On the Escape Velocity in General Relativity and the Problem of Dark Matter

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1 Appled Physcs eseach; Vol. 7, No. ; 5 ISSN E-ISSN Publshed by Canadan Cente of Scence and Educaton On the Escape Velocty n Geneal elatvty and the Poblem of Dak Matte Valey Vaslev & Leond Fedoov Insttute of Poblems n Mechancs, ussan Academy of Scences, Venadskoo, Moscow 956, ussa Coespondence: Valey Vaslev, Insttute of Poblems n Mechancs, Venadskoo, Moscow 956, ussa. Tel: E-mal: vvvas@dol.u eceved: Apl 6, Accepted: May 5, 5 Onlne Publshed: May 9, 5 do:.559ap.v7np8 UL: Abstact The pape s concened wth analyss of the escape velocty fo a sphecal sta wthn the famewok of the Classcal Gavtaton Theoy (CGT) and the Geneal Theoy of elatvty (GT). In GT, two possble esults coespondn to the classcal and the enealzed Schwazschld solutons ae obtaned. Fo the latte soluton whch, n contast to the fome one, s not snula, the ctcal adus of the sta fo whch the patcle cannot leave the suface even f the ntal velocty of the patcle s equal to the velocty of lht s found and s assocated wth the adus of the Dak Sta ntoduced n the 8-th centuy by J. Mchel and P. Laplace. As shown, the adus of the laest vsble sta,.e., the ed supeant UY Scutt, s much lae than the obtaned ctcal adus. If the Dak Stas whose ad ae smalle than the ctcal adus exst, they ae hypothetcally not vsble and can concentate the matte that cannot be obseved. Keywods: escape velocty, eneal elatvty, dak matte. Intoducton To detemne the escape velocty v e coespondn to CGT, consde a sold sphee wth adus, constant densty μ and mass m = πμ () o Assume that a elatvely small patcle wth mass m moves fom the sphee n the adal decton unde the acton of the Newton avtaton foce n accodance wth the follown equaton: d mm e m = G () dt whee G = 6.67 N m k s the classcal avtaton constant and t s tme. Intoducn the so-called avtaton adus Gm = o () c n whch c s the velocty of lht, we can educe Equaton () to d c + = () dt Ths equaton s nteated unde the follown ntal condtons: d t ( = ) =, vt ( = ) = = v (5) dt t = whee v s the ntal velocty of the patcle wth whch t stats movn fom the sphee suface =. The fst nteal of Equaton () d v = C c dt = + (6) 8

2 Appled Physcs eseach Vol. 7, No. ; 5 ncludes the nteaton constant C whch can be found fom the second condton n Equatons (5),.e., c C = v (7) Substtutn C fom Equaton (7) n Equaton (6), we et v = v + c (8) Futhe nteaton of Equaton (6) unde the fst condton n Equatons (5) specfes the follown dependence between the adal coodnate and tme: c C + c + Cc ( + C) t = ln ( c + C ) ( c + C ) C C C c C( c C) + + C + (9) Two ways can be used to detemne the escape velocty. Fst, as follows fom Equaton (9), the moton s eal f C. Then, Equaton (7) yelds v ve = c () To apply the second appoach, put n Equaton (8) to et v = v c () and specfy v e as the ntal velocty v fo whch v =. Then, Equaton () ves the same expesson fo v that follows fom Equaton (). e Takn v = v (o C e = ) and nteatn Equaton (6), we ave at the follown dependence of the adal coodnate on tme: ve = + t As follows fom Equaton (), fo the sphee wth adus = the escape velocty becomes equal to the velocty of lht. Accodn to the ntepetaton of ths esult poposed by J. Mchel n 78 and P. Laplace n 796 (Thone, 99) the sta wth adus can be efeed to as the Dak Sta whch s not vsble. Late, the concepton of Dak Stas based on the Classcal Gavtaton Theoy and the copuscula model of lht has been abandoned by Laplace and the subsequent authos (Thone, 99). Howeve, the dea of Dak Stas becomes attactve n the lht of the moden poblem of Dak Matte. The damatc dscepancy between the mass of the obseved space objects and the mass follown fom the avtaton effects coespondn to these objects allows us to suppose the exstence of hdden mass o Dak Matte whch s estmated to consttute about 85% of the total matte n space. The pesent pape contans an attempt to evve the dea of Dak Stas wthn the famewok of the Geneal Theoy of elatvty.. Equaton of the adal Moton n GT To detemne the escape velocty n GT, ntoduce the sphecal coodnate fame x x = ct and the coespondn sphecally symmetc fom of the lne element =, x = θ, x = φ, ds = d + ( dθ + sn θdφ ) c dt () Hee, the components of the metc tenso j depend on the adal coodnate only. Consde the equaton fo eodesc lnes n the emannan space (Wenbe, 97) k l d x dx dx kl + Г = () ds ds ds 85

