Hongguang Liu
email: liu.hongguang@cpt.univ-mrs.fr

Xin Wang

Haida Li

Yongge Ma

Distinguishing f (R) theories from general relativity by gravitational lensing effect

Keywords: f (R) theory, post-Newtonion approximation, gravitational lens PACS numbers 04.50.Kd, 04.25.Nx, 98.62.Sb

The post-Newtonian formulation of a general class of f (R) theories is set up to 3 rd order approximation. It turns out that the information of a specific form of f (R) gravity is encoded in the Yukawa potential, which is contained in the perturbative expansion of the metric components. Although the Yukawa potential is canceled in the 2 nd order expression of the effective refraction index of light, detailed analysis shows that the difference of the lensing effect between the f (R) gravity and general relativity does appear at the 3 rd order when f ′′ (0)/f ′ (0) is larger than the distance d 0 to the gravitational source. However, the difference between these two kinds of theories will disappear in the axially symmetric spacetime region. Therefore only in very rare case the f (R) theories are distinguishable from general relativity by gravitational lensing effect at the 3 rd order post-Newtonian approximation.

I. INTRODUCTION

Recently modified gravity theories have received increasingly attention in issues related to "dark energy" [1][2][3], "dark matter" [4][5][6][7], as well as non-trivial tests on gravity beyond general relativity (GR) [START_REF] Will | Was Einstein Right? A Centenary Assessment[END_REF]. Historically, Einstein's GR is the simplest relativistic theory of gravity with correct Newtonian limit. To pursue new physics, Weyl and Eddington even began to consider modifying GR just after it was established [START_REF] Weyl | [END_REF][START_REF] Eddington | The mathematical theory of relativity[END_REF]. From the viewpoints of perturbutive quantum gravity, GR is non-renormalizable [START_REF] Utiyama | [END_REF][12][START_REF] Buchbinder | Effective action in quantum gravity[END_REF][START_REF] Vilkovisky | [END_REF], while higher order gravity theories might alleviate the problem. From the phenomenological viewpoints, there are many ways to modify GR, and some empirical approaches seem to have promising prospect, such as Dvali-Gabadadze-Porrati gravity [15], tensorvector-scalar theory [START_REF] Fujii | The scalar-tensor theory of gravitation[END_REF] and Einstein-Aether theory [START_REF] Jacobson | [END_REF]. Among such extended theories, particular attention has been devoted to the so-called f (R)-gravity. This kind of theories is based on a generalization of the Einstein Hilbert Lagrangian to nonlinear functions f (R) of the Ricci scalar [18]. f (R)-gravity covers a lot of characteristics of higher order gravity and is convenient to be operated. Hence, f (R) theories provide an ideal tool to study the possible extension of GR. f (R) theories of gravity can also be non-perturbatively quantized by loop quantum gravity approach [19,20].

To confront f (R)-gravity with observations in Solar System, one can get constraints on the theories from different measurements, such as the EötWash experiment [21], the geodesic precession of gyroscopes measured by Gravity Probe B [22] and the precession of the binary pulsars PSR J0737-3039 [23]. At cosmological scales one would expect to employ f (R) theories to account for the problems of "dark energy" [24][25][26] and "dark matter" [27][28][29] needed in GR. If f (R) gravity could account for dark matter, besides matching the rotation curves of galaxy clusters, it should also match the measurements on gravitational lensing effect [30]. However, it is shown in [31] that, at 2 nd order post-Newtonian approximation, a rather general class of f (R) theories is indistinguishable from GR in gravitational lensing effect. Nevertheless, we will show in this paper that a class of f (R) theories is indeed distinguishable from GR in gravitational lensing effect at 3 rd order post-Newtonian approximation. However, the possibility to account for the dark matter problem with f (R) theory in lensing effect is highly suppressed due to this tiny 3 rd order difference.

This paper is organized as follows. In section II, we briefly review the field equations of metric f (R)-gravity. In section III the post-Newtonian approximation of a class of f (R) theories is formulated to the desired order. In section IV we introduce the gravitational lensing effect in metric theories of gravity and show how the f (R) gravity can be distinguishable from GR at 3 rd -order post-Newtonian approximation. The difference of the lensing reflection indexes is discussed in an example. Finally, conclusions and remarks are given in Sec. V. Throughout the paper, the metric tensor g µν takes the signature (-, +, +, +).

II. FIELD EQUATIONS OF f (R) THEORY

In metric f (R) theories of gravity, the action of gravity coupled to matter fields is given by

S = 1 2χ f (R) √ -gd 4 x + S M , (2.1)
where g is the determinant of the metric tensor g µν , χ = 8πG/c 4 with G and c being the Newtonian gravitational constant and the vacuum speed of light respectively, R = g µν R µν is the Ricci scalar, f (R) is a nonlinear function and S M is the standard matter action. The variation of action (2.1) with respect to the metric g µν yields the Euler-Lagrange equations

f ′ (R)R µν - 1 2 f (R)g µν -∇ µ ∇ ν f ′ (R) + g µν g f ′ (R) = χT µν , (2.2)
where ∇ µ is the covariant derivative for g µν , g := ∇ µ ∇ µ , and T µν = (-2c/ √ -g)(δS M /δg µν ) is the energy momentum tensor of matter. Taking the trace of Eq. (2.2) we can get

3 g f ′ (R) + f ′ (R)R -2f (R) = χT , (2.3)
where T is the trace of T µν . Using Eq. ( 2.3), we can rewrite Eq. (2.2) as

R µν = 1 f ′ (R) ( 1 3 g µν f ′ (R)R - 1 6 f (R)g µν + ∇ µ ∇ ν f ′ (R) + χ(T µν - 1 3 g µν T )) .
(2.4)

III. POST-NEWTONIAN EXPANSION

The matter constituents in the universe are usually well approximated by a perfect fluid with mass density ρ and pressure p [START_REF] Weinberg | Gravitation and cosmology: Principle and applications of general theory of relativity[END_REF]. Hence we assume that the Newtonian potential U of the mass distribution, the typical velocities v and the pressure of the fluid obey such approximation respectively. In the post-Newtonian approximation, we can further expand the dynamical variables in the field equations perturbatively in powers of 1/c, since we have the the following order relation [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF][START_REF] Clifton | [END_REF][35]

U c 2 ∼ v 2 c 2 ∼ p ρc 2 ∼ Π ∼ O(2) , (3.1)
where Π is the ratio of the energy density to the rest-mass density. We consider the case that the gravitational field is weak and assume that in absence of a gravitational field the background space-time is flat [30]. We also assume that f (0) = 0, which neglects the contribution of a possible cosmological constant and excludes some form of f (R) theories, e.g, f (R) = 1/R. Note that actually the contribution of a possible cosmological constant can be equivalently substituted by the corresponding contribution of a energy-momentum tensor. Moreover, the f (R) form which is unable to get weak field solution is useless here. In a weak field regime the metric tensor can be expanded about the Minkowski metric η µν in its Lorentzian coordinate system as

g µν = η µν + h µν , (3.2)
where |h µν | ≪ 1. Up to 3 rd order the components of the metric tensor can be written as [31,[START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]36]:

g 00 = -1+ (2) h 00 + O(4) , g 0i = (3) h 0i + O(5) , g ij =δ ij + (2) h ij + O(4) , (3.3)
where the left upper index (n) means the order O(n). Using Eq. (3.3) we can get the components of the Ricci tensor as 2) h 00,ij - (2) h kk,ij + (2) h ik,kj + (2) h kj,ki ) + O(4) .

R 00 = - 1 2 ∇ 2(2) h 00 + O(4) , R 0i = 1 2 (-∇ 2(3) h 0i - 1 c (2) h jj,0i + (3) h j0,ij + 1 c (2) h ij,0j ) + O(5) , R ij = 1 2 (-∇ 2(2) h ij + (
(

Assuming f (R) to be analytic at R = 0, to the 2 nd order the Ricci scalar and thus f (R) and f ′ (R) read respectively as

R = (2) R + O(4) , f (R) = f ′ (0) (2) R + O(4) , f ′ (R) = f ′ (0) + f ′′ (0) (2) R + O(4) . (3.5)
To the leading order the components of the energy-momentum tensor of matter fields read

χT 00 = χ (-2) T 00 + O(4) , χT 0i = χ (-1) T 0i + O(4) , χT ij = O(4) . (3.6) Note that if f ′′ (0) = 0, Eq. (3.5) implies that f ′ (R) is constant. Then Eq. (2.
3) yields at the 2 nd order (2) R = -χ (2) T 00 /f ′ (0), which is consistent with the equation of GR at the same order. Thus in this approach, GR is nothing else but the first term of the Taylor expansion of a more general f (R) theory. As one can see from the action (2.1), we ask f (R) to carry the same dimension as R's.

