Are Singularity and Dark Energy Consequences of Vacuum Solution of Field Equations?

Transcription

1 Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume- Issue-11 Novembe 017 Pages 7-55 Ae Singulaity and Dak Enegy Consequenes of Vauum Solution of Field Equations? Banko M Novakovi Abstat- Moden physis is tying to solve poblems like singulaity dak enegy and univese expanding by using sophistiated methods in patile and quantum physis But the mentioned poblems should also be solved in the lassial Geneal Relativity Theoy (GRT) As it is well known Shwazshild solution of the Einstein s field equations is in fat a vauum solution Unfotunately this vauum solution geneates appeaanes as singulaity and dak enegy with unknown soue Hee we show that the singulaity and dak enegy with unknown soue ae diet onsequene of the vauum solution of the Einstein s field equations In that sense we employ new Relativisti Alpha Field Theoy (RAFT) that poposes solution of the field equations without appeaanes of singulaity and offes the soue of dak enegy This is the onsequene of the following peditions of RAF theoy: a) no a singulaity in a gavitational field and b) the gavitational foe beomes positive (epulsive) if (GM/ ) > 1 that ould be the soue of a dak enegy Index Tems : Relativisti alpha field theoy (RAFT) Vauum solution Singulaity Soue of dak enegy I INTRODUCTION As it is well known vauum solution of the Einstein s field equations pedits a singulaity in a gavitational field This solution limits the appliations of GRT 1-6 to the elatively weak field as we have in ou sola system Futhe vauum solution annot explain what is eally soue of the enegy that expands ou univese even in the aeleation ate Theefoe this enegy has been alled dak enegy but with unknown soue While the vauum solution of Einstein s field equations is poved to be oet in the elatively weak field in ou sola system 1-6 it annot be applied to the extemely stong field at the Plank s sale Hee we pesent that the new Relativisti Alpha Field Theoy (RAFT) 7-10 gives the solution of the full fom of the Einstein s field equations exluding the osmologial onstant (Λ=0) This solution is without singulaity and an be applied to the extemely stong fields inluding of the Plank s sale In the weak gavitational field this solution is edued to the well-known vauum solution of the Einstein s field equations One of the impotant disovey of the RAF theoy is that gavitational foe beomes positive in the egion min < Hee min is the minimal adius in a gavitational field ( min = GM/ ) and is the adius whee negative aeleation is hanges into the positive one and vie vesa ( = GM/ ) Theefoe gavitational adius less than min is not possible and it potets fom the singulaity point at the zeo adius at = 0 In the egion ( < < ) gavitational aeleation is Banko Novakovi FSB Univesity of Zageb Luieva 5 POB Zageb Coatia negative as we know fom ou todays knowledge Sine ou univese is expanding at the aeleation ate 7-8 the pesent univese adius should be in the egion min < It has been theoetially poved that the metis of the line element in RAF theoy is egula at the Shwazshild adius This means that no a singulaity at the Shwazshild adius in RAF theoy On the othe side in the vauum solution thee no natual limitation to the gavitational adius that it deease to the singula point at zeo adius This is theoetial poof that the vauum solution podues singulaity in a gavitational field Futhe it has been theoetially poved that at the minimal adius min the all field paametes in RAF theoy ae egula This means that the metis of RAF theoy is egula in the egion min < In the ase that the adius is less than the minimal adius min the metis in RAF theoy beomes imaginay This poves the pedition of RAF theoy that thee exists a minimal adius at = (GM/ ) that pevents singulaity at = 0 It seems that the existene of the minimal adius tells us that the natue potet itself fom the singulaity In the ase of the vauum solution the minimal adius is in fat the singula point at zeo adius This poves that the singulaity is the onsequene of the vauum solution of the field equations In ode to theoetially pove that the positive gavitational foe ould be a soue of a dak enegy one an stat with the time evolution of the sale fato α ( t ) This equies the Einstein s field equations togethe with a way of alulation of density ( t ) suh as a osmologial equation of state If the enegy-momentum tenso T