Strong Gravity and the BKL Conjecture

Transcription

1 Introducton Strong Grvty nd the BKL Conecture Dvd Slon Penn Stte October 16, 2007 Dvd Slon Strong Grvty nd the BKL Conecture

2 Introducton Outlne The BKL Conecture Ashtekr Vrbles Ksner Sngulrty 1 Introducton Outlne The BKL Conecture Ashtekr Vrbles Ksner Sngulrty 2 Strong Grvty Constrnts Equtons of Moton 3 New Clculus Exmple Full Theory The BKL Reducton 4 Concluson References Dvd Slon Strong Grvty nd the BKL Conecture

3 Introducton Outlne The BKL Conecture Ashtekr Vrbles Ksner Sngulrty The Conecture (Belnsk, Khltnkov, Lfshtz) (1971) Ner to sngulrty sptlly seprted ponts decouple, nd the role of most forms of mtter s neglgble. Problems Does not pper geometrc Mthemtcl mplctons vgue Only conecture - no nlytc proof. Motvton for pplyng the BKL conecture The Ensten eld equtons re hrd to solve: Smlr to homogenous or sotropc cosmology - ttempt to solve the EFEs n smple cse Numercl evdence supports the BKL conecture: Recently lot of numercl work hs been done (Grnkle, Uggl, Elst, Ells, Wnwrght, Curts, Moncref, Berger...) provdng support for the BKL conecture. Dvd Slon Strong Grvty nd the BKL Conecture

4 Why Ashtekr Vrbles Introducton Outlne The BKL Conecture Ashtekr Vrbles Ksner Sngulrty Why Use Ashtekr Vrbles? Smple formulton Bsc Vrbles e E ; A - no nverses. ADM Vrbles need q b nd q b. Not well dened t sngulrty. Cnddte for quntum theory Sngulrty Resoluton Our work s done n self dul vrbles - re-evluton n rel vrbles to come lter. Sngulrty Resoluton Am s to resolve lrge clss of cosmologcl sngulrtes. Smlr poston to rst ndng sngulrtes Frst specc sngulrtes found Dsgreement bout how generc sngulrtes re Sngulrty proven Dvd Slon Strong Grvty nd the BKL Conecture

5 Ashtekr Vrbles Introducton Outlne The BKL Conecture Ashtekr Vrbles Ksner Sngulrty A bref recp of Ashtekr vrbles wth unts We work wth denstzed trd e E nd connecton A. Erly letters (,b,c..) denote sptl ndces, lter (,,k...) nternl. ee e E b = p qq b (1) A = 1 G ( K ) (2) ee K b = qk b (3) fa (x); e E b (y)g = b 3 (x y) (4) ee q b 1 (5) M LT (6) A Dvd Slon Strong Grvty nd the BKL Conecture

6 The Ksner Sngulrty Introducton Outlne The BKL Conecture Ashtekr Vrbles Ksner Sngulrty Bnch I Soluton The Ksner spce-tme tkes the form: ds 2 = d 2 + 2p1 dx 2 + 2p2 dy 2 + 2p3 dz 2 (7) p 1 + p 2 + p 3 = 1 = p p2 2 + p2 3 (8) p 2 R (9) One degree of freedom Homogeneous vcuum soluton Unstble n perturbtons Addng sclr eld (st ud) gves rse to stble sector Dvd Slon Strong Grvty nd the BKL Conecture

7 Introducton Strong Grvty s Toy Model Strong Grvty Constrnts Equtons of Moton The Strong Couplng Lmt We exmne the lmt G! 1 whlst keepng E e nd A xed. b Reltonshp wth BKL Conecture Consder the dervtve opertor D h h + G c k A h k (10) From 0 s equvlent to G! 1. The strong couplng lmt forms toy model of the BKL conecture - we gnore sptl dervtves. Not the entre pcture - generlly we see sptl dervtves suppressed rther thn gnored. Dvd Slon Strong Grvty nd the BKL Conecture

8 Introducton Constrnts n Strong Grvty Strong Grvty Constrnts Equtons of Moton Full Constrnts es = 2c E e E e b [A b]k + 2G E e k E e b V = c e E b Reduced Constrnts (2@ [A b] + G A G = E e + G k A E e k A b k A [ A b ] (11) k) (12) (13) es = 2G E e E e b V = G e E b A G = G k A e E k [ A b ] (14) k A A b k (15) (16) fe S ; V g = 0 fe S ; G g = 0 fg ; V g = 0 (17) Dvd Slon Strong Grvty nd the BKL Conecture

9 Smpler Vrbles Introducton Strong Grvty Constrnts Equtons of Moton New Vrble e M We cn ntroduce new vrble to smplfy thngs: M e = E e f e M fe E Constrnts n terms of e M e (x); M l (y)g = ( e k M l k A + M e k l)3 (x y) (18) (x); e k M (y)g = e E 3 (x y) (19) k es = G (( M e ) 2 e M e M ) (20) = G (Tr( e M) 2 Tr( e M 2 )) (21) G = G k M k (22) Dvd Slon Strong Grvty nd the BKL Conecture

