RAF Theory Extends the Applications of GRT to the Extremely Strong Gravitational Fields

Size: px

Start display at page:

Download "RAF Theory Extends the Applications of GRT to the Extremely Strong Gravitational Fields"

Shon Scott
5 years ago
Views:

1 Intenational Jounal of New Technology and Reseach (IJNTR) ISSN:5-116 Volume-3 Issue-11 Novembe 017 Pages 56-6 RAF Theoy Extends the Alications of GRT to the Extemely Stong Gavitational Fields Banko M Novakovic Abstact- Moden hysics is tying to solve some oblems in the extemely stong gavitational field by using sohisticated methods in aticle and quantum hysics But we also should solve the mentioned oblems in the classical Geneal Relativity Theoy (GRT) As it is the well-known GRT cannot be alied to the extemely stong gavitational field The main eason fo it is an aeaance of the elated singulaity in that field Hee we show that Relativistic Alha Field Theoy (RAFT) extends the alication of GRT to the extemely stong fields including of the Planck s scale This is the consequence of the following edictions of RAF theoy: a) no a singulaity at the Schwazschild adius and b) thee exist a minimal adius at = (GM/c ) that events singulaity at = 0 ie the natue otects itself It has been theoetically oved that the metics of RAF theoy at the Schwazschild adius as well as at the minimal adius and at the Planck s scale ae egula Index Tems : Relativistic alha field theoy (RAFT) No singulaities in gavitational field Extemely stong gavitational fields Planck scale I INTRODUCTION As it is well known Geneal Relativity Theoy (GRT) 1-6 cannot be alied to the extemely stong gavitational fields including of the Planck s scale The main eason fo this is the aeaance of the elated singulaity in a gavitational field Hee we have used a new theoy that is called Relativistic Alha Field (RAF) theoy 789 It has been showed that RAF theoy extends the caability of the GRT to the alication to the extemely stong fields including of the Planck s scale Namely this is the consequence of the following edictions of RAF theoy: a) no a singulaity at the Schwazschild adius and b) thee exist a minimal adius at = (GM/c ) that events singulaity at = 0 ie the natue otects itself In this ae we stated with the esentation of the solution of the field aametes in RAF theoy This solution is based on the assumtion that the field aametes should connect geomety of the line element with otential enegy of a aticle in an alha field In that sense the concet of the two dimensionless (unit less) field aametes α and α has been intoduced These aametes ae scala functions of the otential enegy of a aticle in an alha field This new solution of the field aametes is the key oint in this theoy In ode to solve the field aametes α and α we stated with the deivation of the elative velocity of a aticle in an alha field v This elative velocity is deived fom the line element in an alha field that is given by the nondiagonal fom with the Riemannian metics Thus the elative velocity Banko Novakovic FSB Univesity of Zageb Luciceva 5 POB Zageb Coatia of a aticle in an alha field v is descibed as the function of the field aametes α and α and a aticle velocityv in the total vacuum (without any otential field) This stuctue of the elative velocityv diectly connects the line elements of the Secial and Geneal Relativity Namely in the case of the total vacuum (without any otential field) field aametes α and α become equal to unity and consequently the elative velocity v becomes equal to the aticle velocity v in the total vacuum This is a diect tansition of the line element fom the Geneal to the Secial Theoy of Relativity The main oint in this ae is the theoetical confimation that Relativistic Alha Field Theoy (RAFT) eally can extends the alication of GRT to the extemely stong gavitational fields including of the Planck s scale In that sense it has been esented that the metics at the Schwazschild adius as well as at the minimal adius ae egula Futhe the minimal adius of the Planck s mass is deived which is equal to the half of the Planck s length Thus the Planck s length is a diamete of the Planck s mass The metic at the Planck s scale is also egula This ae is oganized as follows In Sec II we show deivation of the elative velocity of a aticle in an alha field v as the function of the field aametes α and α Solution of the field aametes α and α in a geneal fom as the function of the otential enegy U is esented in Sec III Solution of the field aametes α and α in gavitational field is consideed in Sec IV Deivation of enegy-momentum tenso fo gavitational field is ointed out in Sec V The theoetical oofs that RAF theoy extends alications of GRT to the extemely stong gavitational field ae esented in Sec VI Finally the elated conclusion and the efeence list ae esented in Sec VII and Sec VIII esectively II DERIVATION OF RELATIVE VELOCITY V α RAF theoy is based on the following two definitions 7: Definition 1 An alha field is a otential field that can be descibed by two dimensionless (unit less) scala aametes α and α To this categoy belong among the othes electical and gavitational fields Definition Field aametes α and α ae descibed as the scala dimensionless (unit less) functions of the otential enegy U of a aticle in an alha field In ode to solve the field aametes α and α we stated with the deivation of the elative velocity of a aticle in an alha field v Poosition 1 If the line element in an alha field is defined by the nondiagonal fom with the Riemannian metics 56 wwwijntog

