Klasse fu Mathematik, Physik, und Tehnik, p.44. (96b): Übe das Gavitationsfeld eines Massenpunktes nah de Einsteinshen Theoie, Sitzungsbeihte de Deutshen Akademie de Wissenshaften zu Belin, Klasse fu Mathematik, Physik, und Tehnik, p.89. Soaes, D. S. L. (009): vesion 4. Newtonian Gavitational Defletion of Light Revisited, Woking Pape