is similaly assumed to be isotopi and homogenous then one obtains the well-known Fiedmann equations Following this appoah we obtain the elated equations of the univese motion in RAF theoy These equations show that the aeleating expansion of ou univese ould be diven by the positive gavitational foe This fat theoetially onfims that the positive (epulsive) gavitational foe ould be the soue of the so alled dak enegy Of ouse this should be onfimed by the elated expeiments On the othe hand vauum solution of the field equations annot povide any soue of the expansion enegy Theefoe this enegy has been named as dak enegy This onfims that the dak enegy is the onsequene of the vauum solution of the field equations This pape is oganized as follows Enegy-momentum tenso as funtion of the field paametes in gavitational field is deived in Se II The theoetial poofs that singulaity and dak enegy with unknown soue ae diet onsequene of the vauum solution of the Einstein s field equations ae pesented in Se III Finally the elated onlusion and 7 wwwijntog

2 Ae Singulaity and Dak Enegy Consequenes of Vauum Solution of Field Equations? efeene list ae pesented in Se IV and Se V espetively II ENERGY-MOMENTUM TENSOR AS FUNCTION OF FIELD PARAMETERS IN GRAVITATIONAL FIELD In this setion it has been shown the poedue fo automatially geneation of the enegy-momentum tenso as the funtion of the field paametes in gavitational field In that sense we stated with the geneal line element ds in an alpha field given in the fist pat of RAF theoy 7 ds dt dt dx dt dy dt dz dx dy dz z (1) Hee and ae field paametes and 1 Following the well-known poedue 1-6 this line element an be tansfomed into the spheial pola oodinates in the nondiagonal fom ds dt dt d d () d sin d The line element () belongs to the well-known fom of the Riemanns type line element 10-1 g dx g dx ds g dx g dx dx g dx Hee gij x y () ae the omponents of the elated meti tenso Compaing the equations () and () we obtain the oodinates and omponents of the ovaiant meti tenso valid fo the line element (): 0 1 dx dt dx d dx d dx d ( ) g 00 g01 g g g g sin Stating with the line element () we employ fo the onvenient the following substitutions: / (5) In that ase the nondiagonal line element () is tansfomed into the new elation ds dt dt d d d sin d Using the oodinate system () the elated ovaiant meti tenso g μη of the line element (6) is pesented by the matix fom g (7) sin () (6) This tenso is symmeti and has six non-zeo elements as we expeted that should be The ontavaiant meti tenso g μη of the nondiagonal line element (6) is deived by invesion of the ovaiant one (7) 1 / ( ) / ( ) 0 0 / ( ) / ( ) 0 0 g / / sin (8) The deteminant of the tenso (7) is given by the elation det g sin (9) On the othe hand the deteminant of the tenso (8) has the fom 1 det g sin (10) Now we stat with the seond type of the Chistoffel symbols of the meti tensos (7) and (8) These symbols an be alulated by employing the well-known elation 1-6 g g g g 0 1 (11) Thus employing (6) (7) (8) and (11) we obtain the seond type Chistoffel symbols of the spheially symmeti non-otating body: / D / D / D / D ( sin ) / D / D / D / D / D ( sin ) / D 1 1 sinos 1 1 tg D t t (1) 0 1 Fo a stati field the Chistoffel symbols 00 and 00 ae edued to the simplest foms: (1) In the stati field the othe Chistoffel symbols in (1) ae emaining unhanged As it is well known the deteminant of the meti tenso of the line element (6) should satisfy the following ondition 11-1 det g sin 1 (1) Inluding the nomalization of the adius = 1 and the angle θ = 90 in (1) we obtain the impotant elations between the paametes ν and λ: wwwijntog 8

3 Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume- Issue-11 Novembe 017 Pages ' ' '' ' '' (15) If we take into aount the elations (15) then the Chistoffel symbols in (1) and (1) beome the funtions only of the paamete Fo alulation of the elated omponents of the Riemannian tenso R and Rii tenso R of the line element (6) we an employ the following elations 1-6: R R R R 0 1 (16) Applying the Chistoffel symbols (1) to the elations (16) we obtain the elated Rii tenso fo the stati field of the line element (6) with the following omponents: ' R 00 1 ' '' ' R01 R 10 ' '' (17) ' R 11 ' '' R ' R ' sin The othe omponents of the Rii tenso ae equal to zeo The elated Rii sala fo the stati field is detemined by the equation: R g R 0 1 ' ' (18) R ' '' In ode to alulate