2 RAF Theoy Extends the Alications of GRT to the Extemely Stong Gavitational Fields 10-13: ds c dt cdt dt cdt dy cdt dz dx dy dz z (1) then the elative velocity of a aticle in an alha field v can be descibed as the function of the field aametes α and α c v v () In the evious equation v is a aticle velocity in the total vacuum (without any otential field) c is the seed of the light in a vacuum and is a constant Poof if the Poosition 1 This oof has been esented in the efeence 7 Poosition The elation in () satisfies the well-known condition fo the metic tenso of the line element (1) 1 1 x det g (3) Poof if the Poosition This oof has been esented in the efeence 7 Poosition 3 Let d and dt ae diffeentials of the oe time and coodinate time (esectively) of the moving aticle in an alha field Futhe let H is a tansfomation facto as an invaiant of an alha field and v is a aticle velocity in that field given by () Fo that case the tansfomation facto H has the following fom 1 / 1 / v cv dt v H 1 d c c c () Poof of the Poosition 3 This oof has been esented in the efeence 7 Futhemoe if a aticle is moving in a total vacuum (without any otential field) then we have = ' = 1 and the elation () is tansfomed into the tansfomation facto valid in the Secial Relativity 1 / dt v dt ' 1 H 1 d c dt' d dt' III SOLUTION OF THE FIELD PARAMETERS IN AN ALPHA FIELD Poosition Let m 0 is a est mass of a aticle U is a otential enegy of a aticle in an alha field c is the seed of the light in a vacuum and ( i ) is an imaginay unit In that case the field aametes α and α can be descibed as dimensionless (unit less) functions of the otential enegy U of a aticle in an alha field Thee ae fou solutions fo both aametes α and α in an alha field that can be esented by the following elations: y (5) f (U ) U / m c U / m c 1 i f (U ) 1 i f (U ) 1 i f (U ) 1 i f (U ) 1 1 i f (U ) 1 i f (U ) i f (U ) 1 i f (U ) Poof of the Poosition This oof has been esented in the efeence 7 The fou solutions of the field aamete α in (6) can be esented in the fom U U 1 1 i U U 3 1 i The elated fou solutions of the field aamete α in (6) can be esented with the following elations U U 1 1 i U U 3 1 i Thus the fou solutions of the field aametes α and α can be obtained by the unification of the two aamete stuctues given by (7) and (8): 0 0 f (U ) U / m c U / m c 1 i f (U ) 1 i f (U ) i f (U ) 1 i f (U ) 3 3 Futhe it is easy to ove that all ii ais fom (9) ae ceating an invaiant ' U ii = 1 i 1 3 ' (10) mc 0 Fo calculation some of the quantities in an alha field we often need to know the diffeence of the field aametes (α-α ): i f (U ) i f (U ) 1 1 i f (U ) i f (U ) 3 3 ( ) ( ) ( ) ( ) (6) (7) (8) (9) (11) The obtained elations in (9) (10) and (11) ae valid geneally and fo thei calculation we only need to know otential enegy U of the aticle in the elated otential field Remaks 1 Fom the equations (9) (10) and (11) we can see that thee ae thee vey imotant oeties of the 57 wwwijntog