10 Equtons of Moton Introducton Strong Grvty Constrnts Equtons of Moton Tme Evoluton of M e nd E e _ ee = N( M e k k _ em e M )e E (23) = 0 (24) Commuttvty of Strong Lmt nd Equtons of Moton Full Constrnt?? yequton of Moton Lrge G lmt! Reduced Constrnt?? yequton of Moton Full Equton of Moton Lrge G lmt! Reduced Equton of Moton Dvd Slon Strong Grvty nd the BKL Conecture

11 Equtons of Moton Introducton Strong Grvty Constrnts Equtons of Moton Relty Condtons q 2 R! _q 2 R! e M 2 R (25) M s symmetrc (from Guss Constrnt) rel mtrx nd s therefore dgonlzble n some bss of our nternl spce. Solutons We cn solve our smple dgonl equtons n the lpse N = 1 cse: ds 2 = d 2 + 2f1 ( o E 1 1 )2 dx 2 + 2f2 ( o E 2 2 )2 dy 2 + 2f3 ( o E 3 3 )2 dz 2 (26) Wth sclr constrnt mplyng X X f = f 2 = 1 (27) Dvd Slon Strong Grvty nd the BKL Conecture

12 A New Clculus Introducton New Clculus Exmple Full Theory The BKL Reducton The BKL Lmt We wll exmne the followng scenro: X! 0 where p Det(q)T hs lmt 8T 2 X A e ; E 2 X Det(q)! 0 Unresonble Assumpton: = 0 Why Use Ths Model? Smlr to \lmost FL" models of Uggl, Elst et l. Numercl evdence from Grnkle Gowdy X suppressed Allows f f bounded. From Ksner model we see E e ; M hve ths behvour. Dvd Slon Strong Grvty nd the BKL Conecture

13 An Exmple Introducton New Clculus Exmple Full Theory The BKL Reducton A non-sngulr exmple Consder Mnkowsk spce n coordntes dpted to hyperbolod: ds 2 = d [d 2 + cosh 2 ()(d 2 + cos 2 ()d 2 )] (28) Here we see! 0 s tkng Det(q)! 0 Dvd Slon Strong Grvty nd the BKL Conecture

14 Introducton New Clculus Exmple Full Theory The BKL Reducton Non-Reduced Constrnts nd Equtons of Moton Constrnts Full constrnts the sme s n the toy model Multple of the Guss constrnt dded to Hmltonn - keeps evoluton rel: H = Equtons of Moton Z d 3 x[ 1 2 N e S N V + (e E D N)G ] (29) _fe b = Ne E _fm = 2 Ne E b D e E b k k (30) E k [A b]l kl Ne k e E b l e E b (D N E e ) e E b D kl Ak b E e N l kl A b (31) (32) + N e M k e M[k] (33) Dvd Slon Strong Grvty nd the BKL Conecture

15 Introducton Reduced Equtons of Moton New Clculus Exmple Full Theory The BKL Reducton Equtons of Moton G = Re[ k M k ] (34) _ ee = N( M e k e k M (35) _ em )e E = 0 (36) Commuttvty of BKL Reducton nd Equtons of Moton BKL Reducton!? yequton of Moton Full Constrnt Reduced Constrnt?? yequton of Moton Full Equton of Moton BKL Reducton! Reduced Equton of Moton Dvd Slon Strong Grvty nd the BKL Conecture

16 Introducton Rentroducng Sptl Curvture New Clculus Exmple Full Theory The BKL Reducton Relxng = 0 We cn relx our = 0 condton e M s no longer symmetrc mtrx _fm = N e M k e M[k] Generl nlytc solutons not found 'Bounces' pper n numercl smulton Perturbtons to symmetrc mtrx grow Dvd Slon Strong Grvty nd the BKL Conecture

17 Conclusons Introducton Concluson References Results BKL Conecture cn be ppled n Ashtekr Vrbles Resultng spcetme hs Ksner sngulrty n 'worst cse' Corresponds to known numercl results Open Issues Work done n self-dul vrbles - chnges n rel vrbles? Sptl curvture s non-zero n generl for bounce behvour Need to dd mtter Dvd Slon Strong Grvty nd the BKL Conecture

18 References Introducton Concluson References Uggl et l. \Pst Attrctor n Inhomogeneous Cosmology" gr-qc/ Andersson + Rendll \Quescent Cosmologcl Sngulrtes" Comm. Mth. Phys. 218 (2001) Berger et l. \Osclltory Approch to Sngulrty" Phys Rev. D 64 (2004) Grnkle \Numercl Smultons of St Flud Grvttonl Sngulrtes" gr-qc/ Grnkle \Numercl Smultons of Generc Sngulrtes" gr-qc/ Rendll \The Nture of Spcetme Sngulrtes" gr-qc/ Dvd Slon Strong Grvty nd the BKL Conecture