the enegy-momentum tenso T η fo the stati field one should employ Rii tenso (17) Rii sala (18) and the Einstein s field equations 1-6 without a osmologial onstant ( = 0): 1 8G R gr kt k 0 1 (19) Hee G is the Newton s gavitational onstant is the speed of the light in a vauum and T η is the enegy-momentum tenso Thus employing of the Einstein s field equations (19) we obtain the following elations fo alulation of the omponents of the enegy-momentum tenso T μη : ' ' kt 00 1 kt 01 kt 10 ' 1 ' kt 11 kt ' '' ' 8G kt sin ' '' k (0) Fo alulation of the omponents of the enegy-momentum tenso T μη by the elations (0) we should know the paamete and its deivations ' and '' fo the elated stati field Paamete is defined by (5) as the funtion of the field paametes α and α / / 1 (1) Fo alulation of the paamete in a gavitational field we need to know the diffeene of the field paametes (α-α ) given by the geneal fom in the fist pat of RAF theoy 7 U U i 1 1 m0 m0 The seond pai is given by the elation U U i () m0 m0 () In the pevious elations U and m 0 ae potential enegy and est mass of a patile in a gavitational field espetively Poposition 1 If the gavitational stati field is desibed by the line element (6) then the solution of the Einstein field equations gives the enegy momentum tensot of that field in the following fom T T T T T T T GM 1 sin 8G () Hee G and M ae the gavitational onstant and the gavitational mass espetively Poof of the poposition 1 In ode to pove of the poposition 1 we should stat with the geneal elations given by (1) () to (1) Fo detemination of the field paametes α and α in a gavitational field one need to know the potential enegy of the patile in that field Thus if a patile with est mass m 0 is in a gavitational field then the potential enegy of the patile in that field U g is desibed by the well-known elation 1-6 m0gm Ug = m0 Vg m0 A g0 (5) Hee V g = A g0 is the sala potential of the gavitational stati field G is the gavitational onstant M is a gavitational mass is a gavitational adius and m 0 is a est mass of the patile that is pesent in a gavitational stati field Fo alulation of the diffeene of the field paametes (α-α ) we an use the geneal foms given by () and () Inluding the substitution U = U g fom (5) into () and () we obtain the diffeene of the field paametes (α-α ) fo a patile in a gavitational stati field: GM GM 1 1 GM GM (6) Applying the esults fom (6) to the elations in (1) we obtain the two solutions of the paamete in a gavitational stati field 9 wwwijntog

4 Ae Singulaity and Dak Enegy Consequenes of Vauum Solution of Field Equations? GM GM (7) Inluding (7) to (0) we obtain the all items needed fo alulations of the omponents of the enegy-momentum tenso T μη in a gavitational stati field: GM GM GM GM GM ' / GM GM GM ' ' GM GM GM GM GM ' GM '' ' ' (8) Now applying the elations (8) to the equations (18) and (0) we obtain the omponents of the enegy-momentum tenso and Rii sala valid fo the gavitational stati field: GM GM 00 1 kt GM 8G kt01 kt 10 k GM GM 11 1 kt kt GM kt sin ' ' R ' '' GM 0 GM (9) Fom the pevious elations we an see that the Rii sala is equal to zeo Finally inluded paamete k into the elations (9) we obtain the omponents of the enegy-momentum tenso in the gavitational stati field T T T T T T T GM 1 sin 8G (0) Beause the elation (0) is equal to the elation () the poof of the poposition 1 is finished Remaks 1 The pevious elations show that the field paametes (6 7) satisfy the full fom of the Einstein s field equations with a osmologial onstant = 0 In the ase of a stong stati gavitational field the quadati tem GM / geneates the elated enegy-momentum tenso T η fo the stati field Fo that ase we do not need to add by hand the elated enegy-momentum tenso T η on the ight side of the Einstein s field equations In the ase of a weak stati gavitational field like in ou sola system we obtain that the quadati tem Fo that GM / 0 ase the field paametes (6 7) satisfy the Einstein s field equations in a vauum (T η = 0 = 0) This oesponds to the well-known Shwazshild vauum solution of the line element But as we know the Shwazshild vauum solution geneates appeaanes as singulaity and dak enegy with unknown soue Theefoe it is not allowed to neglet this quadati tem in the weak gavitational field beause in that ase we obtain the wong onlusions about singulaity and soue of the dak enegy Thus RAF theoy is valid in both weak and stong gavitational