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Immirzi Parameter and Noether Charges in First Order Gravity

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Immirzi Parameter and Noether Charges in First Order Gravity Immirzi parameter and Noether charges in first order gravity Remigiusz Durka Institute for Theoretical Physics, University of Wroc law, Pl. Maksa Borna 9, 50-204 Wroc law, Poland E-mail: [email protected] Abstract. The framework of SO(3, 2) constrained BF theory applied to gravity makes it possible to generalize formulas for gravitational diffeomorphic Noether charges (mass, angular momentum, and entropy). It extends Wald’s approach to the case of first order gravity with a negative cosmological constant, the Holst modification and the topological terms (Nieh-Yan, Euler, and Pontryagin). Topological invariants play essential role contributing to the boundary terms in the regularization scheme for the asymptotically AdS spacetimes, so that the differentiability of the action is automatically secured. Intriguingly, it turns out that the black hole thermodynamics does not depend on the Immirzi parameter for the AdS–Schwarzschild, AdS–Kerr, and topological black holes, whereas a nontrivial modification appears for the AdS–Taub–NUT spacetime, where it impacts not only the entropy, but also the total mass. 1. Black hole thermodynamics The discovery of the laws of black hole dynamics [1] has led to uncovering a remarkable analogy between gravity and thermodynamics. This is especially clear in the case of the first law κ dM = dA +ΩdJ (black hole dynamics) dE = T dS + dW (thermodynamics) , (1) 8πG arXiv:1111.0961v2 [gr-qc] 8 Nov 2011 where on the left we have black hole mass M, angular momentum J, angular velocity Ω, area of the event horizon A and surface gravity κ. The seminal works of Bekenstein [2, 3] and Hawking [4], relating the area of the event horizon with the entropy Area Entropy = , where l = G¯h/c3 , (2) 4l2 p p q and the surface gravity of a black hole with its temperature κ T emperature = ¯hc , (3) 2π have strongly suggested that this analogy might be in fact an identity. Yet, it still lacks better and deeper understanding [5, 6, 7]. Immirzi parameter and Noether charges in first order gravity 2 A major step in this direction has been taken by Robert Wald, who showed that gravitational quantities from (1) could be obtained as the Noether charges [8, 9]. The entropy can be calculated in various approaches to quantum gravity. In particular, within the Loop Quantum Gravity approach, it turns out that the key ingredient, the Immirzi parameter [10, 11], is explicitly present in the black hole entropy formula γ Area S = M , (4) LQG γ 4G where γM is a numerical parameter valued between 0.2 and 0.3. This result, coming from microscopic description and counting microstates, agrees with Bekenstein’s entropy only when γ is fixed to get rid off a numerical prefactor (for recent review see [12]). Such transition is poorly understood and must be further explored (see however [13]). This paper focuses on Wald’s procedure applied to first order gravity in the setting, where the Holst modification and the Immirzi parameter are present in the action we start with. The constrained SO(3, 2) BF theory allows us to check if Noether approach is able to reproduce the result obtained in the LQG framework. After deriving generalized formulas for the gravitational charges we will extend results of [25], and investigate a few AdS spacetimes (Schwarzschild, topological black holes, Kerr, Taub–NUT) discussing outcome in the context of the Immirzi parameter. 2. Wald’s approach and first order gravity Emmy Noether’s theorem concerning differentiable symmetries of the action of a physical system and the resulting conservation laws, holds well deserved place in theoretical physics. If one considers a variation of the action, then, except field equations (f.e.) multiplied by variated field, a boundary term Θ arises δ (ϕ, ∂ϕ) = (f.e.) δϕ + dΘ . L · µ For any diffeomorphism δξϕ = Lξϕ being generated by a smooth vector field ξ , we can derive a conserved Noether current J[ξ]=Θ[ϕ, Lξϕ] Iξ , − L where Lξ denotes the Lie derivative in the direction ξ and the contraction operator Iξ acting on 1 µ ν1 νp−1 a p-form is given by Iξαp = ξ α 1 p−1 dx ... dx . (p−1)! µν ...