Gravitational Noether charges and Immirzi parameter § ¤ ¦ ¥
Remigiusz Durka § ¤ ¦ ¥
6th Quantum Gravity Colloquium - Erlangen October 2011 § ¤ 1 / 19 N ¦ ¥ Content of the talk
1 Gravitational Noether charges
2 MacDowell-Mansouri gravity
3 Constrained SO(2,3) BF theory
4 Generalized thermodynamics and Immirzi parameter
Work is supported by the: -National Science Center grant N202 112740, -European Human Capital Program.
2 / 19 N Gravitational Noether charges Black hole thermodynamics
1st law of thermodynamics 1st law of black hole dynamics
dE = TdS + dW dM = dA + dJ 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 ¦ ¥ c4 = 2 r r (for example for Schwarzchild = 4GM ). p 3 Beckenstein and Hawking black hole entropy (lp = G~=c ) Area Entropy = ; Temperature = 2 4lp 2
3 / 19 N Gravitational Noether charges Noether currents and charges
Emmy Noether's theorem Any dierentiable (smooth) symmetry of the action of a physical system has a corresponding conservation law.
L('; @') = E ¡ ' + d¢
For any dieomorphism generated by a smooth vector eld , one can derive a conserved Noether current 3-form J by £
¢ J[] = ¢['; L'] IL ¡ where L denotes the Lie derivative in the direction and the contraction operator I acting on a p-form is I = 1 dx 1 ^ ::: ^ dx p 1 . p (p 1)! 1:::p 1 On shell the Noether current is closed, and can be written in terms of the Noether charge 2-form Q by the relation J = dQ. § ¤ 4 / 19 N ¦ ¥ Gravitational Noether charges R. Wald 1993 Killing vectors as generators of dieomorphism symmetry.
timelike @t , rotational @' and generator of the horizon (@t + @')
Gravitational Noether charges 8 8 Q[ ] Mass <> t 1 <> Q[']1 ! Angular momentum :> :> Q[t + ']H Temperature ¡ Entropy
However for tetrad formulation... Z Z 1 32GS = Palatini + cosmological £ 2`2 § ¤ ¢ ¡ Noether charge for Schwarzschild-AdS black¦ hole ¥ M r 3 Mass = Q(@t ) = + lim 2 r!1 2G`2 5 / 19 N Gravitational Noether charges Aros, Contreras, Olea, Zanelli 2000 Euler term as boundary term Z Z Z 1 32GS = Palatini + cosmological + Euler4 £ 2`2 § ¤ § ¤ ¢ ¡ ¦ ¥ ¦ ¥ Noether charge for Schwarzschild-AdS black hole M 2 r 3 2 Mass = Q(@t ) = 1 + + lim 1 2 `2 r!1 2G`2 `2
= `2 cures the result. Notice that this is MacDowell-Mansouri model! § 2 ¤ Z Z Z 1 `2 ¦ 32¥GS = Palatini + cosmological + Euler MM 2 £ 2` § ¤ 2 £ ¢ ¡ ¦ ¥ ¢ ¡ 6 / 19 N MacDowell-Mansouri gravity
IJ A connection of the SO(2; 3) MacDowell and Mansouri gravity
8 ab ab > A = ! a; b = (1; 2; 3; 4) < £ 1 AIJ ! with = ; > 1 3 `2 : Aa5 = ea I ; J = (1; 2; 3; 4; 5) `
1 1 1 = !ab M + ea P = A IJ M M = P A 2 ab ` a 2 IJ a5 a
Geometric interpretation of this construction:
SO(2; 3) connection A = (!; e) encodes the geometry of the spacetime M by "rolling anti-de Sitter manifold along M" [Wise]
7 / 19 N MacDowell-Mansouri gravity MacDowell-Mansouri 1977
Connection 1-form AIJ § ¤ 0 1 ¦ ¥ ab 1 a ! ` e AIJ = @ A 1 b ` e 0
Curvature 2-form F IJ = dAIJ + AIK ^ A J § K ¤ 0 1 ¦ ab 1 a b 1 ¥a R + `2 e ^ e ` T F IJ = @ A 1 b ` T 0
Bianchi identity: D AF IJ = 0 § ¤ 8 / 19 ¦N ¥ MacDowell-Mansouri gravity MacDowell-Mansouri 1977
F IJ ! F^ IJ = F ab F ab = Rab + 1 ea ^ eb § ¤ `2 ¦ ¥ General Relativity as gauge symmetry breaking theory § ¤ ¦ Z ¥ `2 ¡ S (A) = tr F^ ^ ?F^ MM 64G Z `2 1 1 S (A) = Rab + ea ^ eb ^ Rcd + ec ^ ed MM 64G `2 `2 abcd Z Z Z 1 `2 32GS = Palatini + cosmological + Euler MM 2`2 2 4 £ § ¤ § ¤ 1 Equations of motion¢ : R¡ab ^ ec + ea ^ eb ^ ec = 0 ; T a = 0 2`¦2 ¥abcd ¦ ¥
9 / 19 N MacDowell-Mansouri gravity Boundary conditions
Boundary term: Euler term Boundary condition: AdS asymptotics at the in nity
1 (Rab (!) + ea ^ eb ) = 0 `2 1
This provides dierentiability of the action!
