Complete solution of spherically symmetric gravitational field

Transcription

1 Complete solution of spheially symmeti avitational field Mioslav Súkeník, Jozef Šima Slovak nivesity of ehnoloy, FCHP, Radlinského 9, 8 7 Batislava, Slovakia sukenik@nexta.sk; jozef.sima@stuba.sk Abstat: hee is a paadim statin that avitational field is of non-loalizable and stationay natue. Contay, in ou model of Expansive Nondeeleative nivese it is hypothesized that avitational field is always loalizable and nonstationay. his assumption allows loalizin its eney density. In this ontibution, a solution of the issue is offeed. his solution an be extapolated to both the past and futue times. Of the solution, the existene of osmoloial membe and, in tun, the the question of existene of dak eney follows. Moeove, eah blak hole epesents an independent nivese bein a subsytem of the infinite Multivese. : INRODCION: Any aspet of avitational field an be expessed by oespondin metis. A sphei symmeti field an be desibed by Shwazshild meti, otatin haed massive objets by Ke Newman meti, nonstationay avitational field by Vaidya meti, sinulaities of the Shwazshild meti ae eliminated usin Finkelstein meti et. All the mentioned metis desibe ston avitational fiel. Neithe of the metis (exludin Vaidya one) is able to loalize avitational field eney density. Cuent osmoloial models do not omply with the Vaidya meti. We suppose, the model of Expansive Nondeeleative nivese (EN) miht onfom to Vaidya meti. Sinulaities pesent in Shwashild meti may be eliminated usin Finkelstein o Kuskal meti, esultin, howeve, in a meti of flat spae (with a body fallin feely ito blak hole). he body an thus falls to one point loated in the ente of the blak body whih lea to a futhe sinulaity. Intoduin a time-vayin nonstationay avitational field, the sinulaities an be eliminated and, also othe poblems of the uent osmoloy, suh as dak eney issue, an be solved. : NONDECELERAIVE NIVERSE MODEL Ou model of the nivese (Expansive Nondeeleative nivese, EN) [], [] is based on a simple pemise that the ate of the nivese expansion is onstant and equal to the speed of liht. Moeove, the nivese mean eney density is idential to its itial eney density. hee ae thee limitin onditions haateizin the EN model, namely whee is the osmoloial onstant, k = () whee k is the uvatue, and a = t () whee a is the sale fato, is the speed of liht in the vauum, t is the osmoloial time. hei pesent EN-based values ae followin: a =.9 6 m; t =.7 y. Within the lassi models of the nivese, the flat nivese is equied to adually deeleate its expansion. It is a ase whee the avitational foe affets the nivese GLOBALLY. Contay, in the EN, the avity affets it only LOCALLY. ()

2 he dynami natue of the EN is desibed by Fiedman equations. Intoduin a dimensionless onfom time, the equations an be expessed as follows: d. da G a ( p ) d a d 8. da G a k a d () (5) whee is the eney density, p is the pessue and the sale fato a is expessed as da a (6) d Intoduin the onditions () to () into elations () and (5), we et 8 G a (7) p (8) he eney density an be expessed also in the fom m a whee m is the mass of the nivese ( m Combinin of (7) and (9) one obtains G m 8. 67x a () It follows dietly fom () that a time evolution of the matte must ou. An amount of the mass eated in one seond is dm m dt t G It means that an amount of the matte eated in ou nivese in a seond is equal to about 5 Sun mass. In the inflationay model, the same amount of matte is emein fom beyond the hoizon. It is not too muh matte if the nivese dimensions ae taken into aout. Fo the sake of illustation, it epesents a poton in a ube of km within a yea. hee is no lobal sale avity in the EN whih ould deeleate the nivese expansion. he EN model is this in ompliane with a Hawkin s statement that the total mass-eney of ou nivese must equal peisely to. It means that the matte, epesentin the positive omponent of the eney, is just ompensated with the avitational field, epesentin the neative omponent of the eney. he onsevation laws ae theefoe obeyed. Howeve, the eation an be undestood also diffeently. We suppose that the mass of the elementay patiles deeases in time and the deease is ompansated thouh an inease in thei quantity. We ae able to eiste only this inease whih appea as the matte eation in spite of pesevation of thei total mass. hee must thus exist a nonstationay avitational field in the EN model. Intoduin the tansfomation m m t () 5 k). (9) ()