3 solutions of the field aametes α and α : a) aametes α and α ae dimensionless (unitless) field aametes b) thee ae fou solutions of the field aametes α and α that eminds us to the Diac s theoy 1 and c) the quantity αα is an invaiant elated to the fou solutions of the field aametes α and α Intenational Jounal of New Technology and Reseach (IJNTR) ISSN:5-116 Volume-3 Issue-11 Novembe 017 Pages 56-6 gavitational field like in ou sola system the field aametes (13) satisfy the Einstein s field equations in a vacuum (T η = 0 = 0) The geneal metics of the elativistic alha field theoy 10 has been alied to the deivation of dynamic model of nanoobot motion in multiotential field 3 IV SOLUTION OF THE FIELD PARAMETERS IN GRAVITATIONAL FIELD If a aticle with the est mass m 0 is in a gavitational field then the otential enegy of the aticle in that field U g is descibed by the well-known elation 1-6 m GM U m V m A 0 g = 0 g 0 g0 (1) Hee V g = A g0 is a scala otential of the gavitational field G is the gavitational constant M is a gavitational mass and is a gavitational adius The fou solutions of the field aametes α and α fo the aticle in a gavitational field can be obtained by the substitution of the otential enegy U g fom (1) into the geneal elations in (9): i f (U ) GM / c GM / c g g GM c GM / c i f (U ) GM / c (13) The fist thee lines in equations (13) descibe a stong gavitational field If the quadatic tem ( GM / c ) 0 then the field aametes (13) descibe a weak gavitational field as we have in ou sola system The diffeences of the field aametes (α-α ) fo a aticle in a gavitational field have the foms: GM GM ( 1 1 ) c c GM GM c c ( ) (1) Remaks In the efeences 8915 it has been shown that field aametes (13) and (1) satisfy the Einstein s field equations with a cosmological constant = 0 In the case of a stong static gavitational field 16-0 the quadatic tem ( GM / c ) in (13) and (1) geneates the elated enegy-momentum tenso T η fo the static field Fo that case 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 The second inteetation could be that the quadatic tem ( GM / c ) geneates the cosmological aamete as a function of a gavitational adius 1 fo T η = 0 It has been shown that this solution of is valid fo both Planck s and cosmological scales In the case of a weak static V ENERGY-MOMENTUM TENSOR FOR GRAVITATIONAL FIELD The basic oblem of this section is to detemine the enegy-momentum tensos fo gavitational field in the Einstein s fou-dimensional sace-time (D) In that sense we stated with the geneal line element ds given by the elation (1) Following the well-known ocedue 1-6 this line element can be tansfomed into the sheical ola coodinates in the nondiagonal fom ds c dt c dt d d d sin d (15) The line element (15) belongs to the well-known fom of the Riemanns tye line element g dx g33 dx ds g dx g dx dx g dx (16) Comaing the equations (15) and (16) we obtain the coodinates and comonents of the covaiant metic tenso valid fo the line element (15): dx c dt dx d dx d dx d ( ) g 00 g01 g g g g sin (17) Stating with the line element (15) we emloy fo the convenient the following substitutions: / (18) In that case the nondiagonal line element (15) is tansfomed into the new elation ds c dt cdt d d d sin d (19) Using the coodinate system (17) the elated covaiant metic tenso g μη of the line element (19) is esented by the matix fom g sin (0) This tenso is symmetic and has six non-zeo elements as we exected that should be The contavaiant metic tenso g μη 58 wwwijntog

