Representation of gravity field equation and solutions, Hawking Radiation in Data General Relativity theory

Transcription

1 epesentation of gavity field equation and solutions Hawking adiation in Data Geneal elativity theoy Sangwha-Yi Depatment of Math aejon Univesity ABSAC In the geneal elativity theoy we find the epesentation of the gavity field equation and solutions. We teat the epesentation of Shwazshild solution eissne- Nodstom solution e-newman solution obetson -Walke solution. Speially obetson -Walke solution is an uniqueness. We found new geneal elativity theoy (we all it Data Geneal elativity heoy;dg). We teat the data of Hawking adiation by Data geneal elativity theoy. his theoy has to apply blak hole (speially Pimodial Massive Blak Hole; PMBH) beause blak hole(pmbh) is an idealisti stutue. PACS Numbe: e ey wods:geneal elativity theoy he othe solution Shwazshild solution eissne- Nodstom solution e-newman solution obetson-walke solution Data Geneal elativity theoy Hawking adiation Pimodial Massive Blak Hole el:

2 . Intodution In the geneal elativity theoy ou atile s aim is that we find the epesentation of the gavity field equation and solutions. We found new geneal elativity theoy(we an all theoy). We moe obtain data of Hawking adiation by Data geneal elativity theoy. Fist the gavity potential g is it Data Geneal elativity ds g () In gavity potential ds' f g we intodue tenso f and sala. f g 0 x g g' ' ' g f' x x' x x' ' ' ' ' g' x x g x' x' f ' x x f x' x' In invese gavity potentialg f f In Chistoffel symbol ' g g () ( g )( g) f g (3) f f f f ( ) x x x g g g ( g )( ) x x x g g ( x g x g ) x () heefoe in the uvatue tenso ' ' x ' x ' ' ' '

3 x x x x x x ( ) (5) In ii tenso In uvatue sala g ' ' ' f g ' g Hene in the gavity field equation of Einstein ' f ' g g( ) 8G g ( 8G g In Newtonian appoximation Enegy-momentum tenso is ) (8) (6) (7) f 00 g 00 8G 00 8G ' ' 00 (9) Hene 8G 8 g00 g G (0) 00

4 ' heefoe ised Einstein s gavity field equation is f 8 G 8G ' ' g 8G 8G ( g ) ' g () () heefoe ised gavity field equation of tenso equation. heefoe g g is able to edue Einstein s gavity field 8G [ g ] g [ g ] g Hene 8G g 8G 8G 8G 8G (3) ii tenso is 8G 8G ( ) ( ) g g (5) () he pope distane is ds g ds ' f g (6). Weak gavity field appoximation. Weak gavity field appoximation is g Aoding to Eq(5) h h 8G ( ) g (7)

5 g h he tenso of weak gavity field is 8G h 8G ( ) g (8) 8G S S S S (9) he solution is G S ( ' ') t x x x h ( t x) d x' x x' d 3 x00 M G h ( t x ) S ( ' ') t x x x d x' x x' 3 d x00 d x 00 As 3 M M G 3 GM h00( x ) d x'[ 00 00] G GM h ( ) 3 ij x d x'[ 00] ij ij he pope distane is ds d (0) () () GM GM i j g ( ) dt ( ) ij (3) he pope distane is in this theoy ds' ds d g GM GM i j ( ) dt ( ) ij

6 GM GM i ( ) dt ( ) ij GM GM i j ( ) dt ( ) ij g i i j j t t x x x x M M () 3. he othe epesentation in Shwazshild solution eissne-nodstom solution e-newman solution and obetson-walke solution Shwazshild solution (vauum solution) is ds GM 0 GM ( ) dt d sin d (5) he othe epesentation of Shwazshild solution is ds' f g ds GM GM ( ) dt d sin d GM ( ) dt d sin d GM j GM GM ( ) dt d sin d g t t M M (6) eissne-nodstom solution is ds g GM kgq GM kgq ( ) dt d sin d (7)

7 he othe epesentation of eissne-nodstom solution is ds' f g g ds GM kgq GM kgq ( ) dt d sin d GM kgq ( ) dt d sin d GM kgq GM kgq GM kgq ( ) dt d sin d t t M M Q Q (8) e-newman solution is ds g GM kgq a sin ( ) dt ( MG kgq ) dtd d GM a kgq sin[ a ( GM kgq) a sin ] d a os (9) he othe epesentation of e-newman solution is ds' f g ds GM kgq a sin ( ) dt ( MG kgq ) dtd d GM a kgq sin[ a ( GM kgq) a sin ] d

8 G M kgq a sin ( ) dt ( MG kgq ) d td g d( ) d G M a kgq a sin sin[ a ( G M kgq ) ] d GM kgq a sin ( ) dt ( MG kgq ) dtd GM a sin[ a kgq d a ( GM kgq ) sin ] d a os a os t t M M Q Q a a J Ma Ma J (30) In this time we obtain the data of the time t the distane the mass M the hage Q and the angula momentum J. obetson-walke solution is ds g dt ( t )[ d sin d ] (3) k he othe epesentation of obetson-walke solution is by the othe sala ' ds' f ' dt dt ' g ' ds ' ( t )[ ' k ' ( t )[ k ' d ' d ' sin d ] sin d ]