3 he mass depen on the osmoloial time ( m t ). It hol: dm t dt m t () a In this situation it is possible to loalize the eney density of avitational field. : LOCALISAION OF WEEK GRAVIAIONAL FIELD As a statin point, Einstein equation 8G () R R is taken. Diveene of this equation lea to avitational eney density ε in the fom R 8 G whee R is the sala uvatue. Vauum sala uvatue is equal to zeo. It hol: R (5) (6) is Newton potential. Gavitational foe bein a fa-eahin foe ats in piniple up to infinity, it is measuable, howeve, only to a etain distane alled effetive ane ef. Its meanin lies in a postulate that in the EN, the effet of avitation an be displayed only in suh a distane, in whih the absolute value of the avitational eney density is hihe than the itial eney density of the nivese. ef a Non-elativisti avitation potential an be thus expess a exp (8) ef Within the distanes shote that the effetive ane, this potential is almost idential to Newton potential. At distanes > ef, the potential appoahes zeo value. Fo weak avitational field the followin membes of meti tenso apply: Gmt Gmt Gmt Gmt,,, dia (9) It hol: h () In ou ase: dia,,, () (7)

4 Gmt Gmt Gmt Gmt h,,, dia () It must then hold fo sala uvatue: R d Gm t () d a he idential esult is obtained usin Vaidya meti, and Einstein o olman pseudotenso [], []. he sala uvatue an be expessed also by anothe way applyin Yukawa potential. It hol: R Gm t h () a whee,,, avitational field:. Combinin elations (5), () and () it follows fo the eney density of weak d m 8G G d a t h. (5) At the same time, the identity must hold: d d o h d h d (6) (7) : COMPLEE SOLION FOR SPHERICALLY SYMMERIC FIELD With ead to (), Shwazshild meti an be ewitted eithe in the fom o Gm t dt d sin. d (8) Gmt d t t dt e d d sin. d e (9) It hol: m t >. Calulatin membes of Rii tenso and intoduin thei in Einstein equations lead to the followin elations 8G e t / () 8 G / e t ()

5 8G / / / / / // e e he time index at and was not iven in the bakets of equations () - () fo the sake of simpliity. 8 G e t Outside the ental body it hol () 8Ga hese membes ae not sinifiant beyond the hoison. 8G a. (6) he omponent epesents avitational eney density. It miht be supposed that the positive and neative values of the momentum-eney tenso will mutually anelled. hen, m, and e e. In suh a ase the meti on the hoison adopts the followin fom It hol t dt d ( d sin d ) (7) d (8) It should be pointed out that is popotional to the positive eney omponent on the hoison. In blak holes thee is a matte eation, in the nivese it is expansion. If, theefoe, the nivese is put idential to blak hole, the elation (8) must hold and is adually ineasin. Ineasin the positive eney omponent, the neative omponent must inease as well and the total eney on the hoison is thus equal to zeo. / Similaly, the omponents, //, /,, and ae equal to zeo. hen On the hoison thee is thus no impat of the avitational field. In the spae unde the blak hole hoison the sins in the meti (8) must be evese. nde the hoison. his eion epesents the past and the hane of sins is undestood as a hane in the phase. It is supposed that the eney field density hee is positive while beyond the hoison it is neative. At the hoison all ontibutions ae mutually anelled and the eney density is thus of zeo value. he hoison behaves as a flexible body. It ties to esist both stethin (neative field eney density the effet of avitation) and ompesin (positive field eney density atin of epulsion). he meti unde hoison is then as follows: Gm t dt d sin. d (9) Gmt d () () (5)