3 Appled Physcs eseach Vol. 7, No. ; 5 and pefom some tansfomatons. Fst, wte Equaton () fo the adal coodnate x =. Second, usn the pocedue descbed elsewhee (Wenbe, 97), chane the vaable s to the tme coodnate t. The esultn equaton s whee Г d d = ( Г Г ) c Г dt dt =, Г =, Г () = (5) and (...) = d(...) d. The coeffcents of the metc tenso whch ente Equatons (5) can be found fom the j basc GT equatons fo the components of the Ensten tenso E whch fo the poblem unde study can be pesented n the follown fom: E = + (6) E = Fo the empty and nfnte extenal space suoundn the sphee wth adus (.e., fo ), E = E =. Intoducn a new functon () as = () and puttn ( ) =, we can nteate Equatons (6), (7) and expess the metc coeffcents and n tems of () as (Vaslev & Fedoov, ) (7) ( ) =, = (8) Fo ( ) =, whch s the case fo the solutons dscussed futhe, and, Equatons (8) educe to the coespondn solutons of CGT. Note that the functon () n Equatons (8) can be appopately chosen as dscussed by Vaslev and Fedoov (5).. Schwazschld Soluton The classcal Schwazschld soluton follows fom Equatons (8) f we take = and has the follown fom (Syne, 96): =, = (9) n whch s specfed by Equaton (). Fo the metc coeffcents n Equatons (9), the moton Equaton () educes to d c + = dt ( ) dt d () Fo the avtaton feld wth elatvely low ntensty, we can nelect the ato n compason wth unty and educe Equaton () to Equaton () coespondn to CGT. To solve Equaton () whch ncludes the unknown functon t (), ntechane the vaables and tansfom t to the follown equaton fo the functon t (): Intoducn a new vaable u () as (Kamke, 959) c ( ) + = d t dt dt d d ( ) d 86

4 Appled Physcs eseach Vol. 7, No. ; 5 dt d = u ()exp d ( ) we can educe ths equaton to the fst ode dffeental equaton du c d = ( )exp u d ( ) apply sepaaton of vaables and ave afte athe cumbesome tansfomatons at the follown fst nteal: d = c f( ) dt () n whch f() = + C and C s the nteaton constant. Equaton () specfes the so-called coodnate velocty. The actual velocty can be found wth ad of Equatons (9) and () as (Landau & Lfshtz, 96) Takn n accodance wth ntal condtons v ( = ) = v, we et v = = cf() () d dt C v = c () and educe Equaton () to the follown fnal fom: v v = c + c Inteaton of Equaton () unde the ntal condton t ( = ) = yelds ( )[ + + C( ) + f( )] t = C[ f( ) f( )] ln + c ( )[ + + C ( ) + f ( )] + C( ) + f( ) C + (C ) ln C + C( ) + f( ) C () (5) As follows fom ths soluton, the moton s eal f C. Then, Equaton () ves v ve = c (6) Fomally, ths esult concdes wth Equaton () fo CGT. Howeve, n contact to Equaton () whch s vald fo =, Equaton (6) does not follow n ths case fom Equaton (), because the denomnato of ths equaton becomes zeo fo =. Ths means that the escape velocty cannot be found fo the Shwazschld soluton f the sphee adus s equal to the avtaton adus. 87