fields without singulaity and gives the soue of dak enegy The seond intepetation of the pevious esults ould be that the quadati tem GM / geneates the osmologial paamete as a funtion of a gavitational adius 18 fo T η = 0 It has been shown in 19 that this solution of is valid fo both Plank s and osmologial sales Futhe the metis of RAF theoy 10 has been applied to the deivation of the genealized elativisti Hamiltonian 0 and dynami model of nanoobot motion in multipotential field III PROOFS OF THE RAF THEORY PREDICTIONS In this setion we pesent the theoetial poofs that no singulaity in a gavitational field and that the positive gavitational foe ould be the soue of the dak enegy At the same time it has been shown that vauum solution geneates singulaity in gavitational field and gives no any soue of the dak enegy A Poof that no a singulaity at the Shwazshild adius RAF theoy pedits that no a singulaity at the Shwazshild adius In ode to pove this pedition we alulate the solution of the paametes and at the Shwazshild adius These paametes ae given by (15) and (7) and ae valid fo the line element (6): GM GM 1 1 GM GM GM GM 1 sh sh sh (1) Following the elations in (1) we an see that at the Shwazshild adius sh paametes and ae egula Thus we an say that the metis of the line element in (6) is egula at the Shwazshild adius On that way the wwwijntog 50

5 Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume- Issue-11 Novembe 017 Pages 7-55 theoetial poof of the popositions A (that no a singulaity at the Shwazshild adius) is finished In the ase of the vauum solution the quadati tem GM / does not exists and the elations in (1) ae tansfomed into the new ones: GM GM GM sh 0 sh sh < < 0 > ; 0 sh (1a) Fom the elations in (1a) we an see that no natual limitation to deeasing of the gavitational adius to the singula point at zeo adius This is theoetial poof that vauum solution podues singulaity in a gavitational field B Poof that thee exists a minimal adius at = (GM/ ) RAF theoy also pedits that thee exists a minimal adius at = (GM/ ) This adius pevents singulaity at = 0 ie the natue potets itself In ode to pove this pedition we alulate paametes and in a gavitational stati field at the minimal adius: GM GM 1 1 GM GM GM GM min 1 0 min im GM 1 0; 0 1 () Fom () we an see that at the minimal adius paametes and ae egula and fo min paamete beomes imaginay numbe im Thus at the minimal adius metis of the line element (6) is egula This poves the pedition that thee exists a minimal adius at = (GM/ ) that pevents singulaity at = 0 It seems that the existene of the minimal adius tells us that the natue potet itself fom the singulaity On that way the theoetially poof of the poposition B is finished In the ase of the vauum solution the elations in () ae tansfomed into the new ones: GM GM GM min 0 (a) Fom the elations (a) we an see that the minimal adius in the vauum solution is in fat the singula point at zeo adius Fom the pevious onsideation we an see that the metis of the line element (6) in RAF theoy is egula fo a gavitational field in the egion min The situation about blak holes in the egion min of a gavitational field should be onsideed late In that sense it is vey inteesting to know what Hawking onditionally said about existene of the blak holes : Thee would be no event hoizons and no fiewalls The absene of event hoizons means that thee ae no blak holes in the sense of egimes fom whih light annot esape to infinity The eent investigations of the natue of blak holes ae also pesented in the efeenes 56 C Poof that thee exists a positive gavitational foe Fo the time-invaiant (o vey slowly hanged) alpha field the gavitational foe elations fo the gavitational stati field ae deived in the thid pat of RAF theoy 9: m0gm GM x 0 Fx m0 x 1 m0gm GM y Fy m0 y 1 m0gm GM z Fz m0 z 1 () The gavitational foe elations in () geneally desibe the inteations in the stong fields But it an also be used in the weak field As we an see fom the elations () the gavitational foe beomes positive (epulsive) if the tem ( GM / ) 1 This is the ase at the extemely stong gavitational fields On the othe hand the gavitational foe () beomes negative (attative) in the ase whee( GM / ) < 1 This is the ase in the elatively weak gavitational field Thus in ou sola system the tem ( GM / ) is too small ompae to 1 and an be negleted Theefoe ou sola system belongs to the elatively weak gavitational field Fo an