Hamiltonian reduction of Einstein s equations without isometries

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1 Hmiltonin reduction of Einstein s equtions without isometries Jong Hyuk Yoon 1, 1 Deprtment of Physics, Konkuk University, 120 Neungdong-ro, Gwngjin-gu, Seoul 05029, Kore Abstrct. I pply the Hmiltonin reduction procedure to 4-dimensionl spcetimes without isometries nd find privileged spcetime coordintes in which the physicl Hmiltonin is expressed in terms of the conforml two metric nd its conjugte momentum. Physicl time is the re element of the cross section of null hypersurfce, nd the physicl rdil coordinte is defined by equipotentil surfces on given spcelike hypersurfce of constnt physicl time. The physicl Hmiltonin is locl nd positive in the privileged coordintes. Einstein s equtions in the privileged coordintes re presented s Hmilton s equtions of motions obtined from the physicl Hmiltonin. 1 Introduction It is well-known tht the true grvittionl degrees of freedom of generl reltivity reside in the conforml two metric of the sptil cross section of null hypersurfces[1, 2]. Eliminting unphysicl degrees of freedom by identifying rbitrrily specifible spcetime coordintes with certin functions in phse spce nd thereby presenting the theory in terms of physicl degrees of freedom in the privileged coordintes, free from constrints, is known s the ADM Hmiltonin reduction[3 5]. Prof. Kuchř nd others pplied this procedure to spcetimes tht dmit two commuting Killing vector fields, known s midi-superspce[6 8], nd showed tht Einstein s theory ws equivlent to cylindricl mssless sclr field theory propgting in the 1+1 dimensionl Minkowski spcetime. In this pper, we pply the ADM Hmiltonin reduction to generl spcetimes of 4 dimensions under no symmetry ssumption[9 13]. The re element of the sptil cross section of null hypersurfce emerges s the physicl time, nd the physicl rdil coordinte is defined by equipotentil surfces on given spcelike hypersurfce of constnt physicl time. We present the fully reduced physicl Hmiltonin in these privileged coordintes[14], which turns out to be locl nd positive. The momentum constrints turn out to be simply the defining equtions of the physicl liner nd ngulr momentum densities in term of the conforml two metric nd its conjugte momentum[6]; hence, there remin no constrints to solve, nd the theory becomes constrint-free. Moreover, we find tht our Hmiltonin reduction is self-consistent becuse Hmilton s equtions of motion obtined through this Hmiltonin reduction re identicl to the Ricci-flt equtions in the privileged coordintes. As by-product of this Hmiltonin reduction, we found n independent proof of topologicl censorship[15 20], which followed e-mil: yoonjh@konkuk.c.kr directly from one of the Einstein s equtions in these coordintes. 2 The ction in the (2+2) formlism Let us recll tht the metric in the (2+2) decomposition[1, 2, 9 13] of 4 dimensionl spcetimes cn be written s ds 2 = 2dudv 2hdu 2 ( + τρ b dy + A+ du + A dv ) ( dy b + A+ b du + A b dv ). (1) The vector fields ˆ ± defined s where ˆ ± := ± A ±, (2) + = / u, = / v, = / y ( = 2, 3), (3) re horizontl vector fields orthogonl to the twodimensionl spcelike surfce N 2 generted by. The inner products of the horizontl vector fields re given by < ˆ +, ˆ + >= 2h, < ˆ +, ˆ >= 1, < ˆ, ˆ >= 0, (4) which tells us tht ˆ is null vector field, nd tht ˆ + hs norm 2h, which cn be either positive, negtive, or zero, depending on the sign of h. In this pper, we choose the sign 2h > 0 so tht v = constnt is spcelike hypersurfce. The metric on N 2 is τρ b, where τ is the re element of N 2 nd ρ b is the conforml two metric with det ρ b = 1. As ws shown in Ref.[13], the Einstein s equtions cn be obtined from the vritionl principle of the following ction integrl: S = dvdud 2 y{π τ τ + π h ḣ + π Ȧ+ + π b ρ b 1 C 0 C + A C }, (5) The Authors, published by EDP Sciences. This is n open ccess rticle distributed under the terms of the Cretive Commons Attribution License 4.0 (