5 Appled Physcs eseach Vol. 7, No. ; 5 To poceed, assume that the ntal velocty has the maxmum possble value,.e., that v = c. Then, Equaton () yelds C =. Takn C = n Equaton (5), we ave at Solvn ths equaton fo, we et t = + ln c ct+ ( ) e = ( ) e (7) Puttn hee =, we fnd that = =. Thus, the patcle cannot leave the suface of the sphee wth adus = even f ts ntal velocty s equal to the velocty of lht. The sphee wth adus = s efeed to as the Black Hole. To evaluate the mass of the sphee, consde the ntenal space ( ). Fo a sold sphee wth constant densty μ, the component of the Ensten tenso n Equaton (7) s Takn 8π G E = χμc, χ = (8) c =, we ave at the follown equaton = χμc The soluton of ths equaton whch satsfes the eulaty condton at the sphee cente = s (Syne, 96) = χμc (9) Index coesponds to the ntenal space. The metc coeffcent must be contnuous on the sphee suface,.e., ( ) = ( ) n whch s specfed by the fst Equaton (9). As a esult, we have χμ c = () and Equaton (9) takes the follown fnal fom: = () As follows fom Equatons (9) and (), the obtaned soluton s snula - and become nfntely hh on the sphee suface = f =. Equaton () allows us to fnd the sphee mass. Indeed, substtutn fom Equaton () and χ fom the second Equaton (8), we ave at Equaton () coespondn to the Eucldean space. Howeve, ths esult does not coespond to the basc GT concept accodn to whch the space nsde the sphee s emannan. Patculaly, fo the metc coeffcent n Equaton (), we have π 9 m= πμ d = μ sn = m () 56 The last pat of ths equaton s the decomposton of the second pat nto the powe sees wth espect to. As follows fom Equaton (), m= m only f =. Thus, fo the mass of the sphee wth adus =, we have two expessons follown fom Equatons () and (),.e., m = πμ, m = () πμ whch do not concde. It should be noted that both esults ae of elatvely low value, because the Black Hole wth snula eomety can hadly be smulated wth a sphee of constant densty. 88

6 Appled Physcs eseach Vol. 7, No. ; 5. Genealzed Soluton To deve the enealzed soluton, assume that the metc coeffcents of the extenal space ae specfed by Equatons (8) n whch s an unknown functon of. Then, the moton Equaton () takes the fom d d c + = dt ( ) dt Applyn the tansfomaton used to solve Equaton (), we fnally ave at the follown soluton whch s vald fo an abtay functon () : d = c + C dt Note, that ths soluton concdes wth Equaton () f we chane to. The actual velocty of the patcle smla to Equaton () s d d v = = = c + C dt dt () Assume that the sphee suface = coesponds to =Ρ,.e., that ( ) =Ρ. Then, the nteaton constant C can be found fom the ntal condton v( Ρ ) = v,.e., v C = Ρc Ρ (5) Equaton (5) fo t () s also vald f we chane, and C to, Ρ and C. As eale, the moton s eal f C and the escape velocty follows fom the follown expesson: v ve = c Ρ As eale, ths esult s vald f Ρ>. To study the case Ρ=, apply the appoach descbed n Secton,.e., take v = c. Then, Equaton (5) yelds C = and the follown moton equaton smla to Equaton (7) specfes the dependence between and t : ct +Ρ ( ) e = ( Ρ ) e (6) As can be seen, the stuaton s the same that fo the Black Hole dscussed n Secton. Fo Ρ=, we have = =Ρ and the patcle cannot leave the sphee suface =Ρ even f ts velocty s equal to the velocty of lht. Thus, the poblem s to detemne the functon () and to fnd Ρ= ( ). To solve ths poblem and to avod the nonunque value of mass n Equatons () found fo the Schwazschld soluton, assume that avtaton, chann Eucldean eomety nsde the nteactve solds to emannan eomety n accodance wth the basc GT concept, does not affect the solds mass (Vaslev & Fedoov, 5). Fo the sphee wth constant densty μ, the ntenal eomety s specfed by Equaton (7) n whch E s ven by Equaton (8). Takn =, we ave at the follown soluton (Vaslev & Fedoov, ): The sphee mass can be expessed as = ( ) (7) χμc 89