example on the sufae of ou Sun the alulation of the amount of this tem 6 is( GM / ) On the sufae of ou planet Eath the elated influene of the Sun to this tem 8 is ( GM / ) Inluding mass and adius of the planet Eath in this tem we obtain that the elated gavitational influene of the planet Eath on its 9 sufae is ( GM / ) The pesented amounts of the tem(gm / )in ou sola system ould be the answe to the question: why ou expeiene is that gavitational foe is only negative (attative) foe? Futhe if the tem( GM / ) 1 then the gavitational foe is equal to zeo This is happened at the gavitational adius ( GM / ) This adius sepaates the attative and epulsive foes in a gavitational field At the minimal adius ( GM / ) gavitational foe is positive min (epulsive) 0 F ( m / GM ) We an say that the min 51 wwwijntog

6 Ae Singulaity and Dak Enegy Consequenes of Vauum Solution of Field Equations? natue potets itself fom the singulaity by poduing positive (epulsive) gavitational foe F at minimal min adius min Geneally we an see that the gavitational foe is positive (epulsive) in the egion min (see the seond pat of RAF theoy 8 ) At the Shwazshild adius sh ( GM / ) the gavitational foe is negative (attative) sh 0 8 and belongs to the F ( m / GM ) negative (attative) set of gavitation foes in the egion At the Plank s sale one an substitute M by Plank s mass M p and by the Plank s length L p Thus at the Plank s sale the tem (GM p / Lp ) 1 Fom the elations in () we have that(gm p / pm ) 1 Thus the minimal adius of the Plank s mass is pm = L p / The last elation tells us that the Plank s length is in fat minimal diamete of the Plank s mass Following the elations () we an onlude that the gavitational foe of the Plank s mass M p is positive (epulsive) if the adius of the Plank s mass is in the egion pm Lp On the othe hand if the adius is geate than the Plank s length L p then the gavitational foe of the Plank s mass M p is negative (attative) in the egion L Fo the ase whee p gavitational adius is equal to the Plank s length L p L the gavitational foe of the Plank s p pm p mass is equal to zeo Hee p is the adius whee the negative aeleation (aused by the Plank s mass) is hanges into the positive one and vie vesa As it is the well known the elated elations fo the vauum solution of gavitational foes have the fom 1-6: m0gm x 0 Fx m0 x (a) m0gm y m0gm z Fy m0 y F z m0 z Fom the elations (a) we an onlude that the gavitational foes an only be negative (attative) foes The all expeiments in ou sola system onfim the elations (a) But as it is pesented befoe ou sola system belongs to the elatively weak gavitational field In suh a field the elations () ae edued to the vauum solution fom (a) On the othe hand the vauum solution (a) an not be applied to the extemely stong gavitational fields beause it will give inoet esults The pevious onsideation theoetially onfims the pedition of the RAF theoy: the gavitational foe beomes positive (epulsive) if (GM/ ) > 1 that ould be the soue of the dak enegy Of ouse this should be poved by the elated expeiments On the othe hand vauum solution of the field equations tells us nothing about the soue of the dak enegy D Poof that a soue of dak enegy ould be a positive gavitational foe In ode to theoetially pove that the positive gavitational foe ould be a soue of a dak enegy one an stat with the time evolution of the sale fato α( t ) of the univese expansion This equies the Einstein s field equations togethe with a way of alulation of density ( t ) suh as a osmologial equation of state If the enegy-momentum tenso T is similaly assumed to be isotopi and homogenous then the Fiedmann equations has the fom 7-7 8: α 8G α α α α 8G p α α α () Hee α( t ) is the sale fato with elated time deivations α and α is spatial uvatue paamete while and p ae fluid mass density and pessue espetively The pesented equations () ae the basis of the standad big bang osmologial model inluding the uent ΛCDM model Following the mentioned assumption that the univese is isotopi and homogenous the model () an be used as a fist appoximation fo the evolution of the eal lumpy univese beause it is simple fo alulation Futhemoe the models whih alulate the lumpiness in the univese an be added to this model as extensions The pai of the equations in () is equivalent to the following pai of the equations: α 8G α α G p α (5) The fist equation in (5) has been deived fom the 00 omponent of the Einstein