2 where the overdot =, nd 1, 0, nd A re Lgrnge multipliers tht enforce the constrints C = 0, C + = 0, nd C = 0, which re given by (i) C := 1 2 π hπ τ h 4τ π2 h 2τ π hd + τ + 1 2τ 2 ρb π π b τ 8h ρb ρ cd (D + ρ c )(D + ρ bd ) 2hτ ρ bρ cd π c π bd 2h πc D + ρ c τr (2) + D + π h (τ 1 ρ b π b ) = 0, (6) (ii) C + := π τ D + τ + π h D + h + π b D + ρ b 2D + (hπ h + D + τ) + 2 (hτ 1 ρ b π b + ρ b b h) = 0, (7) (iii) C := π τ τ + π h h + π bc ρ bc 2 b (ρ c π bc ) D + π (τπ τ ) = 0, (8) respectively. Here, R (2) is the sclr curvture of N 2, nd the diffn 2 -covrint derivtive[13] of tensor density q b with weight s is defined s D + q b := + q b [A +, q] Lb = + q b A+ c c q b q cb A+ c q c b A+ c s( c A+ c )q b, (9) where [A +, q] Lb is the Lie derivtive of q b long A + := A+. For instnce, the diffn 2 -covrint derivtives of the re element τ nd the conforml metric ρ b, which re sclr density nd tensor density with weight 1 nd 1 with respect to the diffn 2 trnsformtions, respectively, re given by D ± τ = ± τ A± τ ( A± )τ, (10) D ± ρ b = ± ρ b A± c c ρ b ρ cb A± c ρ c b A± c +( c A± c )ρ b. (11) On the other hnd, h is sclr under the diffn 2 trnsformtions, whose diffn 2 -covrint derivtive is given by D ± h = ± h A ± h, (12) nd the diffn 2 -covrint field strength F + is defined s F+ := + A A+ [A +, A ] = + A A+ A+ b b A + A b b A+. (13) The diffn 2 -covrint derivtives of the conjugte moment π τ, π h, π, nd π b, which re tensor densities of weights 0, 1, 1 nd 2, respectively, re given by D ± π τ = ± π τ A± π τ, (14) D ± π h = ± π h A± c c π h ( c A± c )π h, (15) D ± π = ± π A± c c π π c A± c ( c A± c )π, (16) D ± π b = ± π b A± c c π b + π cb c A± + π c c A± b 2( c A± c )π b. (17) We note tht becuse ρ b hs unit determinnt, its derivtives stisfy the conditions ρ bc ± ρ bc = ρ bc ρ bc = ρ bc D ± ρ bc = 0, (18) nd the conjugte momentum π b is trceless, ρ b π b = 0. (19) 3 Hmiltonin reduction I: τ s physicl time Let us define potentil function R nd its conjugte momentum π R s + R := hπ h, (20) π R = + ln( h), (21) respectively. The trnsformtion from (h,π h ) to (R,π R ) is cnonicl trnsformtion, s it chnges the ction integrl by totl derivtives only. If we impose the constrints C + = 0 nd C = 0 nd choose the Lgrnge multiplier A = 0, then the ction in Eq. (5) becomes S = dvdud 2 y{π τ τ + π R Ṙ + π Ȧ+ + π b ρ b C } + totl derivtives, (22) where the constrint C in the new vribles is given by C = 2h π τ + R 4hτ ( +R) hτ (D +τ)( + R) h D +( + R) + 1 h ( +R) + ln( h) h ( +R)A + ln( h) τr (2) + 1 2τ 2 ρb π π b (τ 1 ρ b π b ) 2hτ ρ bρ cd π c π bd τ 8h ρb ρ cd (D + ρ c )(D + ρ bd ) 2h πc D + ρ c = 0. (23) The function h still ppers in Eq. (23), but s h is relted to R nd π R by Eqs. (20) nd (21), the constrint function C given by Eq. (23) my be viewed s function of the new vribles (τ, Q I ; π τ, Π I ), where Q I = (R, A+,ρ b ) nd Π I = (π R,π,π b ). Notice tht the first term in Eq. (23) is liner in π τ, nd tht ll the remining terms re independent of π τ. Thus, the eqution of motion for τ is given by τ = dud 2 y δc δπ τ = 2h +R. (24) Now, recll tht τ = τ(v, u,y ). If we solve this eqution for v, then v my be viewed s function of (τ, u,y ) nd consequently, (R, A+, ρ b ) my be regrded s functions of (τ, u,y ). Therefore, Ṙ = τ τ R, Ȧ + = τ τ A +, ρ b = τ τ ρ b, C = ( 2h + R ) τc, (25) becuse u/ v = y / v = 0. Then, the ction in Eq. (22) becomes S = dvdud 2 y τ{π τ + π R τ R + π τ A+ = = +π b τ ρ b + ( 2h + R )C } dτdud 2 y{π R τ R + π τ A+ + π b τ ρ b C (1) } dτdud 2 y{ Π I τ Q I C (1) }, (26) I 2