7 Appled Physcs eseach Vol. 7, No. ; 5 m = πμ d (8) Because t s assumed that avtaton does not affect the sphee mass, wheeas n the absence of avtaton the mass s specfed by Equaton (), suppose that fo the ntenal space = (9) Thus, Equaton (8) educes to Equaton () and m= m. Then, Equaton () s vald and we can tansfom Equaton (7) to the fnal fom ( ) = () Usn Equatons (9) and (), we ave at the follown equaton fo () : = () The soluton of ths equaton whch satsfes the eulaty condton at the sphee cente s F( ) = () n whch F( ) = sn Takn = and puttn =Ρ n Equaton (), we obtan the follown equaton fo P : F ( P) = () Ths equaton allows us to fnd Ρ f the ato s known. Howeve, ths s not the case yet. As follows fom the analyss of the Schwazschld soluton n Equatons (9), cannot be lowe than. To establsh the analoous constant fo the soluton unde study, consde the extenal space fo whch the metc coeffcents ae specfed by Equatons (8). The man poblem s to povde the bounday condtons on the sphee suface accodn to whch ( = ) = ( = ) =Ρ, ( Ρ) = ( Ρ) () Assume that fo the extenal space the follown equaton whch s smla to Equaton (9) s vald = (5) Matchn Equatons (9) and (5), we can conclude that f the fst of the condtons n Equatons () s satsfed, then the second condton s satsfed automatcally. Thus, we need to satsfy only the fst bounday condton n Equatons (). Substtutn fom the fst Equaton (8) n Equaton (5), we ave at the follown equaton fo () coespondn to the extenal space: = (6) The soluton of ths equaton whch satsfes the fst bounday condton n Equatons () s 9

8 Appled Physcs eseach Vol. 7, No. ; ( ) + ln (7) Ρ 5Ρ 5 5 Ρ Ρ + + Ρ( Ρ ) ln + = ( ) 8 8 Thus, the enealzed soluton s specfed by Equatons () and (7) whch allow us to fnd the functons () and () fo the ntenal and extenal spaces. Asymptotc and numecal analyss of ths soluton (Vaslev & Fedoov, 5) shows that t educes to the Schwazschld soluton fo = and. As follows fom Equaton (7), the obtaned soluton s eal f the mnmum value of the functon whch s ( ) =Ρ s not less than,.e., f Ρ. In the lmtn case, substtutn =Ρ n Equaton (), we ave at the equaton fo mnmum possble value of the sphee adus whch yelds = =.5. Applyn Equaton (), we can conclude that the soluton exsts f the sphee adus satsfes the follown condton: mg =. (8) c Fo <, the soluton becomes manay. 5. The Poblem of Dak Matte Detemnn the natue of Dak Matte dscussed n the Intoducton s one of the most mpotant poblems n moden cosmoloy. Up to now, the stuctue of Dak Matte emans speculatve. Accodn to the exstn hypotheses, Dak Matte can be concentated n heavy nomal objects such as Black Holes, neuton stas and whte dwafs, o can be composed of a not yet dentfed type of patcles whch nteact wth odnay matte va avty only. The smplest possble ntepetaton of ths phenomenon was poposed n the 8-th centuy by J. Mchel and P. Laplace lon befoe the poblem of Dak Matte appeaed. epoducn the espectve easonn (Laplace, 799) consde Equaton () fo the escape velocty and tansfom t wth the ad of Equaton () fo the avtaton adus to 8 ve = πμg (9) Fo Eath wth aveae densty μ E = 55 k m and adus E = 67m, Equaton (9) yelds ve = m s whch s 77 tmes less than the velocty of lht. Because the escape velocty n Equaton (9) s popotonal to the adus, nceasn E up to.6 m (whch s 9.6 tmes lae than the adus of Sun, 8 S = 6.96 m), we ave at ve = c. Ths calculaton allowed Laplace to conclude that the sta wth the densty of Eath and adus whch s about 5 tmes lae than the adus of Sun s not vsble and can be efeed to as the Dak Sta. Based on the Newton avtaton theoy, the dea of Dak Stas has been late abandoned. Howeve, Equaton (8) follows fom GT. Fo the sta wth the densty μ E, we et = S whch s close to the esult obtaned by P. Laplace. As follows fom the analyss assocated wth Equaton (6), a patcle cannot leave the suface of the sphee wth adus = even ts ntal velocty s equal to the velocty of lht. As shown n Secton, the Black Hole possesses the same popety and, as known, s not vsble (Thone, 99). So, we can suppose that the sta wth adus < s also not vsble and follown J. Mchel and P. Laplace denote t as the Dak Sta. In contast to the Black Hole, the eomety of the Dak Sta s not snula. Applyn Equatons (8), () and (6), we have =, = (5) n whch () and () ae specfed by Equatons () and (7). Equatons (5) ae vald f and no snulaty s obseved f = (Vaslev & Fedoov, 5). Fo the sphee wth adus <, the soluton becomes manay whch means that ths sphee cannot be consdeed wthn the famewok of GT. In contast to Black Holes, no avtatonal collapse can be assocated wth Dak Stas. Two hypothetcal scenaos can be consdeed. Assume that the sta wth the adus whch satsfes the nequalty n Equaton (8) exsts and s vsble. Possessn hh avtaton, the sta attacts the matte fom space whch esults n the owth of the sta mass and n espectve ncease of n Equaton (8). Fst, assume that the addtonal mass nceases the densty of the sta, wheeas ts adus does not chane and becomes close to the 9