Thus both f ′ (R) and the term f ′′ (0) (2) R in Eq. (3.5) are dimensionless. Since the term f ′′ (0) (2) R is required to be of order O(2), the expansion of Eq. (3.5) will break down if (2) R≥f ′ (0)/f ′′ (0) [31].

To derive neat equations in the post-Newtonian approximation, we impose the gauge conditions [31,36]

g ij,j - 1 2 (g jj -g 00 ) ,i - f ′ (R) ,i f ′ (R) = O(4) , (3.7)
g 0j,j - 1 2c g jj,0 - f ′ (R) ,0 cf ′ (R) = O(5) . (3.8)
With the gauge conditions, we get from Eqs. (2.3) and (2.4) 2) h 00,0i .

∇ 2(2) R = - 1 3f ′′ (0) χ (-2) T 00 + f ′ (0) 3f ′′ (0) (2) R , (3.9)
∇ 2(2) h 00 = - 4χ 3f ′ (0) (-2) T 00 + 1 3 (2) R , (3.10)
∇ 2(2) h ij = -( 2χ 3f ′ (0) (-2) T 00 + 1 3 (2) R)δ ij , (3.11)
∇ 2(3) h 0i = 2χ f ′ (0) (-1) T 0i - 1 2c (
(3.12)

For the sake of physics and simplicity, we consider the case of f ′ (0) > 0 and f ′′ (0) > 0 and define α 2 := f ′ (0)/(3f ′′ (0)). Note that in this case the constant 1/f ′ (0) can be absorbed into the gravitational constant G if necessary. Then from Eq. (3.9) we can get [START_REF] Clifton | [END_REF]36] (

2) R = 2α 2 c 2 V (x, t) , (3.13)
with the Yukawa potential

V (x, t) := G c 2 (-2) T 00 (x ′ , t)e -α|x-x ′ | |x -x ′ | d 3 x ′ . (3.14)
Note that the information of a specific form of f (R) gravity is encoded in the parameter α in the potential V , and we only consider the solution with α > 0. It should be noticed that, for the other solution with α < 0, the potential V would tend to be divergent at infinity. It is easy to show by using Eq. (3.9) that Eq. (3.10) can be written as

∇ 2(2) h 00 = - χ f ′ (0) (-2) T 00 + f ′′ (0) f ′ (0) ∇ 2(2) R . (3.15)
Thus we get

∇ 2 ( (2) h 00 - 1 3α 2 (2) R) = - χ f ′ (0) (-2) T 00 . (3.16)
Using Eq. (3.13), the solution of Eq. (3.16) can be given by (2)

h 00 (x, t) = 2 c 2 (U (x, t) + 1 3 V (x, t)) , (3.17)
where the Newtonian potential U reads

U (x, t) := G c 2 (-2) T 00 (x ′ , t) |x -x ′ | d 3 x ′ . (3.18)
It is obvious that the Newtonian potential U remains unchanged for different forms of f (R) gravity.

Similarly, the solution of Eq. (3.11) reads (2)

h ij (x, t) = 2δ ij c 2 (U (x, t) - 1 3 V (x, t)) . (3.19)
From Eqs. (3.10) and (3.11), we have

∇ 2 ( (2) h 00 -2 (2) h ii ) = - 4χ 3f ′ (0) (-2) T 00 + 1 3 (2) R + 2( 2χ 3f ′ (0) (-2) T 00 + 1 3 (2) R) = (2) R . (3.20)
Using Eqs. (3.17) and (3.19), it is easy to get

∇ 2 ( (2) h 00 -2 (2) h ii ) = 2 c 2 ∇ 2 (V -U ) . (3.21)
Thus from Eq. (3.13) we have

V = 1 α 2 ∇ 2 (V -U ) . (3.22)
Then Eq.(3.17) can be written as

(2) h 00 (x, t) = 1 c 2 ∇ 2 (-ψ + 2 3α 2 (V -U )) , (3.23)
where the potential ψ is defined as

ψ(x, t) := - G c 2 (-2) T 00 (x ′ , t)|x -x ′ |d 3 x ′ , (3.24)
such that ∇ 2 ψ = -2U . Hence Eq. (3.12) can be written as

∇ 2 ( (3) h 0i + 1 2c 3 (-ψ + 2(V -U ) 3α 2 ) ,0i ) = 2χ f ′ (0) (-1) T 0i . (3.25)
The solution of Eq. (3.25) reads

(3) h 0i (x, t) = 1 c 3 (-4Y i (x, t) + 1 2 ψ(x, t) ,0i + Z(x, t) ,0i ) , (3.26)
where

Y i (x, t) := G c (-1) T 0i (x ′ , t) |x -x ′ | d 3 x ′ , Z(x, t) := 1 3α 2 (U (x, t) -V (x, t)) .
(3.27)

So up to 3 rd order post-Newtonian approximation, the final form of the metric components reads

g 00 = -1 + 2 c 2 U + 2 3c 2 V , g 0i = 1 c 3 (-4Y i + 1 2 ψ ,0i + Z ,0i ) , g ij = (1 + 2 c 2 U - 2 3c 2 V )δ ij .
(3.28)

In contrast, the metric components to the same order approximation in GR reads [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]

g 00 = -1 + 2 c 2 U , g 0i = 1 c 3 (-4Y i + 1 2 ψ ,0i ) , g ij = (1 + 2 c 2 U )δ ij . (3.29)
Hence the difference between the f (R) gravity and GR comes from the Yukawa-like potential V and Z ,0i . In the limit f ′′ (0) → 0, we get α → ∞ and V → 0. Then the solution (3.28) of f (R) gravity goes back to (3.29) of GR. On the other hand, it is straightforward to see that, in the limit f ′′ (0) → ∞, we have α → 0 and hence get the most obvious departure of f (R) gravity from GR.

IV. GRAVITATIONAL LENSING

A gravitational lens refers to a distribution of plates (such as a cluster of galaxies) between a distant source (a background galaxy) and an observer, that is capable of bending the light from the source, as it travels towards the observer. The lensing effect can magnify and distort the image of the background source [START_REF] Schneider | Gravitational lensing statistics[END_REF]. According to Fermat's principle, the world line of a light should extremize its arrival time T with respect to an observer under the variation of γ. In metric theories of gravity, this principle implies that the world line of a light coincides with a null geodesic in the spacetime. In the Lorentzian coordinate system of the flat background spacetime, let dl 2 = δ ij dx i dx j be the spatial Euclidean line element. Up to a constant, the travel time of light on a null geodesic γ is given by

T = γ dt = γ dt dl dl = 1 c γ n dl , (4.1)
where we defined the effective refraction index of light as

n := c dt dl . (4.2)
Then Eq.(4.1) takes the form similar to that of the propagation of a light through a medium in Newtonian space and time.

A. 2 nd Order Expansion

At the 2 nd order post-Newtonian approximation, the only nonzero perturbative metric components are h 00 and h ij in Eqs. (3.17) and (3.19). For a null geodesic, we have [31,[START_REF] Schneider | Gravitational Lensing: Strong, Weak and Micro: Saas-Fee Advanced Course[END_REF][START_REF] Blanchet | [END_REF]]

ds 2 = -(1 -(2) h 00 )c 2 dt 2 + (δ ij + (2) h ij )dx i dx j = 0 . (4.3)
Hence we can get

c 2 dt 2 = δ ij + (2) h ij 1-(2) h 00 dx i dx j = 1 + (2) h 1 -(2) h 00 dl 2 , (4.4)
where h is defined such that Eq. (3.19) can be written as (2) h ij = (2) hδ ij . Using Eq. (4.2), we can obtain the effective index of refraction as

n = c dt dl = 1 + (2)h 1 -(2)h 00 , (4.5)
which at the 2 nd order reads

n = 1 + 1 2 ( (2) h 00 + (2) h) . (4.6)
By Eqs. (3.17) and (3.19), it equals to

n = 1 + 2 c 2 U (x, t) , (4.7)
which illustrates that at 2 nd order, the effective refraction index n is only determined by the Newtonian potential U . As shown in Eqs. (3.28) and (3.29), the difference between f (R) gravity and GR comes from the potential V rather than U . Hence one can not distinguish f (R) theories from GR at 2 nd order approximation by the gravitational lensing effect [31].

