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Local symmetries of non-expanding horizons

Local symmetries of a non-expanding horizon has been investigated in the 1st order formulation of gravity. When applied to a spherically symmetric isolated horizon only a U(1) subgroup of the Lorentz group survives as residual local symmetry that one can make use of in constructing an effective theory on the horizon.

In this note we explore local symmetries of a nonexpanding horizon (NEH) in the first order formulation of gravity. For a detailed definition of NEH see [1,2]. For our present purpose it is sufficient to characterize NEH to be a lightlike hypersurface ∆ imbedded in spacetime such that the unique (up to scaling by a function) lightlike, real vector field l tangential to ∆ is expansion, shear and twist-free. Since l is also normal to ∆, it is geodesic as well. These properties of ∆ are independent of the scaling of l [1,3]. Let us further assume that ∆ is topologically equivalent to S × R where S is a 2-sphere.
In the first order formulation Einstein's theory of gravity is invariant, apart from diffeomorphisms, under the local Lorentz group. Here our specific interest is primarily to find out the residual local symmetry of a NEH. Then based on the residual gauge group we wish to propose an effective theory on the horizon whose subsequent quantization would yield the quantum states of a black hole. There is a recent upsurge of interest in such effective theories, where an SU (2) Chern-Simons theory has been proposed [4] as the effective quantum theory on the horizon in contrast to a U (1) theory proposed earlier [5][6][7]. In the canonical formulation of loop quantum gravity one gauge fixes the full Lorentz group to its rotation subgroup SU (2) and the canonical theory reduces to a SU (2) gauge theory. This is the main reason of suspecting that a SU (2) gauge theory (expectedly a topological one) may play a role as the effective theory on the horizon in this case [8][9][10][11][12][13]. However, for a NEH our result goes in favour of the U (1) theory, as we exhibit below.
The Lorentz matrices associated with the transformations (1)-(3) are respectively and the corresponding generators are respectively where B, R generate (1) and (2) respectively and P, Q generate (3). A straightforward calculation gives their Lie brackets where [R, B] IJ = R IK B K J −B IK R K J and so on. This is the Lie algebra of ISO(2)⋉R where the symbol ⋉ stands for the semidirect product; R, P, Q generate ISO(2) and B generates R.
Clearly, the NEH boundary conditions are invariant only under a subgroup of the local Lorentz group. We should keep note of the fact that the group ISO(2) ⋉ R is non-semisimple; its Cartan-Killing metric K is doubly degenerate Let us consider the Palatini connection A IJ and in the interior of the spacetime let us expand A IJ in the internal Lorentz basis where W, V, N, U are connection 1-forms; as defined, W is real, V is imaginary and N, U are complex (in all, there are six of them associated with the six generators). For the rest of our analysis we will fix an internal Lorentz frame for which l I , n I , m I ,m I are constants. However, our results will be unaffected by such a choice. The pull-back of the Palatini connection to the NEH ∆ is of the form where W, V, U are respectively the pull-backs of W, V, U. Clearly, the 1-form N , which is the pull-back of N, vanishes on ∆ by the NEH boundary conditions. Proof: The simplest way to show this is to relate the connection 1forms to the Newman-Penrose coefficients (the constant l I , n I , m I ,m I basis simplifies these relations): So only four independent connection 1-forms W, V, U survive on ∆. This is consistent with our earlier result that the residual gauge group on ∆ is ISO(2) ⋉ R that has only four generators. However, below we present an independent analysis for the connection to prove this.
Under the local Lorentz transformations (1)-(4) the Palatini connection (18) transform as where Λ IJ are the associated Lorentz matrices (9)- (11) for (1) A lengthy but straightforward calculation shows that under the Lorentz transformations (9)-(11) the connection 1-forms transform as Since N transforms homogeneously, its pull-back N 0 in one frame implies that it vanishes in all Lorentz frames related by (9)- (11). However, under (25), the connection 1-forms transform as As a result, dV and dU are proportional to the 2-sphere area 2-form and dW = 0. However, since Y depends on b, one can ask is there any b for which dY 0? The answer is explicitly verifiable and one easily finds that dY 0 if and only if b = 0. Since Y is not closed, acting d once more on the equation db = bY one gets which yields the unique solution b = 0. This shows that the connection (19) is indeed an ISO(2) ⋉ R connection.
Here we wish to remark that one could also arrive at (26)-(29) directly using the relations (20)-(23) and the appropriate Lorentz transformations of the Newman-Penrose coefficients [14].
It is to be noted that unlike the Palatini connection, the Hölst connection H IJ := A IJ − 1 2 γ B ǫ IJ KL A Kl , where γ B is the Barbero-Immirzi parameter, does not transform as a connection under any of the local Lorentz transformations (1)- (4).
For later convenience we expand (19) in the basis (12)-(15) of the Lie algebra iso(2) ⋉ R: The connection 1-forms A B , A R , A P , A Q will turn out to be more useful in the context of an effective theory on the horizon.
Let us now turn our attention to the symplectic structures. The Hölst action [15] gives rise to the symplectic current 3-form (in units of 4πGγ B = 1 and E I is the spacetime tetrad 1-form) Its pull-back to ∆ gives the boundary symplectic current. For simplicity, we take ∆ for the rest of our analysis to be a spherically symmetric isolated horizon. A straightforward calculation gives the pull-back current J where 2 ǫ is the area 2-form of some spherical cross-section of ∆. In the derivation of the symplectic current it is sufficient to assume that the spherical cross-section foliates ∆ and is not necessarily a geometric 2-sphere. However, for the rest of our analysis we will restrict ourselves to the unique foliation of ∆ in which each leaf is a geometric 2-sphere; this is possible if and only if the isolated horizon ∆ is spherically symmetric. For such a horizon with a fixed area A = 2 ǫ the 1-form W is closed and dV is proportional to 2 ǫ [16,17] dW 0, dV 4πi where d is the exterior derivative intrinsic to ∆. Using (35) we find that the symplectic current 3-form is exact on ∆ J(δ 1 , δ 2 ) dj(δ 1 , δ 2 ) where j(δ 1 , δ 2 ) = − A 8π δ 1 (iV + γ B W ) ∧ δ 2 (iV + γ B W ). (36) It is to be noted that in the iso(2) ⋉ R basis the 1-form iV + γ B W = −2(A R − γ B A B ). This gives a boundary symplectic structure (putting back 4πGγ B = 1) where S is the unique spherical cross-section of ∆ and The form (37) suggests that on a spherically symmetric isolated horizon one can take the effective boundary theory as a U (1) Chern-Simons theory. Two distinct cases of U (1) arise: i) If either the pull-back of A B vanishes on S [18] or one restricts the gauge freedom (1) to a constant class (ξ = constant, as has been the original choice [6]) then one gets a compact U (1), ii) In general, if no restrictions are imposed, then one gets a noncompact U (1).