3 R p Y Y = constnt τ = constnt spcelike hypersurfce R = constnt equipotentil surfce N 2 Figure 1. On R = constnt surfce N 2 on, Y = constnt is chosen to be norml to N 2 t ech point p on. In this cse the shift" vector A + is zero t p. where we replced dv τ by dτ in the second line, nd C (1) is defined s C (1) = ( 2h + R )C π τ. (27) Notice tht if we impose the constrint C = 0, the function C (1) becomes C (1) = π τ. (28) 4 Hmiltonin reduction II: Fixtion of u = R nd y = Y such tht A + = 0 The second step in the Hmiltonin reduction consists of identifying rbitrrily specifible coordintes u nd y s u = R, (29) y = Y (30) such tht the shift" vector A + is zero, i.e., A + = 0. (31) For the clss of spcetimes whose sptil topology is either N 2 R or N 2 S 1, where N 2 is compct two-dimensionl spce with the genus g, the condition in Eq. (31) cn lwys be met by relbeling N 2 such tht Y = constnt is norml to u = constnt t ech point on. If we continue to lbel N 2 this wy, then the two dimensionl shift" A+ cn lwys be mde zero for the clss of spcetimes under considertion (see Fig. 1). Thus, we will work in the coordintes X A := (τ, R, Y ), which stisfy the coordinte conditions X A X B = δ B A. (32) Then, it follows from Eq. (24) tht τ = > 0, (33) 2h which mens tht τ increses monotoniclly long the outgoing null vector field. Hmilton s equtions of motion follow from the vritionl principle of the ction integrl in Eq. (26): τ Q I = dud 2 y δc (1) u=r,y δπ =Y, (34) I τ Π I = dud 2 y δc (1) δq I u=r,y =Y, (35) where Q I = (R, A+,ρ b ), Π I = (π R,π,π b ), nd is spcelike hypersurfce defined by τ = constnt. 5 Min results In the following, we present the min results of this pper. Einstein s evolution equtions I 1. R = 0 τr (2) = 1 2 τ 2 ρ b π π b (τ 1 ρ b π b ) (36) (topologicl censorship) 2. A + = 0 τ 1 π = ln( h) (superpotentil) (37) 3. = 1 (trivil) (38) 4. ln( h) = H (superpotentil) (39) τ 5. π = 2τ 1 π + (π bc + τ 2 ρbd ρ ce R ρ de ) ρ bc b (2π bc ρ c + τρ bc R ρ c ) (40) (evolution eqution of π ) 6. π τ = 1 2 τ 2 + τ 2 ρ b ρ cd π c π bd 4 ρb ρ cd ( R ρ c )( R ρ bd ) 2τ 2 (hρ b π b ) (41) (evolution eqution of π τ ) Einstein s constrint equtions 7. C = 0 π τ = H + 2 R ln( h) 2h{τR (2) 2τ 2 ρb π π b + Y (τ 1 ρ b π b )} (42) (definition of physicl Hmiltonin) 8. C + = 0 π R = π b R ρ b (43) (definition of physicl liner momentum) 9. C = 0 τ 1 π = π bc ρ bc + 2 b (π bc ρ c ) τ (H + π R ) (44) (definition of physicl ngulr momentum) Superpotentil ln( h) 10. τ ln( h) = H τ 1 (45) 11. R ln( h) = π R (46) 12. ln( h) = τ 1 π (47) Integrbility conditions 13. R (τ 1 π ) = π R (48) 14. τ π R = R H (49) 15. τ (τ 1 π ) = H. (50) In the bove equtions, H is defined by H = 1 τ ρ bρ cd π c π bd τρb ρ cd ( R ρ c )( R ρ bd ) +π c R ρ c + 1 2τ 1 2τ. (51) The evolution equtions of ρ b nd π b cn be found from the reduced ction principle S = drd 2 Y{π b τ ρ b C(1) }, (52) 3

4 where C (1) is the restriction of C (1) to the coordintes u = R nd y = Y : C(1) := C (1) u=r,y =Y = π τ = H 2 R ln( h). (53) Einstein s evolution equtions II 16. ρ b Στ = drd 2 Y δc (1) (54) δπ b 17. πb = Στ drd 2 Y δc (1) δρ b. (55) The spcetime metric in these privileged coordintes becomes ds 2 = 4hdRdτ 2hdR 2 + τρ b dy dy b. (56) 6 discussion Let us discuss some key fetures of the Hmiltonin reduction discussed in this rticle. (i) First of ll, let us mention tht the whole set of equtions summrized in Section 5 is identicl to the vcuum Einstein s equtions R AB = 0 (57) for spcetimes whose metric is given by Eq. (56). Thus, the whole procedure of the Hmiltonin reduction respects generl covrince, s it must, even though the finl theory is written in the privileged coordintes. (ii) The integrl of Eq. (36) over closed two surfce N 2 becomes N 2 d 2 Yτ 2 ρ b π π b = 16π(1 g) 0, (58) where g is the genus of N 2. This identity sttes tht, s long s the out-going null hypersurfce forms congruence of null geodesics which dmits cross section, the sptil topology of tht null hypersurfce is either two sphere or