B. 3 rd Order Expansion

We now consider the 3 rd order post-Newtonian approximation which is needed in dealing with light rays in spacetime [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]. At the 3 rd order expansion, the line element of the metric can be written as

ds 2 = -(1 -(2) h 00 )c 2 dt 2 + (δ ij + (2) h ij )dx i dx j + 2 (3) h 0i cdtdx i . (4.8)
For a null geodesic, by using Eq. ( 4.2) we have

0 = -(1 -(2) h 00 )n 2 + (δ ij + (2) h ij ) dx i dx j dl 2 + 2 (3) h 0i dx i dl n . (4.9)
Thus we obtain 2) h 00 , (2) h, (3)

n = - (3) h 0i -1 + (2) h 00 dx i dl ± F ( (
h 0i ) , (4.10)
where

F := ( (3) h 0i dx i ) 2 (1 -(2) h 00 ) 2 dl 2 - (1 + (2) h)(dx i 2 ) (-1 + (2) h 00 )dl 2 = ( (3) h 0i dx i ) 2 (1 -(2) h 00 ) 2 dl 2 + 1 + (2) h 1 -(2) h 00 . (4.11)
At 3 rd order approximation, Eq. (4.11) can be expressed as 2) h 00 , (2) h, (3)

F ( (
h 0i ) =F | 0,0,0 + ∂F ∂ (2) h 00 | (0,0,0) (2) h 00 + ∂F ∂ (2) h | (0,0,0) (2) h + ∂F ∂ (3) h 0i | (0,0,0) (3) h 0i + (4) E( (2) h 00 , (2) h) + O(5) =1 + 1 2 (2) h 00 + 1 2 (2) h + O(4) , (4.12)
where (4) E( (2) h 00 , (2) h) represents the expansion terms at fourth order. It is obvious that the "-" sign in front of the function F in Eq.(4.10) should be neglected, since otherwise the refraction index n would become negative. Hence Eq. (4.10) becomes

n = - (3) h 0i dx i dl -1 + (2) h 00 + (1 + 1 2 (2) h 00 + 1 2 (2) h) = (3) h 0i dx i dt n c + (1 + 1 2 (2) h 00 + 1 2 (2) h) . (4.13)
It is easy to see that Eq. (4.13) can be solved as

n = 1 + 1 2 (2) h 00 + 1 2 (2) h + 1 c (3) h 0i dx i dt = n 2 + 1 c (3) h 0i u i , (4.14)
where u i = dx i /dt is the components of the coordinate speed of light, and n 2 := 1 + 2U (x, t)/c 2 is the refraction index at 2 nd order. Therefore, at 3 rd order post-Newtonian approximation, the effective refraction index of light is obviously dependent on the 3 rd order metric components h 0i . From Eqs. (3.27), (3.28) and (3.29) one can see that, in contrast to the case of GR, in f (R) gravity h 0i is effected also by the Yukawa potential V . Hence, f (R) theories are in principle distinguishable from GR by gravitational lensing effect at 3 rd order post-Newtonian approximation.

C. Differences: An example

Although the difference of lensing effect between f (R) gravity and GR is encoded in the 3 rd order terms, it is still unclear whether the difference can actually be detected at this order and in which case the departure become most obvious. To answer these questions, we first recall from Eq.(4.14) that the difference at 3 rd order effect is contained in the difference of the metric components h 0j between f (R) gravity and GR, which reads

∆n := u j c ∆n j = u j c 4 Z ,0j u j := 1 3α 2 c 4 (U (x, t) -V (x, t)) ,0j . (4.15)
A straight-forward calculation leads to

∆n j = G c 5 (-2) T 00 (x ′ , t)v ′ j |x -x ′ | g(α|x -x ′ |)d 3 x ′ - G c 5 (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j |x -x ′ | 3 h(α|x -x ′ |)d 3 x ′ , (4.16)
where

g(α|x -x ′ |) := 1 3α 2 |x -x ′ | 2 1 -e -α|x-x ′ | (1 + α|x -x ′ |) , h(α|x -x ′ |) := 1 α 2 |x -x ′ | 2 1 -e -α|x-x ′ | 1 + α|x -x ′ | + 1 3 α 2 |x -x ′ | 2 (4.17)
are two monotone decreasing functions. Hence the lensing refraction indexes of f (R) gravity and GR will take the biggest departure in the limit of α → 0, which reads

∆n j ∼ G 6c 5 (-2) T 00 (x ′ , t)v ′ j |x -x ′ | d 3 x ′ - G 6c 5 (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j |x -x ′ | 3 d 3 x ′ = 1 6c 3 ψ ,0i . (4.18)
One may noticed that this is nothing else but the potential ψ ,0i appearing in the 3 rd order post Newtonian approximation of GR. Thus, in the case of the most departure, the difference of the lensing refraction indexes is at the same 3 rd of GR. By noticing that the functions g and h satisfy the relation

h(αd) d 3 = - 1 d ∂[g(αd)/d] ∂d , (4.19)
where d = |xx ′ |, the expression (4.16) can be further simplified. In terms of cylindrical coordinates {r, θ, z}, Eq. (4.16) can be written as

∆n j = G c 5 r ′ dr ′ dz ′ (-2) T 00 (x ′ , t)v ′ j |x -x ′ | g(α|x -x ′ |)dθ ′ + G c 5 r ′ dr ′ dz ′ (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j rr ′ sin(θ ′ ) d dθ ′ g(α|x -x ′ |) |x -x ′ | dθ ′ = r ′ dr ′ dz ′ (-2) T 00 (x ′ , t)v ′ j - d dθ ′ (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j rr ′ sin(θ ′ ) g(α|x -x ′ |) |x -x ′ | dθ ′ , (4.20)
where we used the identity

d θ ′ |x-x ′ | = d θ ′ r 2 + r ′2 -2rr ′ cos(θ ′ ) + (z -z ′ ) 2 = rr ′ sin(θ ′ )/|x-x ′ | and θ = 0.
In an axially symmetric spacetime, it is reasonable to consider the case that the velocities v of the gravitational sources are all tangent to the r -θ plane. Then from Eq.(4.20) one gets

∆n r = r ′ dr ′ dz ′ -(-2) T 00 (x ′ , t)v ′ sin(θ ′ ) + d dθ ′ (-2) T 00 (x ′ , t)(v ′ sin(θ ′ )r)(r -r ′ cos(θ ′ )) rr ′ sin(θ) g(α|x -x ′ |) |x -x ′ | dθ ′ =0 , ∆n θ = r ′ dr ′ dz ′ (-2) T 00 (x ′ , t)v ′ cos(θ ′ ) + d dθ ′ (-2) T 00 (x ′ , t)(v ′ sin(θ ′ )r)(-r ′ sin(θ ′ )) rr ′ sin(θ ′ ) g(α|x -x ′ |) |x -x ′ | dθ ′ =0 . (4.21)
Therefore, in an axially symmetric spacetime region, which coincides in most cases with those of galaxies and compact objects, one can not distinguish f (R) theories from GR by the lensing correction to the 3 rd order term. This result suggests that there is few opportunities to distinguish f (R) theory from GR even at 3 rd order post Newtonian approximation. However the difference ∆n will not vanish in a non-axially-symmetrical spacetime region, thus one could detect the difference in principle. Since the potential appearing in the expression (4.20) without function g are 3 rd post Newtonian terms, the order of ∆n is determined by the order of the functions g(αd) (or g(αd) and h(αd), with the same order as shown in Fig. 1. It is shown that, for α ≤ 1 d , ∆n will be around 10 -1 times the 3 rd GR terms, and thus keeps the same 3 rd order. However for α ≥ 10 d , ∆n will be less than 10 -2 times the 3 rd order terms, and hence is indistinguishable with the 4 th order term. This estimation would approach the exact result for the spacetime region far away from the matter center. Then the functions g and h could be approximated by the values g(αd 0 ) and h(αd 0 ), where d 0 is the distance of the position to the matter center. Thus one could write the first-order approximation of (4.16) as

∆n j ≃g(αd 0 ) G c 5 c (-2) T 00 (x ′ , t)v ′ j |x -x ′ | d 3 x ′ -h(αd 0 ) G c 5 c (-2) T 00 v ′ • (x -x ′ )(x -x ′ ) j |x -x ′ | 3 d 3 x ′ , (4.22)
in which the whole integration is approximated by the integration of the region around the center where most part of the matter locate. Therefore, in highly non-axially symmetric spacetime region, it is possible to distinguish the lensing of GR from those of the f (R) theories satisfying α ≤ 1 d 0 . It also requires that the measurement can approach the 10 -1 precision of the 3 rd order effect.

V. CONCLUDING REMARKS

In this paper, the post-Newtonian approximation of a general class of f (R) theories is formulated up to 3 rd order. At the 3 rd order expansion, the metric components contain not only the Newtonian potential U but also the Yukawa potential V together with the third order potentials. Note that f (R) theories can be transformed into generalized Brans-Dicke theories by suitable conformal transformations. Since the post-Newtonian formulation of Brans-Dicke gravity has been well studied [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF], one can check the consistency of the post-Newtonian formulations between the two kinds of theories. It turns out that, in the limit of α → 0, our result (3.28) of f (R) gravity coincides with the result of Brans-Dicke gravity given in [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF]. The proof will be presented in Appendix A.

