Constrained BF theory as gravity

Transcription

1 Constrained BF theory as gravity (Remigiusz Durka) XXIX Max Born Symposium (June 2010) 1 / 23

2 Content of the talk 1 MacDowell-Mansouri gravity 2 BF theory reformulation 3 Supergravity 4 Canonical analysis 5 Gravitational Noether charges 2 / 23

3 variable: MacDowell-Mansouri gravity General Relativity g metric Hilbert-Einstein Action action: S 1 = d 4 x p g (R 2 ) 16G T = = 0; + variable: so(1; 3)-connection! ab and tetrad e a Palatini Action action: S = 1 64G d 4 x abcd (R abe ke d e ae be ce d ) 3 R ab ab T a = D! e ab +! a c! cb D! e a ; a cb! c! ; R a b = d! a b +! a c ^! c b T a = D! e a = de a +! a b ^e b 3 / 23

4 MacDowell-Mansouri gravity A IJ connection of the SO(2; 3) MacDowell and Mansouri proposal 8 >< A IJ! >: A ab =!ab A a5 = 1` e a with 3 = 1`2 ; a ; b = (1; 2; 3; 4) I ; J = (1; 2; 3; 4; 5) A = 1 2!ab M ab + 1` e a P a = 1 2 A IJ M IJ M a5 = P a For de Sitter group the geometric interpretation of this construction: SO(1; 4) connection A = (!; e) encodes the geometry of the spacetime M by "rolling de Sitter manifold along M" [Wise] 4 / 23

5 MacDowell-Mansouri gravity MacDowell-Mansouri 1977 Connection 1-form A IJ 0 A IJ ab 1 e a ` 1 ` e b 0 Curvature 2-form F IJ = da IJ + A IK ^ A J K 1 A Bianchi identity: F IJ = Rab + 1 `2 e a ^ e b D A F IJ = 0 1` T a 1 ` T b 0 1 A 5 / 23

6 MacDowell-Mansouri gravity MacDowell-Mansouri 1977 F IJ! ^F IJ = F ab General Relativity as gauge symmetry breaking theory S MM (A) = S MM (A) = `2 64G tr ^F ^? ^F `2 R ab + 1`2 64G e a ^ e b ^ R cd + 1`2 e c ^ e d abcd 32G S MM = Palatini 1 `2 + cosmological + Euler 2`2 2 4 R ab ^ e c 1 + 2`2 e a ^ e b ^ e c abcd = 0 ; T a = 0 Equations of motion : 6 / 23

7 BF theory reformulation Perturbed BF theory Introduce independent so(2; 3)-valued 2-form B to the action S = M tr B ^ F G 6 ^B ^ ^B B a5 : F a5 = 0; ^B ab : F ab = G 3 abcd B cd In this form Macdowell-Mansouri gravity has the appearance of the deformation of a topological gauge theory. The symmetry breaking occurs in the last term with dimensionless coecient proportional to: G In some sense general relativity is "not too far pertubatively" from a topological eld theory. 7 / 23

8 Ashtekar connection BF theory reformulation Holst action 1996 Phase space of gravity can be described by self and antiselfdual connections provided by projector (1 i?)! a i =! 0a i i 2 abc! i bc Real Barbero-Immirzi connection! a i =! 0a i + 2 0abc! i bc ; P i a = 4 G abc ijk e b j e c k Holst Action S = 1 32G f! a i (x ); P j b (y)g = (x y)j i a b abcd R ab ^ e c ^ e d G R ab ^ e a ^ e b 8 / 23

9 BF theory reformulation Starodubtsev and Freidel form connection A and independent so(2; 3)-valued 2-form B S = M tr B ^ F 2 B ^ B 4 ^B ^ ^B Action proposed by Starodubtsev and Freidel S = M B IJ ^ F IJ 2 B IJ ^ B IJ 4 abcd5b ab ^ B cd with constants (; `)! (G ; ; ): = ; = 3`2 ; where = G 3(1 + 2 ) ; = G 3(1 + 2 ) 9 / 23

10 S = BF theory reformulation Constrained BF theory d 4 x B IJ F IJ IJ BIJ B 2 IJ IJKL5B 4 B KL Structure standing behind the constrained BF model 32G S = R ab ^ e c ^ e d 1 abcd + e a ^ e b ^ e c ^ e d abcd 2`2 `2 + R ab ^ R cd 2 abcd + R ab ^ e a ^ e b 2 `2 R ab ^ R ab (T a ^ T a R ab ^ e a ^ e b) 32G S = Palatini + 1 `2 cosmological + 2`2 2 Euler 2 + Holst `2 Pontryagin Nieh/Yan 10 / 23

11 Supergravity Supergravity N = 1 Adding gravitino spin-3=2 eld A 1 = 2! ab M ab + 1` e a P a + Q Super-curvature F A i [A ; A 1 ] = F (s)ij M 2 IJ + F Q 11 / 23

12 Supergravity Supergravity N = 1 Supergravity action S 1 sugra = abcd R ab d e c e 64G 3 e a e a d e c e d 4 x G 2 5 a e a D i! 5 ab e a e b d 4 x 4` 12 / 23

13 Canonical analysis Canonical analysis S = dt d 3 x L BF ; H = p _q L L BF = P i IJ _ A IJ i + A IJ 0 D A i P i IJ + B 0i IJ 2 ijk F jk IJ P i IJ i KL 2 IJKL4 P fa IJ i (x ); P j KL g = IJ (x KL y)j i ; where P i IJ 2 ijk IJ B jk 13 / 23