Z Z Z a ab (Palatini + £ + Euler) = (f :e:)a e + (f :e:)ab ! + d¢ = 0 M M M where ab cd ¢ = abcd ! ^ F :
10 / 19 N Constrained SO(2,3) BF theory Perturbed BF theory
Introduce independent so(2; 3)-valued 2-form B to the action Z £ G£ 16 S(A; B) = tr B ^ F ¢ B^ ^¡ £B^ M 6
1 G£ B : F a5 = T a = 0; B^ : F ab = abcd B a5 ` ab 3 cd
SBF (A; B) = SMM (A)
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 coe cient proportional to: = G£=6 10 120 . § ¤ ¦ ¥ 11 / 19 N Constrained SO(2,3) BF theory Starodubtsev and Freidel 2005
1-form connection A and independent so(2; 3)-valued 2-form B £ £ Z ¢ ¡ ¢ ¡ 16 S = tr B ^ F B ^ B B^ ^ £B^ M 2 4
Action proposed by Starodubtsev and Freidel Z IJ IJ ab cd 16 S = BIJ ^ F BIJ ^ B abcd5B ^ B M 2 4 with constants ( ; `) ! (G; £; ):
3 G£ G£ = ; £ = ; where = ; = `2 3(1 + 2) 3(1 + 2)
12 / 19 N Constrained SO(2,3) BF theory Constrained BF theory
Z 1 S = d 4x B F IJ B B IJ B IJ B KL 64 IJ 2 IJ 4 IJKL5
Z Z 1 32GS = Rab ^ e c ^ e d + e a ^ e b ^ e c ^ e d abcd 2`2 abcd Z Z `2 2 + Rab ^ Rcd + Rab ^ e ^ e 2 abcd a b Z Z 2 + 1 `2 Rab ^ R + 2 (T a ^ T Rab ^ e ^ e ) ab a a b
Structure standing behind the constrained BF model § Z Z Z ¤ 1 `2 32¦GS = Palatini + cosmological + Euler¥ 2 Z £ Z 2` § ¤Z 2 £ 2 2 + 1 + Holst¢ `2 ¡ Pontryagin¦ + 2 ¥ Nieh/Yan¢ ¡ £ § ¤ § ¤ 13 / 19 ¢ ¡ ¦ ¥ N ¦ ¥ Generalized thermodynamics and Immirzi parameter Generalized Noether charge Action Z Z Z 1 `2 32GS = Palatini + cosmological + Euler 2 Z £ Z 2` § ¤Z 2 £ 2 2 + 1 + Holst¢ `2 ¡ Pontryagin¦ + 2 ¥ Nieh/Yan¢ ¡ £ § ¤ § ¤ ¢ ¡ with interesting scheme ¦ ¥ ¦ ¥
(Palatini + £) + `2 Euler (Holst+Nieh/Yan) + `2 Pontryagin § 2 ¤ § 2 ¤
More¦ useful form of the action¥ ¦ ¥ Z 1 1 1 S (!; e) = M abcd F ^ F T a ^ T BF ab cd 2 a 16 M 4 ` with use of
M ab = ( ab ab ) cd ( 2 + 2) cd cd 14 / 19 N Generalized thermodynamics and Immirzi parameter Generalized Noether charge
Noether charge from Wald's approach generalized to the case of rst order gravity with Immirzi parameter: Z Z L 1 Q[] = (AIJ ) = (!ab )M F cd IJ abcd @¦ F 32 @¦
Generalized Noether charge Z 2 ¡ ` ab cd ab j k Q[] = ( ! ab ) cd F jk 2 F jk dx ^ dx : 32G @¦
15 / 19 N Generalized thermodynamics and Immirzi parameter SchwarzschildAdS spacetime
ds2 = f (r)2dt 2 + f (r) 2dr 2 + r 2(d2 + sin2 d'2)