9 Appled Physcs eseach Vol. 7, No. ; 5 nceasn ctcal adus. When eaches, the sta becomes nvsble. Second, the buldup of the sta mass can be accompaned by the owth of ts adus, wheeas the sta densty μ does not chane. As follows fom Equatons () and (8), s popotonal to m, wheeas s popotonal to m. So, wth the owth of the sta mass, nceases faste than and the condton = can be satsfed f the sta mass becomes πμ m=.6c G If Dak Stas exst, the ad of vsble stas must be lae than n accodance wth Equaton (8). The laest of the obseved by now vsble stas the ed supeant UI Scut s chaactezed wth adus =.9 m and mass m = 6 k (Aoyo-Toes et al., ). Usn Equaton (8), we can calculate the ctcal adus = 59m whch s much lowe than. 6. Concluson The analyss of the escape velocty undetaken wthn the famewok of the Classcal Gavtaton Theoy and Geneal Theoy of elatvty has shown that the solutons coespondn to both theoes demonstate the hypothetcal exstence of Dak Stas whch concentate the matte that cannot be dectly obseved. In contast to the Black Holes whch follow fom the classcal Schwazschld snula soluton, ae hypotheszed as a esult of avtatonal collapse and whose ntenal stuctue cannot be dentfed wthn the famewok of GT, the Dak Stas ae chaactezed wth eula eomety whch follows fom GT. The ctcal adus of the Dak Sta detemnes the lmt unde whch the GT soluton becomes manay and the theoy can be teated as not vald. The exstence of Dak Stas, n case t takes place n ealty, allows us to popose a easonable explanaton of the Dak Matte phenomenon. efeences Thon, K. S. (99). Black Holes and Tme Waps Ensten s Outaes Leacy. New Yok, London, W.W. Noton and Company. Wenbe, S. (97). Gavtaton and Cosmoloy. New Yok, Wlley. Vaslev, V. V., & Fedoov, L. V. (). On the soluton of sphecally symmetc statc poblem fo a flud sphee n eneal elatvty. Appled Physcs eseach, 6(), Vaslev V. V., & Fedoov, L. V. (5). Possble foms of the soluton fo sphecally symmetc statc poblem n eneal elatvty. Appled Physcs eseach, 7(), Syne, J. L. (96). elatvty: the Geneal Theoy. Amstedam, Noth Holland. Landau, L. D., & Lfshtz, E. M. (96). Feld Theoy. Moscow, Nauka (n ussan). Kamke, E. (959). Dffeentallechunen. Lepz. Aoyo-Toes, D, Wttkovsk, M., Macade, J. M., & Hauschldt, P. H. (). The atmosphec stuctue and fundamental paametes of the ed supeants AH Scop, UY Scut, and KW Satta. Astonomy and Astophyscs, 55(A76), -. Laplace, P. S. (799). Poof of the theoem that the attactve foce of a heavenly body could be so lae that the lht could not flow out of t. Allemene Geoaphsche Ephemeden, Weme, IV, Bd. St. Copyhts Copyht fo ths atcle s etaned by the autho(s), wth fst publcaton hts anted to the jounal. Ths s an open-access atcle dstbuted unde the tems and condtons of the Ceatve Commons Attbuton lcense ( 9