torus. This is remrkbly simple proof of topologicl censorship, s it does not rely on ssumptions such s globl hyperbolicity, symptotic conditions, energy conditions, nd so on, either inside or outside the null hypersurfce, which re normlly ssumed in the literture[15 20], lthough condition tht no custics exist on the null hypersurfce is necessry for the existence of well-defined cross sections of the null congruences. (iii) The sptil integrl of C(1) defined in Eq. (53) is the sought-for physicl Hmiltonin K of vcuum spcetimes, hh K := drd 2 YH 0, (59) which is positive-definite in the privileged coordintes. (iv) The logrithm of the conforml fctor in the (τ, R) subspce is the superpotentil ln( h), whose grdients yield (H τ 1, π R, τ 1 π ) through Eqs. (45), (46), nd (47), respectively. The superpotentil[6] is locl function of X A = (τ, R, Y ), nd is determined by the line integrl ln h(x) h(x 0 ) = X {(H τ 1 )dτ π R dr τ 1 π dy } (60) X 0 long ny contour from X 0 to X in given spcetime. (v) The integrbility conditions in Eqs. (48), (49), nd (50) re the consistency conditions, which follow from the definition of the superpotentil ln( h). (vi) We would like to emphsize tht ll the constrint equtions, C ± = 0 nd C = 0 in the privileged coordintes, re solved in such wy tht the constrints simply turn out to be the defining equtions of the physicl Hmiltonin density π τ, the liner momentum density π R, nd the ngulr momentum density τ 1 π, s is summrized in Eqs. (42), (43), nd (44), respectively. Nmely, π R nd H given by Eqs. (43) nd (51) re functions of (τ; ρ b,π b ), nd the superpotentil ln( h) given by Eq. (60) is function of H, π R, nd τ 1 π, which is gin determined by (τ; ρ b,π b ), up to n overll constnt fctor. Thus, ll the terms on the right-hnd sides of Eqs. (42), (43), nd (44) re functions of (τ; ρ b,π b ). This mens tht the constrints re completely solved for π τ, π R, nd τ 1 π in terms of the free, unconstrined dt (ρ b,π b ). (vii) If we define the totl liner momentum Π R nd totl ngulr momentum Π s Π R := drd 2 Yπ R, Π := drd 2 Yτ 1 π, (61) then we find, by integrting Eqs. (43) nd (44), Π R = drd 2 Yπ bc R ρ bc, (62) Σ τ Π = drd 2 Yπ bc Y ρ bc. (63) Eqs. (62) nd (63) show tht Π R nd Π re the generting functions of trnsltions of ρ b nd π b long R nd / Y. This justifies our interprettions of Π R nd Π s the totl liner nd ngulr momentum crried by the conforml two metric, respectively. Moreover, if no sptil boundries exist, then by virtue of the integrbility conditions in Eqs. (49) nd (50), Π R nd Π re conserved in the time τ. References [1] R. Schs, Proc. Roy. Soc. A 270, 103 (1962). [2] R. A. d Inverno nd J. Stchel, J. Mth. Phys. 19 (12), 2447 (1978). [3] R. Arnowitt, S. Deser, nd C. W. Misner, Phys. Rev. 117, 1595 (1960). [4] C. Rovelli nd L. Smolin, Phys. Rev. Lett. 72, 446 (1994). [5] J. D. Brown nd K. V. Kuchř, Phys. Rev. D 51, 5600 (1995). [6] K. V. Kuchř, Phys. Rev. D 4, 955 (1971). [7] V. Husin, Phys. Rev. D 50, 6207 (1994). 4

5 [8] J. D. Romno nd C. G. Torre, Phys. Rev. D 53, 5634 (1996). [9] Y. M. Cho, Q. H. Prk, K. S. Soh nd J. H. Yoon, Phys. Lett. B 286, 251 (1992). [10] J. H. Yoon, Phys. Lett. B 308, 240 (1993). [11] J. H. Yoon, Phys. Lett. B 451, 296 (1999). [12] J. H. Yoon, J. Koren Phys. Soc. 34, 108 (1999). [13] J. H. Yoon, Phys. Rev. D 70, (2004). [14] V. Husin nd T. Pwłowski, Phys. Rev. Lett. 108, (2012). [15] S. W. Hwking, Comm. Mth. Phys. 25, 152 (1972). [16] J. L. Friedmn, K. Schleich, nd D. M. Witt, Phys. Rev. Lett. 71, 1486 (1993). [17] P. T. Chruściel nd R. M. Wld, Clss. Quntum Grv. 11, L147 (1994). [18] T. Jcobson, nd S. Venktrmni, Clss. Qunt. Grv. 12, 1055 (1995). [19] G. J. Gllowy, Clss. Qunt. Grv. 13, 1471 (1996). [20] G. J. Gllowy, K. Schleich, D. M. Witt, nd E. Woolgr, Phys. Rev. D 60, (1999). 5

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which