9 dt ( t )[ k' d sin d ] g ' t t ( t ) ( t ) ' k ( 0 ) k k' (0 ) ' ' ' (3) Hene ' In this time ds is an uniqueness. his theoy didn t apply the osmology.. Obtaining poess infomation of Hawking adiation Stephen Hawking found blak hole s themodynamis. By Hawking adiation we obtain the new data fom fomulas of blak-hole s themodynamis. We stat the obtaining poess infomations of Hawking adiation. In Wikipedia (Hawking adiation) we know fomulas of Hawking adiation. he adiation tempeatue of Shwazshild blak hole (In this theoy PMBH) 3 8GMk B (33) he adiation tempeatue is in Data Geneal elativity theoy GMk 8G Mk B B (3) he blak hole (PMBH) s entopy ds 8 dq GkBMdM / d(m ) GkB / (35) he blak-hole (PMBH) s entopy S is in Data Geneal elativity theoy. ds 8MdMGk / d(m ) Gk / d(m ) Gk / B dq d( Q) ds S S Q Q (36) Blak-hole (PMBH) adiation s powe P is B B P S H A A s S GM S is onstant (37) Blak-hole (PMBH) adiation s powe P is in Data Geneal elativity theoy.

10 GM G M As S S As S S P P AS H As Stefan-Boltzmann onstant k B 3 60 Blak-hole (PMBH) is a pefet blak body( ) (38) he apoation time t of a blak hole (PMBH) is t 3 50 G M (39) he apoation time t of a blak hole (PMBH) is in Data Geneal elativity theoy. t 50 G M 50 G M 3 3 t he powe of apoation enegy of the blak-hole (PMBH) is P de (0) () dt he powe of apoation enegy of the blak-hole (PMBH) is in Data Geneal elativity theoy. P de dt d( E ) P dt M E E M () 5. Conlusion We find the othe epesentation of solutions in the Geneal elativity theoy. In this time obetson- Walke solution is an uniqueness. We moe obtain the infomation of blak-hole themodynamis in Data Geneal elativity theoy. If we use vaiable A instead of A Data Geneal elativity theoy is edued to nomal geneal elativity theoy. his theoy s emakable thing is if and blak hole (PMBH) s mass M is M M blak hole (PMBH) s distant of gavitation is blak hole (PMBH) s pope time is.if otating blak hole (PMBH) s mass M is to be M M we pedit the angula momentum J of the blak-hole (PMBH) is to be J J. In this time we have to apply only blak-holes (PMBHs) beause blak hole (PMBH) is an idealisti

11 stutue. BH is Blak hole. Appendix A In DG we have to apply only blak-holes (PMBHs). Aoding to [7]Paul H. Fampton Physial Lette B(07) if the mass of sun is M data is. Astophysial Mass sola Peiod Angula Objet masses seonds momentum kg km s.. PIMBH 0 M 0.03s PIMBH 00 M 0.063s PIMBH3 000 M 0.63s PIMBH 0 M 6.3s 7. 0 PIMBH5 5 0 M 63s 7. 0 PSMBH6(M87) M s Aoding to DG BH (PMBH) s mass M is to be M time is to be Angula momentum J is to be J.Hene PIMBH s (fom PIMBH) is 5 PIMBH3 s (fom PIMBH ) is 0PIMBH s(fom PIMBH3) is 0 PIMBH5 s (fom PIMBH) is 0 PSMBH6 s (fom PIMBH5) is 6 0. In this time PIMBH is Pimodial Intemediate Massive Blak hole PSMBH is Pimodial Supe Massive Blak Hole. he hypothesis of the onstituents of dak matte in the galati halo ae PIMBHs is his theoy. Aoding to this theoy [7] Angula momentum of dak matte blak holes 0 M MPIMBH M 5 0 M M PSMBH 0 7 M

12 heefoe alulated data is in DG. Astophysial Mass sola Peiod Angula Objet masses seonds momentum kg km s. PIMBH 0 M 0.03s PIMBH 00 M 0.065s PIMBH3 000 M 0.65s PIMBH 0 M 6.5s PIMBH5 5 0 M 65s PSMBH6(M87) M s efeene []S. Weinbeg Gavitation and Cosmology(John wiley & SonsIn97) []. Eotvos D. Peka and E. Fetkete Ann.Phys.(Gemany)68(9) [3]C. Misne hone and J. Wheele Gavitation (W.H.Feedman & Co.973) []S. Hawking and G. Ellishe Lage Sale Stutue of Spae-ime(Cam-bidge Univesity Pess973) [5]. AdleM.Bazin and M.ShiffeIntodution to Geneal elativity(mgaw-hillin.965) [6]Hawking adiation-wikipedia [7]P. oll. otkov and. Dike Ann. Phys.(U.S.A) 6(96) [8]V. Baginsky and V. Panov Zh. Eksp. & eo. Fiz.6 873(97)(English tanslation Sov. Phys.-JEP 36(97) [9]L.Einsenhat iemannian Geomety(Pineton Univesity Pess 96) [0]J. Shouten ii-calulus(spinge-velag Belin 95) []A. Einstein he Foundation of Geneal elativity Ann. Phys. (Gemany)9(96) []G.Bikoff elativity and Moden Physis (Havad Univesity Pess93)p53

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