In our post-Newtonian formulation, the information of a specific form of f (R) theories is contained in the Yukawa potential. While the Yukawa potential does not show in the 2 nd order expression of the effective refraction index n of light, it does appear in the 3 rd order expression of n. Therefore in principle we could distinguish f (R) gravity and GR. Moreover, detailed analysis shows that a series of f (R) forms, more specific, whose parameter 1/α ∼ f ′′ (0)/f ′ (0) is larger than the distance to the massive center, are distinguishable from GR by the gravitational lensing effect at the 3 rd order post-Newtonian approximation. It should be noted that the conclusion that f (R) theories can lead to the gravitational lensing effect different from that of GR can also be obtained by the approach of Minkowski functionals [START_REF] Chenxiaoji | Distinguishing GR and f(R) gravity with the gravitational lensing Minkowski Functionals[END_REF]. However, it is shown in this paper that, in the axially symmetrical spacetime region, the gap term between these two kinds of theories vanishes and hence they are indistinguishable at 3 rd order.

One of the motivations for developing modified gravity theories is to account for the observed mass profiles in galaxies as well as clusters of galaxies without the inclusion of dark matter. The existence of dark matter in GR is confirmed by the observational data not only from the dynamical analysis, such as rotation curves in spiral galaxies [START_REF] Sofue | [END_REF] and velocity dispersions in early-type systems [42,43], but also from gravitational lensing observations [44,45]. Observations indicate that we need to take into account almost the same large amount of dark matter to explain the gravitational lensing effect as that for the dynamical data like the velocity dispersion or the temperature profile of the X-ray emitting intracluster medium [46,47] in galaxy clusters or spiral galaxies. Up to now, certain f (R) theories are tested by the dynamical data in galaxy clusters and spiral galaxies [27][28][29][48][49][50]. However, concerning the gravitational lensing observations, our results here shows a disfavor of the attempts in this direction. For any f (R) form which could be weakly expanded, the lensing effect correction due to the f (R) from will be at most the 3 rd order, which is at most 10 -2 times of the leading order, i.e., the 2 nd order post Newtonian effect. Moreover, the fact that in axially symmetrical spacetime region there is no difference in lensing effect between these two kinds of theories strongly indicates that most of the lensing observations will not show the difference even at 3 rd order. Thus it is impossible to explain the lensing observations in the pure f (R) theories that we are considering without any dark matter involved.

It is still possible to determine the parameter α 2 := f ′ (0)/(3f ′′ (0)) though the precise observational results in non-axially symmetric system. Thus in near future, precise observations of lensing effect would be useful to distinguish certain f (R) theories from GR. It should be remarked that our result is only valid for the f (R) forms which could be weakly expanded. It is interesting to further study whether the dark matter content can be replaced by other unexpandable f (R) theories or other kinds of modified gravity.

From (3.14) and (3.18), we have

U (x, t) ,0j := G c 2 (-2) T 00 (x ′ , t)v ′ j |x -x ′ | 3 d 3 x ′ - 3G c 2 (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j |x -x ′ | 5 d 3 x ′ , (A4) V (x, t) ,0j = G c 2 (-2) T 00 (x ′ , t)e -α|x-x ′ | v ′ j (1 + α|x -x ′ |) |x -x ′ | 3 d 3 x ′ - 3G c 2 (-2) T 00 (x ′ , t)e -α|x-x ′ | v ′ • (x -x ′ )(x -x ′ ) j (1 + α|x -x ′ |) |x -x ′ | 5 d 3 x ′ (A5) - G c 2 (-2) T 00 (x ′ , t)e -α|x-x ′ | α 2 v ′ • (x -x ′ )(x -x ′ ) j |x -x ′ | 3 d 3 x ′ ,
where v ′ j donates the j components of the velocity at point x ′ . Thus one get

Z 0j := G c 2 (-2) T 00 (x ′ , t)v ′ j |x -x ′ | 3 1 3α 2 (1 -e -α|x-x ′ | (1 + α|x -x ′ |)) + (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j α 2 |x -x ′ | 5 e -α|x-x ′ | 1 + α|x -x ′ | + 1 3 α 2 |x -x ′ | 2 -1 d 3 x ′ .
(A6)

Then we are going to see the limit of α → 0. It is directly to see, lim α→0 V = U . Thus the limit of g 00 and g ij could be read out directly:

g 00 = -1 + 2 U c 2 4 3 , g ij = (1 + 2 U c 2 2 3 )δ ij . (A7)
And one could show that, the limit of α → 0 leads to lim (A8)

Thus for potential Z 0i we simply get

Z 0j := G c 2 (-2) T 00 (x ′ , t)v ′ j |x -x ′ | d 3 x ′ - G c 2 (-2) T 00 (x ′ , t)v ′ • (x -x ′ )(x -x ′ ) j |x -x ′ | 3 d 3 x ′ = 1 6 (Y j -W j ) . (A9)
Since from the standard PPN formalism in GR, we already know that

g GR 0i := 1 c 3 (-4Y j + 1/2ψ ,0i ) = 1 c 3 (- 7 2 Y j - 1 2 W j ) , (A10)
we have

g 0i := 1 c 3 (- 10 3 Y j - 2 3 W j ) . (A11)
After the redefinition of gravitational constant G ′ = 4 3 G based on g 00 , we get

g 00 = -1 + 2 c 2 U , g 0i = 1 c 3 (- 5 2 Y j - 1 2 W j ) , g ij = (1 + 1 c 2 U )δ ij .
(A12)

FIG. 1 :

 1 FIG.1:The evaluation of the functions g(αd) and h(αd) respect to αd

α→0 1 3α 2 ( 1 -, lim α→0 1 α 2 e 3 α 2

 211232 e -α|x-x ′ | ) = |xx ′ | 2 6 -α|x-x ′ | 1 + α|xx ′ | + 1 |xx ′ | 2 = -|xx ′ | 2 6 .

Acknowledgments

This work is supported in part by the NSFC (grant nos.11235003 and 11475023), the Research Fund for the Doctoral Program of Higher Education of China, and the National Undergraduate Training Programs for Innovation of China. and one directly read out γ = 0.5 and α 1 = α 2 = ζ 1 = ζ = 0 respectively, which meet the result of this kind of f (R) form given by C. M. Will [33].

Appendix A: PPN formalism

We will derive explicitly the 3 rd order terms and get its limit of the largest effect in this appendix. Especially, we could get the limit of α → 0 and compare it to the result with the standard parametrized post Newtonian (PPN) form.

First we will write out explicitly 3 rd order g ij term, more precisely, the Z 0i term containing in the g ij expression. The other two terms Y i and ψ ,0i is the usual potential in post Newtonian form which reads

where W i reads


Christian Röken
email: christian.roeken@mathematik.uni-regensburg.de

F Ashtekar

Ashtekar-Barbero Formulations

Kerr Isolated Horizons in Ashtekar and Ashtekar-Barbero Connection Variables

Keywords:

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

In general relativity, a charged, rotating black hole in equilibrium is usually described in terms of the future event horizon of the Kerr-Newman solution of the vacuum Einstein field equations despite the fact that the canonical definition of a future event horizon is a non-local concept, i.e., it refers to future null infinity and requires the existence of a global time-translational Killing vector field for static spacetimes and an asymptotic time-translational Killing vector field at space-like infinity for stationary spacetimes. This concept is overly restrictive because it suffices to consider a horizon structure with a local time-translational Killing vector field, that is, with a time-independent intrinsic geometry and the possibility for a dynamical bulk geometry. Moreover, since the information on the entire spacetime history is needed when one refers to an event horizon, it is in general not well-suited for describing black hole evolution. A more appropriate notion, however, is given by the so-called quasi-local isolated horizons, which account for equilibrium states of black holes and cover all essential local features of event horizons (for reviews on the subject see, e.g., [START_REF] Ashtekar | Isolated and dynamical horizons and their applications[END_REF][START_REF] Diaz-Polo | Isolated horizons and black hole entropy in loop quantum gravity[END_REF]). Isolated horizons employ only local time-translational Killing vector fields and require neither asymptotic structures nor foliations of spacetime. Consequently, they allow for the occurrence and for the dynamics of matter and radiation distributions in the exterior spacetime. In the following, we recall the definition of isolated horizons.

An isolated horizon ∆ is a 3-dimensional submanifold of a 4-dimensional globally hyperbolic, asymptotically flat spacetime (M, g) that is equipped with an equivalence class

[X] ∼ ∶= {X ∈ T M X = c X ′ , c ∈ R >0 } (1)
of normal, future-directed null vector fields and satisfies the conditions:

(i) ∆ is a null 3-surface with topology S 2 × R, which admits a suitable foliation by a family of 2-spheres.

(ii) The outgoing expansion rate θ (X) = q µν ∇ µ X ν of the vector fields X ∈ [X] ∼ , where q is the induced metric on the horizon and ∇ is the covariant derivative on T M, vanishes on ∆.

(iii) All field equations hold on ∆.