4 RAF Theoy Extends the Alications of GRT to the Extemely Stong Gavitational Fields of the nondiagonal line element (19) is deived by invesion of the covaiant one (0) 1 / ( ) / ( ) 0 0 / ( ) / ( ) 0 0 g / / sin (1) The deteminants of the tensos (0) and (1) ae given by the elations: det g sin 1 det g sin () Poosition 5 If the gavitational static field is descibed by the line element (19) 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 (3) Hee G and M ae the gavitational constant and the gavitational mass esectively Poof of the oosition 5 In ode to ove of the oosition 5 we can stat with the second tye of the Chistoffel symbols of the metic tensos (0) and (1) These symbols can be calculated by emloying the well-known elation 1-6 g g g g () Thus emloying (19) (0) (1) and () we obtain the second tye Chistoffel symbols of the sheically symmetic non-otating body: / D / D / D / D ( sin ) / D / D / D / D / D ( sin ) / D sincos ctg D t t (5) and 00 ae Fo a static field the Chistoffel symbols educed to the simlest fom: (6) In a static field the othe Chistoffel symbols in (5) ae emaining unchanged As it is well known the deteminant of the metic tenso of the line element (19) should satisfy the following condition det g sin 1 (7) Including the nomalization of the adius = 1 and the angle θ = 90 in (7) we obtain the imotant elations between the aametes ν and λ: 1 1 ' ' '' ' '' (8) If we take into account the elations (8) then the Chistoffel symbols in (5) and (6) become the only functions of the aamete Fo calculation of the elated comonents of the Riemannian tenso R and Ricci tenso R of the line element (19) we can emloy the following elations 1-6: R R R R (9) Alying the Chistoffel symbols (5) to the elations (9) we obtain the elated Ricci tenso fo the static field of the line element (19) with the following comonents: 33 ' R 00 1 ' '' ' R01 R 10 ' '' ' R 11 ' '' R ' R ' sin (30) The othe comonents of the Ricci tenso ae equal to zeo The elated Ricci scala fo the static field is detemined by the equation R g R ' ' R ' '' (31) In ode to calculate the enegy-momentum tenso T η fo the static field one should emloy Ricci tenso (30) Ricci scala (31) and the Einstein s field equations 1-6 without a cosmological constant ( = 0) 59 wwwijntog

5 1 8G R gr kt k (3) c Hee G is the Newton s gavitational constant c is the seed of the light in a vacuum and T η is the enegy-momentum tenso Thus emloying the Einstein s field equations (3) we obtain the following elations fo calculation of the comonents of the enegy-momentum tenso T μη : ' ' kt kt kt ' 1 ' kt 11 kt ' '' ' 8G kt33 sin ' '' k c (33) Fo calculation of the comonents of the enegy-momentum tenso T μη by the elations (33) we should know the aamete and its deivations ' and '' fo the elated static field Paamete is defined by (18) as the function of the field aametes α and α / / 1 (3) Alying of the solution of the field aametes α and α in a gavitational field (131) to the elation (3) we obtain the two solutions of the aamete in a gavitational static field GM GM c c Intenational Jounal of New Technology and Reseach (IJNTR) ISSN:5-116 Volume-3 Issue-11 Novembe 017 Pages 56-6 (35) Now one can calculate of the all comonents needed fo detemination of the enegy-momentum tenso T μη in a static gavitational field: GM GM GM GM GM ' / c c c c c GM GM GM ' c c c ' GM GM GM GM 3 3 c c c c GM ' 3 GM '' ' ' 3 c c (36) Alying the elations (36) to the equations (31) and (33) we obtain the comonents of the enegy-momentum tenso and Ricci scala valid fo the static gavitational field: GM GM 00 1 kt c c GM 8G kt01 kt 10 k c c GM GM 11 1 kt kt c c GM kt33 sin c ' ' R ' '' GM 0 c c GM (37) Fom the evious elations we can see that the Ricci scala is equal to zeo Finally included aamete k into the elations (37) we obtain the comonents of the enegy-momentum tenso in the static gavitational field T T T T T T T GM 1 sin 8G (38) Because the elation (38) is equal to the elation (3) the oof of the oosition 5 is finished VI PROOFS THAT RAF THEORY EXTENDS APPLICATIONS OF GRT TO EXSTREMELY STRONG GRAVITATIONAL FIELD As it is the well known gavitational fields become stong and extemely stong at the Schwazschild and smalle adiuses In ode to extend the alication of GRT to the extemely stong gavitational fields we have to have the elated theoy with egula line element in that egion Just RAF theoy offes egula line element in that egion by the following edictions: a) no a singulaity at the Schwazschild adius and b) thee exists a minimal adius at the osition min GM / c that events singulaity at = 0 ie the natue otects itself Thus RAF theoy has a egula line element in the egion min In ode to ove edictions a) and b) we stat with the solution of the aametes and in a static gavitational field given by (8) and (35) and valid fo the line element (19): 60 wwwijntog