ν ∧ ∧ Noether current is closed on shell, which allows us to write it in the terms of the Noether charge 2-form Q by the relation J = dQ. In Wald’s approach one applies this construction to the Einstein-Hilbert Lagrangian and its diffeomorphism symmetry [8, 9]. The generators, being Killing vectors associated with the time translations (ξt = ∂t) and spacial rotations (ξϕ = ∂ϕ), by the integration at the infinity, produce Q[ξt]∞ and Q[ξϕ]∞, which are mass and angular momentum, respectively. Charge calculated for the horizon generator Q[ξt +Ωξϕ]H ensures the product of the horizon temperature and the entropy. Outcome of this formalism agrees with other methods in different frameworks, however such a procedure applied directly to the tetrad formulation of gravity, based on Einstein-Cartan action (often called the Palatini action) with a negative cosmological constant Λ = 3/ℓ2 − 1 ab c d 1 a b c d S = R e e + e e e e ǫabcd, (5) 32πG ∧ ∧ 2ℓ2 ∧ ∧ ∧ Z leads to serious problems. Noether charge evaluated for AdS–Schwarzschild metric gives wrong factor before M and also requires background subtraction to get rid off cosmological divergence 1 r3 Mass = Q[ξt]∞ = M + lim . (6) 2 r→∞ 2Gℓ2 Immirzi parameter and Noether charges in first order gravity 3 To deal with this apparent problem it was suggested by Aros, Contreras, Olea, Troncoso and Zanelli [14, 15] to use the Euler term as the boundary term ab c d 1 a b c d ab cd 32πGS = R e e ǫabcd + e e e e ǫabcd + ρ R R ǫabcd , (7) ∧ ∧ 2ℓ2 ∧ ∧ ∧ ∧ Z Z Z so for arbitrary weight ρ the formula (6) changes into M 2 r3 2 Mass = Q[∂t]∞ = 1+ ρ + lim 1 ρ . (8) 2 ℓ2 r→∞ 2Gℓ2 ℓ2 − ℓ2 It is easy to notice, that the fixed weight ρ = 2 cures the result. It simultaneously removes the divergence and corrects the factor before the mass value. Moreover, with the Euler term contributing to a boundary term, adding boundary condition of the AdS asymptotics at infinity 1 (Rab(ω)+ ea eb) = 0 (9) ℓ2 ∧ ∞ and fixing the connection δω = 0 on the horizon (in order to fix a constant temperature and fulfill the zeroth law) ensures differentiability of the action a ab δS ℓ2 = (f.e.)aδe + (f.e.)ab δω + dΘ = 0 , (10) (P alatini+Λ+ 2 Euler) M M M Z Z Z where (f.e.) are field (Einstein and torsion) equations and boundary term ab cd 1 c d Θ = ǫabcd δω R (ω)+ e e = 0. (11) ∂M ∧ ℓ2 ∧ ∂M 3. From MacDowell-Mansouri gravity to BF theory The formulation of gravity with the action 1 ℓ2 32πGS = Palatini + cosmological + Euler (12) MM 2ℓ2 2 Z Z Z was introduced already in the late 70’s by MacDowell and Mansouri [16]. This formulation of gravity based on broken symmetry of a gauge theory has many interesting features and intriguing philosophy behind it. It combines the so(3, 1)-spin connection ωab and the tetrad ea, so the two independent variables in first order gravity, into higher gauge so(3, 2)-connection AIJ through 1 Aab = ωab , Aa4 = ea . (13) µ µ µ ℓ µ Notice that the Lorentz part (with indices a, b = 0, 1, 2, 3) is embedded in the full symmetry group of the anti-de Sitter (for which I, J = 0, 1, 2, 3, 4). The AdS case sets the Minkowski metric to be ηIJ = diag( , +, +, +, ). To make dimensions right a length parameter needs to be introduced and later associated− with− a negative cosmological constant to recover the standard Palatini action Λ 1 = . (14) 3 −ℓ2 One can interpret (see [17]) the geometric construction of so(3, 2)-connection AIJ = (ωab, ea) as a way of encoding the geometry of the spacetime by parallel transport being ”rolling” the anti-de Sitter manifold along the . M M Immirzi parameter and Noether charges in first order gravity 4 The connection 1-form AIJ can be further used to build the curvature 2-form F IJ = dAIJ + AIK A J , (15) ∧ K which directly splits on torsion a4 1 a a b a a b 1 ω a ω a 1 a F = ∂µe + ωµ b e ∂ν e ων b e = D e D e = T , (16) µν ℓ ν ν − µ − µ ℓ µ ν − ν µ ℓ µν and so-called AdS curvature 1 F ab = Rab + ea eb ea eb , (17) µν µν ℓ2 µ ν − ν µ containing standard curvature ab ab ab a cb a cb R = ∂µω ∂ν ω + ω ω ω c ω . (18) µν ν − µ µc ν − ν µ The full curvature F IJ could not be used for constructing an action of the dynamical theory in four dimensions. However, one gets such a formulation with the use of the Hodge star and breaking the gauge symmetry by projecting full curvature down to Lorentz indices 1 F IJ FˆIJ = F ab where F ab = Rab + ea eb . (19) → ℓ2 ∧ Then General Relativity as a gauge symmetry breaking theory emerges from the action 2 2 ℓ ℓ ab 1 a b cd 1 c d SMM (A)= tr(Fˆ ⋆Fˆ)= R + e e R + e e ǫabcd . (20) 64πG ∧ 64πG ℓ2 ∧ ∧ ℓ2 ∧ Z Z The field equations 1 Rab ec + ea eb ec ǫ = 0 , T a = 0 , (21) 2ℓ2 abcd ∧ ∧ ∧ are unaffected by quadratic in curvatures Euler term, because its variation leads to DωRab, which vanishes due to the Bianchi identity.
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