(iv) The stress-energy tensor T on ∆ satisfies the dominant energy condition.

(v) The induced derivative operator D on ∆, which is defined by the horizon pullback of ∇ and the vector fields

Y , Z ∈ T ∆ via Y µ D µ Z ν ∶= Y µ ← ∇ µ Z ν , satisfies -L X , D X = 0 ∀ X ∈ [X] ∼ (weakly isolated horizon condition) -L X , D Y = 0 ∀ Y ∈ T ∆ (isolated horizon condition) ,
where the bracket [⋅ , ⋅] denotes the commutator and L X the Lie derivative along the null vector field X.

Using the Raychaudhuri equation and the dominant energy condition, one can prove that

L X q µν = 0 . (2)
This equation shows that the null normal X is a local time-translational Killing vector field on the horizon. Furthermore, it expresses the time independence of the induced metric. The time independence of the induced derivative operator is directly given by the isolated horizon condition in (v). Thus, the intrinsic isolated horizon geometry can be captured by the pair (q, D). A Carter-Penrose diagram of a spacetime containing an isolated horizon is shown in Figure 1. Isolated horizons can be classified into three different types, namely spherically symmetric horizons, axisymmetric horizons, and general, distorted horizons (for more details, in particular on their symmetry groups, see [START_REF] Ashtekar | Mechanics of rotating isolated horizons[END_REF][START_REF] Lewandowski | Symmetric non-expanding horizons[END_REF]). The basic spherically symmetric type has already been analyzed extensively in both the classical and quantum regimes (see, e.g., [2-5, 7, 15, 27]), whereas studies concerning the more intricate axisymmetric and distorted types [START_REF] Ashtekar | Quantum horizons and black-hole entropy: inclusion of distortion and rotation[END_REF][START_REF] Lewandowski | Geometric characterizations of the Kerr isolated horizon[END_REF][START_REF] Perez | Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy[END_REF] are up to now still incomplete. In the present article, we provide a well-behaved set of connection variables for Kerr isolated horizons, i.e., we explicitly compute expressions for the Ashtekar and Ashtekar-Barbero connections and curvatures with a focus on regularity at the event horizon region. Furthermore, we show that the horizon 2-sphere pullback of the Ashtekar-Barbero curvature is not proportional to that of the Plebanski 2-form, which is in contrast to the associated horizon boundary condition of the Schwarzschild isolated horizon case. As such a condition relates the degrees of freedom of the bulk geometry to the isolated horizon surface degrees of freedom, making it the key constraint in the classical symplectic theory, and because it is essential for an SU(2) Chern-Simons treatment of the gauge degrees of freedom in the horizon boundary in the context of loop quantum gravity, it is of great interest to have it satisfied also for Kerr isolated horizons. Hence, we present a non-standard method for the construction of a new connection whose curvature fulfills this particular form of the horizon boundary condition for Kerr isolated horizons. In more detail, we are concerned with the rotating Kerrtype of the isolated horizon models, which represents one specific symmetric rotator amongst the many rotating isolated horizon geometries. Nonetheless, it is of particular interest since, in the astrophysical context, it accounts for supermassive spinning black holes and the associated accretion disks in active galactic nuclei in good approximation. Besides, recent observations in astrophysics [START_REF] Mcclintock | The spin of the near-extreme Kerr black hole GRS 1915+105[END_REF] indicate that most of the black holes actually appear to rotate quite rapidly around an internal symmetry axis and, thus, probably constitute the most common form existing in nature. The general idea behind Kerr isolated horizons is the following. Every Killing horizon that has the topology S 2 × R is an isolated horizon [START_REF] Ashtekar | Isolated and dynamical horizons and their applications[END_REF]. This implies that in particular the event horizon of Kerr geometry is an isolated horizon. However, working with such a horizon leads to the aforementioned problems, namely that one usually requires an asymptotic time-translational Killing vector field at spacelike infinity and refers to future null infinity. Thus, we relax the first condition by using only a local time-translational Killing vector field at the horizon, given in terms of the equivalence class of local null normal vector fields [START_REF] Ashtekar | New variables for classical and quantum gravity[END_REF]. The latter condition is dropped completely, as we only consider quantities that are intrinsic to the horizon. The starting point of our analysis is Kerr geometry represented by a regular Kinnersley tetrad in horizon-penetrating Kruskal-Szekeres-like coordinates, which covers the exterior as well as the interior black hole region, up to the Cauchy horizon. Solving the first Maurer-Cartan equation of structure in this frame, we determine the spin coefficients. Both the tetrad and the spin coefficients are then adapted to the isolated horizon conditions using local Lorentz transformations. With the resulting quantities, we compute the spin connection on the horizon 2-spheres. This spin connection is subsequently transformed into an orthonormal frame that is fixed in the time gauge. Next, we explicitly calculate the Ashtekar and Ashtekar-Barbero connections on the horizon 2-spheres, which are defined by linear combinations of the components of the spin connection, and the corresponding curvatures. It turns out, as already noted above, that the resulting Ashtekar-Barbero curvature does not comply with the simple form of the horizon boundary condition of the Schwarzschild isolated horizon case, that is, it is not proportional to the Plebanski 2-form projected onto the horizon 2-spheres. Therefore, we introduce a method for the construction of a new connection whose curvature satisfies a horizon boundary condition of the desired form. For illustrative purposes, this method is used in the framework of slowly rotating Kerr isolated horizons, i.e., in the first-order formalism for small angular momenta.

II. PRELIMINARIES

Kerr geometry is described by a connected, orientable and time-orientable, smooth, asymptotically flat Lorentzian 4-manifold (M, g) with topology S 2 × R 2 , in which the metric g is stationary and axisymmetric. The tangent and cotangent bundles T M and T ⋆ M are endowed with the respective bases (e µ ) and (e µ ), µ ∈ {0, 1, 2, 3}, and with the Levi-Civita connection ω. Choosing coordinates (x µ ) = (t = x 0 , x 1 , x 2 , ϕ = x 3 ), where t is the coordinate time and ϕ the azimuthal angle about the axis of symmetry, for which the Killing vectors are given by ∂ t and ∂ ϕ and all metric coefficients g µν = g µν (x 1 , x 2 ) are independent of t and ϕ, the metric becomes

g = g tt dt ⊗ dt + g x 1 x 1 dx 1 ⊗ dx 1 + g x 1 x 2 dx 1 ⊗ dx 2 + dx 2 ⊗ dx 1 + g x 2 x 2 dx 2 ⊗ dx 2 + g tϕ dt ⊗ dϕ + dϕ ⊗ dt + g ϕϕ dϕ ⊗ dϕ .
Note that in order to obtain this particular representation, one additionally requires the metric to be invariant under simultaneous discrete time and azimuthal angle isometries t ↦ -t and ϕ ↦ -ϕ. In the present study, we employ horizon-penetrating Kruskal-Szekeres-like coordinates (Appendix B) and express Kerr geometry in terms of a regular Kinnersley tetrad (Appendix C), i.e., a local Newman-Penrose null frame that is, on the one hand, adapted to the class of principal null geodesics and, on the other hand, regular at the event horizon. (For more details on the Newman-Penrose formalism see Appendix A.) Since Kerr geometry is algebraically special and of Petrov type D, using the Kinnersley frame has the computational advantage that one has four vanishing spin coefficients (κ = σ = λ = ν = 0) and only one non-vanishing Weyl scalar (Ψ 2 = 0). Moreover, one of the elements of this frame coincides with the isolated horizon generator (cf. Appendix D).

III. SPIN CONNECTION FOR KERR ISOLATED HORIZONS

We begin by imposing the isolated horizon conditions on the regular Kinnersley tetrad (C3) (Appendix D). Then, using the corresponding spin coefficients (D4), we compute the pullback of the spin connection to the isolated horizon 2-spheres and apply a transformation into an orthonormal frame that is fixed in the time gauge (Appendix E). The resulting spin connection reads

⇐ ω (0)(1) = a sin (θ) ρ 2 + ⎛ ⎝ a cos (θ) dθ + ρ 2 0+ sin (θ) α + + r + ρ 2 + d φ+ ⎞ ⎠ ⇐ ω (0)(2) = ⇐ ω (1)(2) = - 1 √ 2 ρ 4 + ⎛ ⎝ r 2 + ρ 2 0+ - a 4 sin 2 (2θ) 4 dθ + a sin (2θ)ρ 2 0+ 2 r + -a 2 sin 2 (θ) α + + 2r + ρ 2 + d φ+ ⎞ ⎠ ⇐ ω (0)(3) = ⇐ ω (1)(3) = 1 √ 2 ρ 4 + ⎛ ⎝ r + a cos (θ) ρ 2 0+ + a 2 sin 2 (θ) dθ -ρ 2 0+ sin (θ) r 2 + -a 2 sin 2 (θ) r + α + + r 2 + -a 2 cos 2 (θ) ρ 2 + d φ+ ⎞ ⎠ ⇐ ω (2)(3) = 1 ρ 2 + -r + a sin (θ) dθ + ρ 4 0+ cos (θ) ρ 2 + d φ+ , (3)
where θ ∈ [0, π] and φ+ ∈ [0, 2π) are angular coordinates covering the horizon 2-spheres, aM is the angular momentum and M the mass of the Kerr black hole, r ± ∶= M ± √ M 2 -a 2 denote the event and the Cauchy horizon, respectively, ρ 2

+ = ρ 2 + (θ) ∶= r 2 + + a 2 cos 2 (θ), ρ 2 0+ ∶= r 2 + + a 2 , and α + ∶= (r + -r -) (2ρ 2 0+
). These quantities are the basis for the subsequent analysis of the Ashtekar and Ashtekar-Barbero connection variable formulations of Kerr isolated horizons.