6 RAF Theoy Extends the Alications of GRT to the Extemely Stong Gavitational Fields GM GM 1 1 c c GM GM GM c c c GM 1 3 sch sch sch c GM min 1 0 min im c GM c The elated line elements ae given by the elations: 1 3 ds c dt k cdt d d d sch ds c dt d d min GM / c ds kcdt d d d ds c dt d d d d sin d k 1 (39) (0) Following the elations (39) and (0) we can see that at the Schwazschild adius sch aametes and ae egula This oves the ediction a) no a singulaity at the Schwazschild adius Futhe fom the same elations we also can see that at the minimal adius GM / c aametes and ae also egula and fo min aamete becomes imaginay numbe im This oves the ediction b) thee exists a minimal adius at min GM / c that events singulaity at 0 It seems that the existence of the minimal adius tells us that the natue otect itself fom the singulaity Thus we can say that the metics of the line element (19) is egula fo a gavitational field in the egion min On that way the oof of the oositions a) and b) is finished At the same time it has been oved that RAF theoy extends the alications of GRT to the extemely stong gavitational fields Now we can assume that the Planck s mass M 6 is the sheically symmetic non-otating body Fo that case one can calculate the minimal adius of the Planck s mass m : GM M c m c G M L m ћc / G c L m L m (1) 3 ћg / c G By the elations in (1) we found out that the minimal adius of the Planck s mass M is equal to half of the Planck s length 7 This means that Planck s length is the diamete of the Planck s mass At the minimal adius of the Planck s min mass m = L / aametes and and the elated line element have the following foms: GM G c ( L / )c c G GM GM 1 1 ( L / )c ( L / )c G c G c 1 0 c G c G m ds c dt d d () Fom the elations in () we can see that at the minimal adius m = L / of the Planck s mass aametes and as well as the line element ae also egula This oves ediction a) no a singulaity at the minimal adius of the Planck s mass If the adius is less than the minimal adius m = L / then aamete fo the Planck s mass becomes imaginay numbe im This oves the ediction b) thee exists a minimal adius of the Planck s mass m = L / that events singulaity at = 0 It means that the existence of the minimal adius tells us that the natue otect itself fom the singulaity Thus we can say that the metics of the line element in (19) is also egula fo a gavitational field at the Planck s scale On that way the oofs of the oositions a) and b) at the Planck s scale ae finished At the same time it has been oved that RAF theoy extends the alications of GRT to the extemely stong gavitational fields at the Planck s scale VII CONCLUSION In this ae we show that Relativistic Alha Field Theoy (RAFT) extends the alication of GRT to the extemely stong gavitational fields including of the Planck s scale It has been esented that the metics at the Schwazschild adius as well as at the minimal adius ae egula Futhe the minimal adius of the Planck s mass is deived which is equal to the half of the Planck s length Thus the Planck s length is a diamete of the Planck s mass The metic at the Planck s scale is also egula The esented esults ae the consequence of the vey imotant edictions of RAF theoy: a) no a singulaity at the Schwazschild adius and b) thee exists a minimal adius at min GM / c that events singulaity at 0 ie the natue otects itself If the edictions of RAF theoy ae coect then it could give the new light to the egions like black holes quantum theoy high enegy hysics Big Bang theoy and cosmology ACKNOWLEDGMENTS The autho wishes to thank to the anonymous eviewes fo a vaiety of helful comments and suggestions This wok is suoted by gants ( ) fom the National Scientific Foundation of Reublic of Coatia 61 wwwijntog

Intenational Jounal of New Technology and Reseach (IJNTR) ISSN:5-116 Volume-3 Issue-11 Novembe 017 Pages 56-6 VIII REFERENCES 1 A Einstein Ann Phys 9 769-8 (1916) A Einstein The Meaning of Relativity