2GM r 2 f (r)2 = (1 + ) r `2
There is no Immirzi parameter in the black hole thermodynamics
Q[t ]1 = M Mass
Q[']1 = 0 Angular momentum 4(r 2 +`2) Q[ + ] = ¡ H Temperature ¡ Entropy t ' H 2 4G
Entropy shifted by a constant
Area 4`2 Entropy = + 4G 4G
16 / 19 N Generalized thermodynamics and Immirzi parameter Details of SchwarzschildAdS
ds2 = f (r)2dt 2 + f (r) 2dr 2 + r 2(d2 + sin2 d'2)
2GM r 2 f (r)2 = (1 + ) r `2 Z 4`2 ¡ Q[@ ] = !01 F F 34 dx j ^ dx k t 32G t jk 01 0134 jk Z 4`2 1 @f (r)2 r 2 Q[@ ] = 0134 1 f (r)2 + sin d ^ d' t 32G 2 @r `2 @¦
The charge calculated at the SchwarzschildAdS black hole horizon: Z `2 r 2 4(r 2 + `2) Q[] = 1 + H sin d ^ d' = H H 8G `2 2 4G @¦ The value of charge calculated at the boundary at in nity: Z 1 `2GM 2 Q[]1 = lim M + sin d ^ d' = M r!1 4 r 3 @¦ 17 / 19 N Generalized thermodynamics and Immirzi parameter Immirzi parameter Entropy calculated in LQG framework
ln 2 Area S = p LQG 3 4G
Immirzi parameter might be present in black hole thermodynamics Z @gt' , 0 @¦
Under investigation: AdSTaubNUT metric with NUT charge £ dr 2 ds2 = f (r)2(dt + 4n¢sin2 d')2 + + (n2 + r¡2)(r 2d2 + r 2 sin d'2) 2 f (r)2 r 2 2GMr n2 + (r 4 3n4 + 6n2r 2)` 2 where f (r)2 = n2 + r 2 18 / 19 N Generalized thermodynamics and Immirzi parameter Bibliography
S. W. MacDowell and F. Mansouri, Uni ed Geometric Theory Of Gravity And Supergravity, Phys. Rev. Lett. 38, 739 (1977) [Erratum-ibid. 38, 1376 (1977)].
L. Freidel and A. Starodubtsev, Quantum gravity in terms of topological observables, arXiv:hep-th/0501191.
D. K. Wise, MacDowell-Mansouri gravity and Cartan geometry, Class. Quant. Grav. 27, 155010 (2010) [arXiv:gr-qc/0611154].
R. Durka, J. Kowalski-Glikman and M. Szczachor, Supergravity as a constrained BF theory, Phys. Rev. D 81, 045022 (2010) [arXiv:0912.1095 [hep-th]].
R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, 3427 (1993) [arXiv:gr-qc/9307038].
R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Conserved charges for even dimensional asymptotically AdS gravity Phys. Rev. D 62, 044002 (2000) [arXiv:hep-th/9912045].
R. Durka and J. Kowalski-Glikman, Gravity as a constrained BF theory: Noether charges and Immirzi parameter, Phys. Rev. D 83, 124011 (2011)
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