IV. ASHTEKAR AND ASHTEKAR-BARBERO CONNECTIONS AND CURVATURES

In this section, we explicitly calculate the Ashtekar and Ashtekar-Barbero connections and curvatures on the horizon 2-spheres (for a brief summary of the Ashtekar and Ashtekar-Barbero formulations see Appendix F). The self-dual Ashtekar connection A + and the associated curvature F + are defined as

A (i) + ∶= 1 2 (i) (j)(k) ω (j)(k) + i ω (0)(i) (4)
and

F (i) + ∶= F (i) (A + ) = dA (i) + + 1 2 (i) (j)(k) A (j) + ∧ A (k) + (5)
with i, j, k ∈ {1, 2, 3}. Substituting the spin connection 1-forms (3) into (4) and employing the convention (1)(2)(3) = 1, the Ashtekar connection on the horizon 2-spheres becomes

⇐ A (1) + = - a sin (θ) ς + dθ + ρ 2 0+ ρ 2 + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ cos (θ) -ia sin 2 (θ) α + + 1 ς + ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d φ+ ⇐ A (2) + = -i ⇐ A (3)
+ = 1 √ 2 ς + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ r + + a 2 sin 2 (θ) ς + i dθ + ρ 2 0+ sin (θ) ρ 2 + r + -a 2 sin 2 (θ) α + + 1 ς + d φ+ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (6)
where ς + = ς + (θ) ∶= r + -ia cos (θ). Applying ( 6) in ( 5), we immediately obtain the corresponding Ashtekar curvature ⇐ F

(1)

+ = 2 ρ 2 0+ sin (θ) Ψ 2 r=r+ dθ ∧ d φ+ ⇐ F (2) + = -i ⇐ F (3) + = - 3ia sin 2 (θ) ρ 2 0+ √ 2 ς + Ψ 2 r=r+ dθ ∧ d φ+ (7)
with

Ψ 2 r=r+ = - M ρ 6 + r + r 2 + -3a 2 cos 2 (θ) + ia cos (θ) 3r 2 + -a 2 cos 2 (θ)
being the second Weyl scalar evaluated at the event horizon r = r + . In terms of the self-dual projection of the Plebanski 2-form Σ ∶= e ∧ e, which in the time gauge is given by

Σ (i) = 1 2 (i) (j)(k) e (j) ∧ e (k) ,
the Ashtekar curvature (7) yields

⇐ F (1)
+ = 2 Ψ 2 r=r+ ⇐ Σ (1)
⇐ F

(2) 1) ,

+ = -i ⇐ F (3) + = - 3ia sin (θ) √ 2 ς + Ψ 2 r=r+ ⇐ Σ (
where

⇐ Σ (1) = ρ 2 0+ sin (θ) dθ ∧ d φ+ and ⇐ Σ (2) = ⇐ Σ (3) = 0 .
Next, computing the horizon 2-sphere pullback of the Ashtekar-Barbero connection

A (i) γ ∶= 1 2 (i) (j)(k) ω (j)(k) + γ ω (0)(i) , γ ∈ C {0} , we find ⇐ A (1) γ = - a sin (θ) ρ 2 + r + + γa cos (θ) dθ + ρ 2 0+ ρ 2 + ⎛ ⎝ ρ 2 0+ ρ 2 + cos (θ) -γa sin 2 (θ) α + + r + ρ 2 + ⎞ ⎠ d φ+ ⇐ A (2) γ = 1 √ 2 ρ 4 + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ r + ρ 2 0+ γr + -a cos (θ) -a 3 sin 2 (θ) cos (θ) r + + γa cos (θ) dθ + ρ 2 0+ sin (θ) ⎛ ⎝ r + -a 2 sin 2 (θ) α + + r + ρ 2 + r + + γa cos (θ) - a 3 sin 2 (θ) cos (θ) ρ 2 + γr + -a cos (θ) ⎞ ⎠ d φ+ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⇐ A (3) γ = 1 √ 2 ρ 4 + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -r + ρ 2 0+ r + + γa cos (θ) + a 3 sin 2 (θ) cos (θ) -γr + + a cos (θ) dθ + ρ 2 0+ sin (θ) ⎛ ⎝ r + -a 2 sin 2 (θ) α + + r + ρ 2 + γr + -a cos (θ) + a 3 sin 2 (θ) cos (θ) ρ 2 + r + + γa cos (θ) ⎞ ⎠ d φ+ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . ( 8
)
The associated Ashtekar-Barbero curvature

F (i) γ ∶= F (i) (A γ ) = dA (i) γ + 1 2 (i) (j)(k) A (j) γ ∧ A (k) γ reads ⇐ F (1) γ = 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Re Ψ 2 r=r+ + γ Im Ψ 2 r=r+ + 1 + γ 2 4ρ 6 + r 4 + -a 4 cos 4 (θ) + a 2 ρ 2 0+ 2 3 cos 2 (θ) -1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⇐ Σ (1) ⇐ F (2) γ = 3a sin (θ) √ 2 ρ 2 + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ -γr + + a cos (θ) Re Ψ 2 r=r+ + r + + γa cos (θ) Im Ψ 2 r=r+ + a cos (θ) 1 + γ 2 6ρ 4 + 5r 2 + -a 2 cos (2θ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⇐ Σ (1) ⇐ F (3) γ = 3a sin (θ) √ 2 ρ 2 + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ r + + γa cos (θ) Re Ψ 2 r=r+ + γr + -a cos (θ) Im Ψ 2 r=r+ + r + 1 + γ 2 6ρ 4 + 3r 2 + -a 2 1 + 2 cos (2θ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⇐ Σ (1) . (9)

V. SU(2) CHERN-SIMONS FORM OF THE HORIZON BOUNDARY CONDITION

The pullback of the Ashtekar-Barbero curvature to the horizon 2-spheres ( 9) is of the form

⇐ F (i) γ = f (i) (θ) ⇐ Σ (1) , (10)
where

f (i) (θ) ∶ [0, π] ↦ C, i ∈ {1, 2, 3}
, are disparate complex-valued functions. This relation differs from the one of the Schwarzschild isolated horizon case for which the Ashtekar-Barbero curvature and the Plebanski 2-form, both projected onto the horizon 2-spheres, are proportional

⇐ F (i) γ = α 0 ⇐ Σ (i) , α 0 = const. ( 11
)
As the latter equation is the key constraint in the classical symplectic theory of Schwarzschild isolated horizons, relating the bulk geometry degrees of freedom to the isolated horizon surface degrees of freedom, and, furthermore, in their SU(2) Chern-Simons treatment in the framework of loop quantum gravity [START_REF] Engle | Black hole entropy from the SU(2)-invariant formulation of type I isolated horizons[END_REF], it is desirable to have such an equation satisfied also for Kerr isolated horizons. (The symplectic structure and a first quantum model of Kerr isolated horizons based on an SU(2) Chern-Simons theory can be found in [START_REF] Frodden | Modelling black holes with angular momentum in loop quantum gravity[END_REF].) In the following, we present a method for the construction of a new connection whose curvature satisfies this particular form of the horizon boundary condition. Let ⇐ F γ be a curvature of the form [START_REF] Barbero | Real Ashtekar variables for Lorentzian signature space-times[END_REF]. We consider this curvature as a vector in R 3 with three non-zero components, i.e., 1) ,

⇐ F γ = f ⇐ Σ (
where f = f (θ) = f (i) (θ) ê(i) ∈ R 3 with ê(i)
denoting the basis vectors along the 1, 2, 3-directions, respectively. Note that the components of f can be directly read off from [START_REF] Ashtekar | Quantum horizons and black-hole entropy: inclusion of distortion and rotation[END_REF] and that the 2-form ⇐ Σ (1) acts merely as a multiplicative factor. Besides, this identification implies that γ is restricted to values in R {0}. Since the pullback of the Plebanski 2-form to the horizon 2-spheres ⇐ Σ, which we also treat as a vector in R 3 , has only one non-vanishing component, namely ⇐ Σ (1) = 0, we construct a new curvature ⇐ F γ that fulfills the relation