7 Intenational Jounal of New Technology and Reseach (IJNTR) ISSN:5-116 Volume-3 Issue-11 Novembe 017 Pages 56-6 VIII REFERENCES 1 A Einstein Ann Phys (1916) A Einstein The Meaning of Relativity (Pinceton Univ Pess Pinceton 1955) 3 C Sean Sacetime and Geomety: An intoduction to Geneal Relativiy (Amazoncom Bookshtm Hadcove 003) S Weinbeg Gavitation and Cosmology: Pinciles and Alication of the Geneal Theoy of Relativity (Gebundene Ausgabe RelEsWeinbegdf 197) 5 S W Hawking G F R Ellis The Lage Scale Stuctue of Sace-Time (Univ Pess Cambidge 1973) 6 M Blau Lectue Notes on Geneal Relativity (A Einstein Cente fo Fundamental Physics Univ Ben Ben 01 01) 7 B M Novakovic Relativistic Alha Field Theoy - Pat I: Detemination of Field Paametes Intenational Jounal of New Technology and Reseach (IJNTR) ISSN: Vol 1 Issue (015) Pae ID-IJNTR B M Novakovic Relativistic alha field theoy - Pat II Does gavitational Field Could be Without Singulaity? IJNTR ISSN: Vol 1 Issue (015) Pae ID-IJNTR B M Novakovic Relativistic alha field theoy-pat III Does Gavitational Foce Becomes Positive if (GM/cc) > 1? IJNTR ISSN: Vol 1 Issue (015) Pae ID-IJNTR B M Novakovic Int J of Comut Antici Syst IJCAS 7 93 (01) 11 S Gallot D Hullin and D J Lafontane Riemannian Geomety ( Singe-Velag Belin New Yok ed 3 00) 1 C T J Dodson and T Poston Tenso Geomety Gaduate Texts in Mathematics (Singe-Velag Belin New Yok ed 1991) M T Vaughin Intoduction to Mathematical Physics (Wiley-VCH Velag GmbH & Co Weinheim 007) 1 P A M Diac Diections in Physics (Wiley New Yok 1978) 15 B M Novakovic Relativistic alha field theoy e-book Amazoncom ublished (016) 16 D H Pekins Intoduction to High Enegy Physics (Cambidge Univ Pess Cambidge 000) 17 D Sheman et al Nat Phys (015) 18 J Steinhaue Nat Phys (01) 19 M Meckel et al Nat Phys (01) 0 R Ding et al Phys Rev D 9 (015008) (015) 1 B M Novakovic D Novakovic and A Novakovic in Poceedings of the Sixth Int Conf on Com Antici Syst Liege 003 edited by D Dubois (Univesity of Liege Liege 003) AIP-CP (00) DOI: / B M Novakovic D Novakovic and A Novakovic in Poceedings of the Seventh Int Conf on Com Antici Syst Liege 005 edited by D Dubois (Univesity of Liege Liege 005) AIP-CP (006) DOI: / B M Novakovic Stojastvo 53 () (011) B M Novakovic in Poceedings of the Ninth Int Conf on Com Antici Syst Liege 009 edited by D Dubois (Univesity of Liege Liege 009) AIP-CP (010) DOI: / B M Novakovic Is Positive Gavitational Foce Souce of Dak Enegy? IJNTR ISSN: Vol 1 Issue (015) Pae ID-IJNTR CODATA Values: Planck mass US National Institute of Standad and Technology June (015) (htt://hysicsnistgov/cgi-bin/cuu/value?lkm) 7 CODATA Constants: Planck length US National Institute of Standad and Technology June (015) (htt://hysicsnistgov/cgi-bin/cuu/value?lkl) Banko Novakovic is a Pofesso emeitus at FSB Univesity of Zageb Coatia Pof Novakovic eceived his PhD fom the Univesity of Zageb in 1978 His eseach of inteest includes hysics contol systems obotics neual netwoks and fuzzy contol He is autho of thee books; Relativistic Alha Field Theoy ((RAFT e-book 016) Contol Methods in Robotics Flexible Manufactuing Systems and Pocesses (1990) Contol Systems (1985) and the fist co-autho of a book Atificial Neual Netwoks (1998) He has ublished ove 30 eseach aes in his eseach of inteest 6 wwwijntog

d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c

d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.