14 Canonical analysis Constraints Complicated explicit constraints i = P i 4 ijk D! j e k 0 ` = P i i M F jk ijk 0 K e j e k 0 = 2`2 ijk D! i = 1` ijk K e R i jk 2 ( )`3 ijk e i e e j k 0 where M ( ( ) ), K ( ) ( 1 + ) 14 / 23

15 Canonical analysis Topological terms in the action 32 S topological = G NY G P 4 4G E 4 They all can be expressed as a total derivative! 32 S topological ( something ) d 4 x For constant time surfaces on a manifold R without spacial boundary (@ = 0) all terms with spacial derivatives drop out, which means 32 S topological W (e ;!) ; where W 4 = ijk e i a D! j e a 2 k + ( i )L CS ( +!)+(+i )L CS (!) `2 ( ) (functional of torsion and self and anti-self dual Chern-Simons forms L CS ) 15 / 23

16 Canonical analysis Canonical transformation Canonical transformation P _ Q K (Q; P; t ) + dw dt = [p _q H (q; p; t )] New momenta Pa i = P i a + fp i a ; W (!; e)g; P i ab = P i ab + fp i ab ; W (!; e)g Canonical analysis of constrained BF theory for manifold without boundary is equivalent to analysis of Holst action with shifted denition of the momenta 16 / 23

17 Gravitational Noether charges Black hole thermodynamics 1st law of thermodynamics 1st law of black hole dynamics de = TdS + dw dm = da + dj + dq 8G We identify the surface gravity of a black hole with temperature the area of the event horizon with the entropy Surface gravity =acceleration needed to keep an object at horizon. 2 = 1 2 r r (for example for Schwarzchild = Beckenstein and Hawking black hole entropy (l p = p G =c 3 ) Entropy Area = 4l 2 ; Temperature = p 2 c4 ). 4GM 17 / 23

18 Gravitational Noether charges R. Wald 1993 Emmy Noether's theorem Any dierentiable (smooth) symmetry of the action of a physical system has a corresponding conservation law. Killing vectors as generators of dieomorphism symmetry Gravitational Noether charges 8 8 Mass >< >< >: Q[ t ]1 Q[ ' ]1 Q[ t + ' ] H! >: Angular momentum Temperature Entropy 18 / 23

19 Gravitational Noether charges Euler term as boundary term 32G S = Palatini 1 + cosmological + Euler 2`2 4 Noether charge for Schwarzschild-AdS black hole Mass M = Q(@ t ) = 1 + 2`2 2 + lim r!1 r 3 1 2`2 4G `2 Factor should be equal to the factor from the MacDowell-Mansouri model! Adding boundary condition (AdS asymptotics) at the innity: (R ab 1 (!) e a ^ e b ) = 0 `2 1 provides dierentiability of the action! (it's due to =! ab ^ F ab ): (Palatini + + Euler ) = (f :e :) a e a + (f :e :) ab! ab + M M M d = 0 19 / 23

20 Gravitational Noether charges Generalized Noether charge Structure behind BF theory (Palatini + ) + `2 2 Euler + (Holst+Nieh/Yan) + `2 2 Pontryagin Rewritten form of our action with M ab cd = S BF (!; e) = 16 M 1 4 M abcd F ab ^ F cd ( ) ( ab cd ab ): cd 1 T a ^ T a `2 Noether charge from Wald's approach generalized to the case of rst order gravity with Immirzi parameter: Q[] = ( A IJ ) L 1 F IJ = (! ab )M abcdf cd Generalized Noether charge Q[] = `2 (! ab) ab 32G cdf cd jk 2F jk ab dx j ^ dx k 20 / 23

21 Gravitational Noether charges SchwarzschildAdS spacetime ds 2 = f (r ) 2 dt 2 + f (r ) 2 dr 2 + r 2 (d 2 + sin 2 d ' 2 ) f (r ) 2 = (1 2GM r + r 2 `2 ) There is no Immirzi parameter in the black hole thermodynamics Q[ t ]1 = M Q[ ']1 = 0 Q[ t + '] H = 4(r2 +`2) H 2 4G Mass Angular momentum Temperature Entropy Entropy shifted by a constant! Entropy = Area 4G + 4` 2 4G 21 / 23

22 Gravitational Noether charges Immirzi parameter Entropy calculated in LQG framework S log 2 Area LQG = p 3 4G Immirzi parameter might be present in black g t ' 0 Under investigation: AdSTaubNUT metric with NUT charge ds 2 = f (r ) 2 (dt + 4n sin 2 2 d ')2 + dr 2 f (r ) 2 + (n2 + r 2 )(r 2 d 2 + r 2 sin d ' 2 ) with f (r) 2 = r2 2GMr n 2 + (r 4 3n 4 + 6n 2 r 2 )` 2 n 2 + r 2 22 / 23

23 Bibliography S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38, 739 (1977) [Erratum-ibid. 38, 1376 (1977)]. L. Freidel and A. Starodubtsev, arxiv:hep-th/ R. Durka, J. Kowalski-Glikman and M. Szczachor, Phys. Rev. D 81, (2010) [arxiv: [hep-th]]. R. Durka and J. Kowalski-Glikman, Class. Quant. Grav. 27, (2010) [arxiv: [gr-qc]]. R. Durka and J. Kowalski-Glikman, Phys. Rev. D 83, (2011) [arxiv: [gr-qc]]. D. K. Wise, Class. Quant. Grav. 27, (2010) [arxiv:gr-qc/ ]. S. Holst, Phys. Rev. D 53, 5966 (1996) [arxiv:gr-qc/ ]. R. M. Wald, Phys. Rev. D 48, 3427 (1993) [arxiv:gr-qc/ ]. R. Aros, M. Contreras, R. Olea, R. Troncoso and J. anelli, Phys. Rev. D 62, (2000) [arxiv:hep-th/ ]. 23 / 23