⇐ F γ ∶= R ⇐ F γ = f ê(1) ⇐ Σ (1) = f ⇐ Σ , (12)
in which we align ⇐ F γ with the 1-direction using an SO(3) transformation R. We point out that the choice of rotation is not unique. Here, we use a rotation that transports the head of the vector ⇐ F γ along the shortest path between the initial and the end point on the enclosing sphere (with radius ⇐ F γ ). This is realized by rotating the curvature by an angle

[0, π] ∋ α = arctan ⎛ ⎜ ⎝ f (2) 2 + f (3) 2 f (1) ⎞ ⎟ ⎠ + 0 for f (1) ≥ 0 π for f (1) < 0 (13)
around a (unit) normal to the ⇐ F γ -1 plane given by

n = f × f ê(1) f × f ê(1) = 1
f (2) 2 + f (3) 2 ⎛ ⎜ ⎝ 0 f (3) -f (2) ⎞ ⎟ ⎠ . ( 14
)
Applying this rotation to ⇐ F γ , we get

⇐ F γ = R n (α) f ⇐ Σ (1) = n⋅f n+cos (α) n×f ×n+sin (α) n×f ⇐ Σ (1) = cos (α)f +sin (α) n×f ⇐ Σ (1) . ( 15
)
We now define the connection that corresponds to the new curvature [START_REF] Chandrasekhar | The mathematical theory of black holes[END_REF]. For this purpose, we rotate the connection (8) employing the SU(2) representation ⇐

A γ = i 2 ⇐ A (i)
γ σ (i) and the correspondence

R n (α) = e α n⋅J ←→ G n (α) = e iα n⋅σ 2 = id C 2 cos α 2 + i (n ⋅ σ) sin α 2 ,
where the J i are the generators of SO(3) and the σ i are the Pauli matrices. According to the transformation law

⇐ A γ = G n (α) ⇐ A γ G -1 n (α) -dG n (α) G -1 n (α) , we find ⇐ A (i) γ ∶= ⇐ A (i) γ -sin (α) dn i + ⇐ A γ , n i + 2 sin 2 α 2 [n, dn] i -⇐ A γ ⋅ n n i -n i dα .
Substitution of the rotation angle (13) and the unit normal ( 14) leads to the expressions

⇐ A (1) γ = ⇐ A (1) γ + f (2) ⇐ A (2) γ + f (3) ⇐ A (3) γ f + 1 - f (1) f f (2) df (3) -f (3) df (2) f (2) 2 + f (3) 2 ⇐ A (2) γ = ⇐ A (2) γ - df (3) + f (2) ⇐ A (1) γ f + f (3) df (1) f 2 + 1 - f (1) f f (3) f (2) 2 + f (3) 2 f (2) df (2) + f (3) df (3) f + f (2) ⇐ A (3) γ -f (3) ⇐ A (2) γ ⇐ A (3) γ = ⇐ A (3) γ + df (2) -f (3) ⇐ A (1) γ f - f (2) df (1) f 2 -1 - f (1) f f (2) f (2) 2 + f (3) 2 f (2) df (2) + f (3) df (3) f + f (2) ⇐ A (3) γ -f (3) ⇐ A (2) γ . ( 16
)
The next step is to construct a curvature that satisfies the relation

⇐ γ ∶= α 0 f ⇐ F γ = α 0 ⇐ Σ . (17)
Therefore, to achieve the deformation f → α 0 , we use a simple rescaling transformation on ⇐ F γ . The associated connection ⇐ γ is obtained by the replacement φ+ → α 0 f φ+ that amounts to the substitution

d φ+ → α 0 f d φ+ - φ+ f 2 f (1) df (1) + f (2) df (2) + f (3) df (3)
in the formulas given in [START_REF] Frodden | Modelling black holes with angular momentum in loop quantum gravity[END_REF]. A few concluding remarks are in order. The aim of this section was to find a connection, other than the Ashtekar-Barbero connection, with a curvature that is proportional to the Plebanski 2-form on the Kerr isolated horizon 2-spheres. To this end, we applied an SU(2) gauge transformation and a diffeomorphism solely to the Ashtekar-Barbero connection (not affecting the Plebanski 2-form), in this way constructing a new horizon boundary connection with the desired property. As Eq.( 11) is both gauge and diffeomorphism invariant, it is obvious that we did not use these transformations in order to turn Eq.( 10) into Eq.( 11). The proposed method guarantees that the constructed quantity is still a connection. This can be directly seen from the fact that it is derived from an su 2 (C)-valued 1-form, namely ⇐ A γ , that transforms under SU(2) gauge transformations as a connection and under diffeomorphisms as a 1-form, only using such kind of transformations according to the proper transformation laws. We remark in passing that the proposed strategy to obtain the desired horizon boundary condition is general in the sense that it works virtually on any 2-surface in spacetime and, thus, it is not specific to the isolated horizon 2-spheres under consideration.

VI. SLOWLY ROTATING KERR ISOLATED HORIZONS

We present the first-order formalism of the Ashtekar-Barbero connection variables on the horizon 2-spheres for small angular momenta a r + ≪ 1. Moreover, we explicitly compute the new connection ⇐ γ and the associated curvature ⇐ γ , which satisfies the SU(2) Chern-Simons form of the horizon boundary condition, according to the construction method derived in the previous section. Expanding the Ashtekar-Barbero connection ( 8) and the Ashtekar-Barbero curvature (9) up to the first order in a r + yields ⇐ A (1) γ ≃ -

a sin (θ) r + dθ + cos (θ) - 3γa sin 2 (θ) 2r + d φ+ ⇐ A (2) γ ≃ 1 √ 2 r + γr + -a cos (θ) dθ + sin (θ) r + + γa cos (θ) d φ+ ⇐ A (3) γ ≃ 1 √ 2 r + -r + + γa cos (θ) dθ + sin (θ) γr + -a cos (θ) d φ+ (18)
and

⇐ F (1) γ ≃ γ 2 -1 2 - 3γa cos (θ) r + sin (θ) dθ ∧ d φ+ = 1 r 2 + γ 2 -1 2 - 3γa cos (θ) r + ⇐ Σ (1) 0 ⇐ F (2) γ ≃ 1 γ ⇐ F (3) γ ≃ 3γa sin 2 (θ) 2 √ 2 r + dθ ∧ d φ+ = 3γa sin (θ) 2 √ 2 r 3 + ⇐ Σ (1) 0 , (19)
where ⇐ Σ

(1) 0 ∶= ⇐ Σ (1) (a r + ≪ 1) ≃ r 2 + sin (θ) dθ ∧ d φ+ is the first component of the first-order Plebanski 2-form. Using [START_REF] Kinnersley | Type D vacuum metrics[END_REF] and the components of the vector f

f (1) ≃ 1 r 2 + γ 2 -1 2 - 3γa cos (θ) r + f (2) ≃ f (3) γ ≃ 3γa sin (θ) 2 √ 2 r 3 + ,
which are directly read off from [START_REF]Maximal extension of Schwarzschild metric[END_REF], we find for the first-order connection ( 16)

⇐ A (1) γ ≃ - a sin (θ) r + dθ + cos (θ) - 3γa sin 2 (θ) r + (1 -γ 2 ) d φ+ ⇐ A (2) γ ≃ 1 √ 2 r + ⎛ ⎝ γr + -a cos (θ) 1 + 3γ 2 γ 2 -1 dθ + sin (θ) r + + γa cos (θ) 1 - 3 γ 2 -1 d φ+ ⎞ ⎠ ⇐ A (3) γ ≃ 1 √ 2 r + ⎛ ⎝ -r + + γa cos (θ) 1 - 3 γ 2 -1 dθ + sin (θ) γr + -a cos (θ) 1 + 3γ 2 γ 2 -1 d φ+ ⎞ ⎠ . (20)
The corresponding first-order curvature (15) becomes

⇐ F (1) γ ≃ 1 r 2 + γ 2 -1 2 - 3γa cos (θ) r + ⇐ Σ (1) 0 ⇐ F (2) γ ≃ 1 γ ⇐ F (3) γ = O(a 2 ) ≃ 0 . (21)
Note that these expressions are restricted to values of the Barbero-Immirzi parameter for which both γ 2 -1 ≫ a r + and γ 2 -1 ≥ 12 a γ cos (θ) r + hold. The connection and curvature for the remaining values can be obtained similarly. Next, choosing the constant α 0 such that it reproduces the value of the Schwarzschild isolated horizon case in the limit a ↘ 0

α 0 ∶= γ 2 -1 2r 2 + + a 0 ,
where a 0 is a small first-order correction, the replacement for the calculation of the connection ⇐ γ and the curvature [START_REF] Kerr | Gravitational field of a spinning mass as an example of algebraically special metrics[END_REF] reads