s field equations (19) with 0 On the othe side the seond equation in (5) is deived fom the tae of the Einstein s field equations (19) with 0 It is easy to see that the fist equation in (5) an be ewitten into the fom of the fist equation in () The seond equation in (5) an be obtained by substitution of the tem α/α fom the fist to the seond equations in () This onfims that the pai of the equations in () is equivalent to the elated pai of the equations in (5) Some osmologists all the seond equation in (5) as Fiedmann aeleation equation and eseve the tem Fiedmann equation fo only the fist equation in (5) Now one an employ the time deivative of the fist equation in (5) and ombine it with the seond equation in (5) As the esult one obtains the new pai of the equations that ae also equivalent to the pais of the equations in () and (5): α p α α G p α (6) wwwijntog 5

7 Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume- Issue-11 Novembe 017 Pages 7-55 Hee the spaial uvatue index seving as a onstant of the integation fo the seond equation in (6) The fist equation in (6) an be deived also fom themo-dynamial onsideation and is equivalent to the fist law of themodynamis assuming that the expansion of the univese is an adiabati poess This assumption has been impliitly inluded into the deivation of the Fiedmann Lemaite Robetson Walke meti The seond equation in (6) states that both the enegy density and the pessue ause the expansion ate of the univese α to deease It means that both the enegy density and the pessue ause a deeleation in the expansion of the univese This is the onsequene of gavitation inluding that pessue is playing a simila ole to that of enegy (o mass) density This is of ouse in aodane with the piniples of geneal elativity On the othe side the osmologial onstant Λ auses aeleation in the expansion of the univese Futhe one an intodue the density paamete as the atio of the atual (o obseved) density to the itial density of the Fiedmann univese This atio detemines the oveall geomety of the univese It is well known that in the ealie models whih did not inlude a osmologial onstant tem the itial density was egaded also as the wateshed between an expanding and a ontating Univese Reently the itial density is estimated to be appoximately five atoms (of monatomi hydogen) pe ubi mete wheeas the aveage density of odinay matte in the Univese is believed to be 0 atoms pe ubi mete 7 Meanwhile a muh geate density omes fom the unidentified dak matte Fom the Geneal Relativity we know that both odinay and dak matte ontibute in favo of ontating of the univese But the lagest pat of density omes fom the so-alled dak enegy that aounts fo the osmologial onstant tem Although the total density of the univese is equal to the itial density (exatly up to measuement eo) the dak enegy does not lead to ontation of the univese In fat the dak enegy ontibutes in favo of expanding of the univese Reently the spatial geomety of the univese has been measued by the WMAP spaeaft The esult of that measuing showed that the spatial geomety of the univese is nealy flat Following this esult one an onlude that the univese an be well appoximated by a model without the spatial uvatue paamete (ie = 0) Meanwhile this does not neessaily imply that the univese is infinite It is beause ou measuing is elated to the obsevation pat of the univese that is limited but the univese is muh lage than the pat we an see Futhe we show a new appoah to the desiption of the univese motion This appoah is based on the new Relativisti Alpha Field (RAF) theoy 789 It has been shown that the geneal nondiagonal fom of the line element ds of RAF theoy in the spheial pola oodinates an be desibed by the equation 7: ds dt k dt d d d sin d k / ds dt dt d d d sin d (7) Hee is the speed of the light in a vauum is a adius veto θ is an angle between adius veto and z-axis and is an angle between pojetion of a adius veto on (x-y) plane and x-axis Paamete k = 1 (see 7) The geneal solutions of that line element ae pesented by the elations: GM GM onst 1 GM GM 1 1 ' 1 1 GM GM GM GM RAFT 1 GM GM GM GM 1 1 ' 1 1 (8) Hee RAFT means RAF theoy GM/ is the Newton s onstant of integation G is a gavitational onstant M is a total gavitational mass is a gavitational adius Λ is a osmologial onstant and is the speed of the light in a vauum In RAF theoy it is assumed that the osmologial onstant = 0 and the enegy momentum tenso T 0 If displaement fou-veto dx is defined in fame K by the expession: i 0 1 dx K dtdd d dx i (9) then the elated ovaiant meti tenso of the line element (7) has the fom 789: g (0) sin Hee the non-null omponents of the meti tenso g μ ae given by the elations: GM onst g00 1 GM g01 g10 g 11 1 g g sin GM GM RAFT g 00 1 GM GM g01 g 10 g 11 1g g sin (1) The elated deteminant of the meti tenso (0) has the foms: det g sin 1 det g 1 1 () 5 wwwijntog