6 In this ase: and 8G a. 8Ga () nde the hoison a new sale must be implemented. It hol a and then () 8G () G ( ) 8 In eq. () the osmoloial membe has appeaed () his membe exets a epulsive (antiavitational) effet. At the hoison it hol and this is a eason of why the ontibution (5) and (6) will be equal to zeo, and the ontibutions () and(), as well as () and () will mutually anelled. he hoison enlaement (i. e. the nivese expansion) will ou by a onstant veloity. he meti (8) o (9) with solution (6) and the meti (9) with solutions () o () indiate the aodane with the oneption of advaned and etaded waves [5],[6]. he ondition is in ompliane with the existene of advaned waves. An uptake of advaned waves means the expansion of the spae-time (the nivese). he ondition means the existene of etaded waves. Eah blak holes has its own time t v. It must hold:. t v (5) Note: he hoison has its own units of time and spae. Relation (5) is valid all the time and it is not alteed even by the existene of non-zeo osmoloial membe in the past. An explanation delain that the nivese expantion deeleated in the past and aeleated at pesent is not justifiable. I tis not eal and ationalizable that the effets of past deeleation ae exatly ompansated by the pesent aeleated expansion. Suh a oinidene is absolutely unbelievable. he diffeent values of the omponent pesent just a hane in phase whih is neessay to distinuish the past and the futue. hese ontibution ae fully anelled at the hoison. his is why dak eney is just an illusion. he omponents and exhibit sinulaities at the hoison, but they will be anelled as well. () 5: SINGLARIIES IN HE NONDECELERAIVE MODEL hee is a one issue desevin explanation, speifially elimination of the sinulaities fom Shwazshildovej meti and sinulaities in eneal. In adition, solutions of the metis (8), (7) and (9) should be unified.

7 he mentioned issued an be simply explained via appliation of Yukawa potential (8). In a nomal eime, when ef, Yukawa potential is edued to lassi Newton potential. If ef, the potential shaply deeases to zeo. A diffeent situation elates to the blak bole hoison. In ase when, then ef. It epesents a loseness of the spae-time (the effetive avitational ation adius is equal to zeo fom the viewpoint of the blak hole hoison). It means that at the blak hole hoison also avitational potential deeases to zeo and the meti hanes to a flat spae meti. his avitational potential deease is not adual but it is ealized in a step as an immediate poess. All omponents of momentum-eney tenso will be of zeo value and no sinulaities will exist. Howeve, at the hoison, a hane in the phase of the advaned and etaded waves haateizin the spae-time and matte will ou. 6: CONCLSION In the pesent ontibution, Shwazshild meti with non-stationay avitational field was solved. Based on the mentioned solution, the followin fou fundamental onlustions ae deived: : Gavitational impat is always only of loal natue. It applies to the hoison and blak hole (the nivese) suoundins with ead to the Multivesum. : Sinulaities do not exist. At the hoison, the positive and neative pats of eney density ae mutually anelled. : he nivese Multivesum is infinitive both in the time and spae. Eah blak hole epesents a loal nivese with its own hoison and time. : Dak eney is just an illusion esultin fom non-stationay solution of avitational equations and the neessity to distinuish the past and futue. NOE: hee ae the followin easons to identify the nivese with blak holes. : Dimensions of both Nondeeleative nivese and blak hole an be desibed by idential equation. : Enlaement of the nivese (expansion) means simultaneously enlaement of blak hole (eation). : he Nondeeleative nivese and blak hole an be desibed by idential wavefuntion. : hee is idential elation fo entopy of Nondeeleative nivese and that of blak hole, sine loaithm of the nivese phase volume equals to the hoison sufae devided by the Plank lenth squae. 5: hee is a sinulaity in the blak hole ente and appaent osmoloial sinulaity at the beinnin of the nivese expansion. 6: In the model of expansit Nondeeleative nivese it hol that the meti tenso epesents the phase shift advaned and etaded waves. In avitational field the phase shift deeases and at the hoison is equal to zeo. Beyond the hoison (the futue) and unde it (the past) the phase is evesed and thee is no spae fo sinulaity in the ente of a blak hole. Refeenes: [] Šima, J. and Súkeník, M., Paifi Jounal of Siene and ehnoloy, () (). [] Sukenik, M., Sima, J., Vixa, 5,6 [] Vaidya, P.C., Po.,Indian Aad. Si., A (95) 6. [] Vibhada, K.S., Pamana-J. Phys., 8 (99). [5] Wheele J.A. and Feynman R.P., Rev. Moden Phys., (99) 5. [6] Came J., Rev. Moden Phys., 58 (986) 67.