8 Ae Singulaity and Dak Enegy Consequenes of Vauum Solution of Field Equations? In the pevious elation we use the nomalization fo = 1 = / and the well-known ondition fo the meti tenso of the line element det(g μη ) = -1 As the esult we obtain the simple elation between field paametes ν and λ If the line element in an alpha field is defined by the elations (7) to () then the dynami model of the Univese motion fo = onst is given by the equations 8: 8G onst () G p Compaing the fist equation in () with the fist Fiedmann equation in () and the seond equation in () with the seond Fiedmann equation in (5) one an onlude that both equations in () have the same foms as the Fiedmann equations if we inlude the substitutions = α = α and = α Thus if the equations () desibe the univese motion then the adial oodinate (t) has the oll of the sale fato α (t) The seond equation in () states that both the enegy density and the pessue ause a deeleation in the expansion of the univese This is the onsequene of gavitation inluding that pessue is playing a simila ole to that of enegy (o mass) density This is of ouse in aodane with the piniples of geneal elativity On the othe side the osmologial onstant Λ auses aeleation in the expansion of the univese In that ase the osmologial onstant Λ plays the ole of the dak enegy If the line element in an alpha field is defined by the elations (7) to () then the dynami model of the Univese motion in RAF theoy is given by the equations 8: 8G G RAFT 1 G p G 1 () Compaing the fist equation in () with the fist Fiedmann equation in () one an onlude that both equations have the same foms of the two pats of the left side and the fist pat of the ight side if the substitutions = α and = α ae valid Compaing the seond equation in () with the seond Fiedmann equation in (5) one an onlude that both equations have the same foms of the left side and of the fist pat of the ight side if the substitutions = α and = α ae valid Thus if the equations () desibe the univese motion then the adial oodinate (t) has the oll of the sale fato α (t) The fist pat of the ight side of the seond equation in () states that both the enegy density and the pessue ause a deeleation in the expansion of the univese This is the onsequene of gavitation inluding that pessue is playing a simila ole to that of enegy (o mass) density This is of ouse in aodane with the piniples of geneal elativity On the othe hand the seond pat of the ight side of the seond equation in () auses aeleation in the expansion of the univese This is the onsequene of the positive (epulsive) gavitation foe pesented by RAF theoy in 789 This fat theoetially onfims that the positive (epulsive) gavitational foe ould be the soue of the so alled dak enegy Of ouse this should be onfimed by the elated expeiments IV CONCLUSION In this pape we show that the new Relativisti Alpha Field Theoy (RAFT) poposes solution of the field equations without appeaane of singulaity and offes the soue of dak enegy On that way RAF theoy extends the appliation of GRT to the extemely stong fields at the Plank s sale This onlusion is based on the theoetially poofs that: a) no a singulaity at the Shwazshild adius b) thee exists a minimal adius at = (GM/ ) that pevents singulaity at = 0 ie the natue potets itself and d) a positive (epulsive) gavitational foe ould be the soue of dak enegy On that way we theoetially poved that the singulaity and dak enegy ae onsequenes of the vauum solution of Einstein s field equations ACKNOWLEDGMENTS The autho wishes to thank to the anonymous eviewes fo a vaiety of helpful omments and suggestions This wok is suppoted by gants ( ) fom the National Sientifi Foundation of Republi of Coatia V REFERENCES 1 A Einstein Ann Phys (1916) A Einstein The Meaning of Relativity (Pineton Univ Pess Pineton 1955) C Sean Spaetime and Geomety: An intodution to Geneal Relativiy (Amazonom Bookshtm Hadove 00) S Weinbeg Gavitation and Cosmology: Piniples and Appliation of the Geneal Theoy of Relativity (Gebundene Ausgabe RelEspWeinbegpdf 197) 5 S W Hawking G F R Ellis The Lage Sale Stutue of Spae-Time (Univ Pess Cambidge 197) 6 M Blau Letue Notes on Geneal Relativity (A Einstein Cente fo Fundamental Physis Univ Ben Ben 01 01) 7 B M Novakovi Relativisti alpha field theoy - Pat I: Detemination of Field Paametes Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume-1 Issue-5 Septembe 015 Pages -0 8 B M Novakovi Relativisti alpha field theoy - Pat II: Does a Gavitational Field Could be Without Singulaity? Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume-1 Issue-5 Septembe 015 Pages B M Novakovi Relativisti alpha field theoy - Pat III: Does Gavitational Foe Beomes Positive if (GM/) > 1? Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume-1 Issue-5 Septembe 015 Pages B M Novakovi Int J of Comput Antiip Syst IJCAS 7 p 9 (01) 11 S Gallot D Hullin and D J Lafontane Riemannian Geomety ( Spinge-Velag Belin New Yok ed 00) 1 C T J Dodson and T Poston Tenso Geomety Gaduate Texts in Mathematis (Spinge-Velag Belin New Yok ed 1991) p 10 1 M T Vaughin Intodution to Mathematial Physis (Wiley-VCH Velag GmbH & Co Weinheim 007) 1 D H Pekins Intodution to High Enegy Physis (Cambidge Univ Pess Cambidge 000) 15 D Sheman et al Nat Phys (015) 16 J Steinhaue Nat Phys (01) 17 M Mekel et al Nat Phys (01) wwwijntog 5

9 Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume- Issue-11 Novembe 017 Pages B M Novakovi D Novakovi and A Novakovi in Poeedings of the Sixth Int Conf on Comp Antiip Syst Liege 00 edited by D Dubois (Univesity of Liege Liege 00) AIP-CP 718 p1 (00) DOI: 10106/ B M Novakovi D Novakovi and A Novakovi in Poeedings of the Seventh Int Conf on Comp Antiip Syst Liege 005 edited by D Dubois (Univesity of Liege Liege 005) AIP-CP 89 p1 (006) DOI: 10106/ B M Novakovi in Poeedings of the Ninth Int Conf on Comp Antiip Syst Liege 009 edited by D Dubois (Univesity of Liege Liege 009) AIP-CP 10 p 11 (010) DOI: 10106/ R Ding et al Phys Rev D 9 (015008) (015) B M Novakovi Stojastvo 5 () (011) M Rees Just Six Numbes (Oion Book London p 81 p 8 000) S W Hawking: Thee no Blak Holes in Infomation Pesevation and Weathe Foeasting fo Blak Holes axiv: hep-th Jan 01 5 A Almheii D Maolf J Polhinski J Sully Blak Holes: Complementaity o Fiewalls? J High Enegy Phys 06 (01) 6 D Geosa E Beti Ae meging blak holes bon fom stella ollapse o pevious meges? Phys Rev D (017) 6 June A Fiedmann Übe die Kümmung des Raumes ZPhys 10 (1) pp (in Geman) (19) A Fiedman On the Cuvatue of Spae Geneal Relativity and Gavitation 1 (1) pp (in English) (1999) 8 A Fiedmann Übe die Möglihkeit eine Welt mit konstante negative Kümmung des Raumes ZPhys 1 (1) pp 6- (in Geman) (19) A Fiedman On the Possibility of a Wold with Constant Negative Cuvatue of Spae Geneal Relativity and Gavitation 1 (1) pp (in English) (1999) 9 GLemaîte Expansion of the Univese a Homogeneous Univese of Constant Mass and Ineasing Radius Aounting fo the Radial Veloity of Exta Galati Nebulae Monthly Noties of the Royal Astonomial Soiety 91 pp 8-90 (191) 0 G Lemaîte L Unives en Expansion Annales de la Soiété Sientifique de Buxelles A5 pp (in Fenh) (19) 1 HP Robetson Kinematis and Wold Stutue Astophysial Jounal 8 pp 8-01 (195) HP Robetson Kinematis and Wold Stutue II Astophysial Jounal 8 pp (196) HP Robetson Kinematis and Wold Stutue III Astophysial Jounal 8 pp (196) A G Walke On Milne s Theoy of Wold Stutue Poeedings of the London Mathematial Soiety (1) pp (197) 5 L Begstöm A Gooba Cosmology and Patile Astophysis Spint p 61 nd ed ISBN 509 (006) 6 P Ojeda H Rosu Supesymmety of FRW Baotopi Cosmologies Intenational Jounal of Theoetial Physis 5 (6) pp (006) 7 Wikipedia the fee enylopedia (01) FLRW Metis _Metis Retieved 19 Mah (01) 8 B M Novakovi Is Positive Gavitational Foe Soue of Dak Enegy? Intenational Jounal of New Tehnology and Reseah (IJNTR) ISSN:5-116 Volume-1 Issue-7 Novembe 015 Pages 06-1 Banko Novakovi is a Pofesso emeitus at FSB Univesity of Zageb Coatia Pof Novakovi eeived his PhD fom the Univesity of Zageb in 1978 His eseah of inteest inludes physis ontol systems obotis neual netwoks and fuzzy ontol He is autho of thee books; Relativisti Alpha Field Theoy (RAFT e-book 016) Contol Methods in Robotis Flexible Manufatuing Systems and Poesses (1990) and Contol Systems (1985) He is also o-autho of a book Atifiial Neual Netwoks (1998) He has published ove 0 eseah papes in his eseah of inteest 55 wwwijntog