φ+ → ⎛ ⎝ 1 + 2a γ 2 -1 r 2 + 0 + 3γ cos (θ) r + ⎞ ⎠ φ+ .
Consequently, substituting

d φ+ → ⎛ ⎝ 1 + 2a γ 2 -1 r 2 + 0 + 3γ cos (θ) r + ⎞ ⎠ d φ+ - 6γa sin (θ) r + (γ 2 -1)
φ+ dθ into [START_REF] Lewandowski | Geometric characterizations of the Kerr isolated horizon[END_REF] and ( 21) leads to

⇐ A (1) γ ≃ - a sin (θ) r + 1 + 6γ cos (θ) γ 2 -1 φ+ dθ + ⎛ ⎝ cos (θ) + a γ 2 -1 2r 2 + 0 cos (θ) + 3γ r + 1 + cos 2 (θ) ⎞ ⎠ d φ+ ⇐ A (2) γ ≃ 1 √ 2 r + ⎛ ⎝ γr + - a γ 2 -1 cos (θ) 4γ 2 -1 + 6γ sin 2 (θ) φ+ dθ + sin (θ) r + + a γ 2 -1 γ cos (θ) γ 2 + 2 + 2r 3 + 0 d φ+ ⎞ ⎠ ⇐ A (3) γ ≃ 1 √ 2 r + ⎛ ⎝ -r + + γa γ 2 -1 cos (θ) γ 2 -4 + 6γ sin 2 (θ) φ+ dθ + sin (θ) γr + + a γ 2 -1 cos (θ) 2γ 2 + 1 + 2γr 3 + 0 d φ+ ⎞ ⎠ and ⇐ F (1) γ ≃ γ 2 -1 2r 2 + + a 0 ⇐ Σ (1) 0 ⇐ F (2) γ ≃ 1 γ ⇐ F (3) γ ≃ 0 .
Thus, we have shown that this particular curvature satisfies the first-order horizon boundary condition of the desired SU(2) Chern-Simons form ( 11)

⇐ F (i) γ = γ 2 -1 2r 2 + + a 0 ⇐ Σ (i) 0 with ⇐ Σ (1)
0 given below [START_REF]Maximal extension of Schwarzschild metric[END_REF] and ⇐ Σ

(2)

0 = ⇐ Σ (3)
0 = 0, which concludes the analysis of the first-order formalism.

To determine the spin connection ω, one solves the first Maurer-Cartan equation of structure In the Newman-Penrose formalism, the Ricci rotation coefficients are represented by twelve special symbols called spin coefficients

de µ + ω µ ν ∧ e ν =
κ = γ (2)(0)(0) = γ (2)(0)(3) = 1 2 γ (1)(0)(0) + γ (2)(3)(0) σ = γ (2)(0)(2) µ = γ (1)(3)(2) γ = 1 2 γ (1)(0)(1) + γ (2)(3)(1) λ = γ (1)(3)(3) τ = γ (2)(0)(1) α = 1 2 γ (1)(0)(3) + γ (2)(3)(3) (A2) ν = γ (1)(3)(1) π = γ (1)(3)(0) β = 1 2 γ (1)(0)(2) + γ (2)(3)(2) .
Accordingly, we obtain the first Maurer-Cartan equation of structure in the following form

dl = 2 Re( ) n ∧ l -2 n ∧ Re(κ m) -2 l ∧ Re [τ -α -β] m + 2 i Im( ) m ∧ m dn = 2 Re(γ) n ∧ l -2 n ∧ Re [α + β -π] m + 2 l ∧ Re(ν m) + 2 i Im(µ) m ∧ m dm = dm = (π + τ ) n ∧ l + 2 i Im( ) -n ∧ m -σ n ∧ m + µ + 2 i Im(γ) l ∧ m + λ l ∧ m -(α -β) m ∧ m . (A3)
Finally, we specify the relevant classes of local Lorentz transformations that are applied to the tetrad and the spin coefficients. With the parameters ξ, ψ ∈ R and a, b ∈ C, which are functions of the spacetime coordinates (x µ ), these yield [START_REF] Chandrasekhar | The mathematical theory of black holes[END_REF] Class I ∶

Class III ∶ l ↦ l ′ = l l ↦ l ′ = ξ l n ↦ n ′ = n + a 2 l + a m + a m n ↦ n ′ = ξ -1 n (A4) m ↦ m ′ = m + a l m ↦ m ′ = e iψ m m ↦ m ′ = m + a l m ↦ m ′ = e -iψ m and Class I ∶ κ ↦ κ ′ = κ , σ ↦ σ ′ = σ + a κ , ↦ ′ = + a κ , ↦ ′ = + a κ , τ ↦ τ ′ = τ + a + a σ + a 2 κ , γ ↦ γ ′ = γ + a α + a (β + τ ) + a 2 ( + ) + a 2 σ + a a 2 κ , π ↦ π ′ = π + 2 a + a 2 κ + l a , λ ↦ λ ′ = λ + a (2 α + π) + a 2 ( + 2 ) + a 3 κ + m a + a l a , α ↦ α ′ = α + a ( + ) + a 2 κ , (A5) µ ↦ µ ′ = µ + a π + 2 a β + 2 a 2 + a 2 σ + a a 2 κ + m a + a l a , β ↦ β ′ = β + a + a σ + a 2 κ , ν ↦ ν ′ = ν + a λ + a (µ + 2 γ) + a 2 (τ + 2 β) + a 3 σ + a 2 (π + 2 α) + a a 2 ( + 2 ) + a 2 a 2 κ + ( a 2 l + n + a m + a m) a Class III ∶ κ ↦ κ ′ = ξ 2 e iψ κ , σ ↦ σ ′ = ξ e 2iψ σ , τ ↦ τ ′ = e iψ τ , π ↦ π ′ = e -iψ π , ↦ ′ = ξ , µ ↦ µ ′ = ξ -1 µ , α ↦ α ′ = e -iψ α + i 2 e -iψ m ψ + 1 2 ξ -1 e -iψ m ξ , β ↦ β ′ = e iψ β + i 2 e iψ m ψ + 1 2 ξ -1 e iψ m ξ , (A6) γ ↦ γ ′ = ξ -1 γ + 1 2 ξ -2 n ξ + i 2 ξ -1 n ψ , ↦ ′ = ξ + 1 2 l ξ + i 2 ξ l ψ , λ ↦ λ ′ = ξ -1 e -2iψ λ , ν ↦ ν ′ = ξ -2 e -iψ ν .
is the Regge-Wheeler coordinate, α + ∶= (r + -r -) (2ρ 2 0+ ), and ρ 2 0+ ∶= ρ 2 0 (r = r + ). Note that these particular coordinates do not provide regularity at the Cauchy horizon. Also, we restrict our study to the exterior Kerr geometry including the event horizon r + ≤ r < ∞. The Boyer-Lindquist basis vectors (∂ t , ∂ r , ∂ θ , ∂ ϕ ) and basis 1-forms (dt, dr, dθ, dϕ) transform according to

∂ t = α + L ∂ K + K ∂ L - a ρ 2 0+ ∂ φ+ , ∂ r = α + β ρ 2 0 L 2 -K 2 K ∂ K + L ∂ L , ∂ θ = ∂ θ , ∂ ϕ = ∂ φ+ (B3) and dt = L dK -K dL α + (L 2 -K 2 ) , dr = L dL -K dK α + β ρ 2 0 , dθ = dθ , dϕ = d φ+ + a (L dK -K dL) α + ρ 2 0+ (L 2 -K 2 ) , (B4)
respectively. In terms of the Kruskal-Szekeres-like coordinates (B2), the Kerr metric (B1) becomes

g = Ω 2 ρ 4 ρ 4 0 dK ⊗ dK -dL ⊗ dL -χ 1 L dK -K dL ⊗ L dK -K dL -χ 2 L dK -K dL ⊗ d φ+ -ρ 2 dθ ⊗ dθ - sin 2 (θ) ρ 2 ρ 4 0 -a 2 sin 2 (θ) ∆ d φ+ ⊗ d φ+ (B5)
with the functions

Ω ∶= 1 α + √ β ρ , χ 1 ∶= a 2 sin 2 (θ) β ρ 4 0+ r + r + r -r - ⎛ ⎝ r + r + r -r - + 2ρ 2 + ρ 2 0 1 + a 2 sin 2 (θ) (r 2 -r 2 + ) 2ρ 2 0 ρ 2 + ⎞ ⎠ , χ 2 ∶= 2aα + sin 2 (θ) ρ 2 0+ ρ 2 0 r + r + r -r - + ρ 2 + , and β ∶= e 2α+r (r -r -) 1+(r 2 -+a 2 ) ρ 2 0+ .
The event horizon is determined via the implicit equation

L 2 -K 2 = ∆ β ,
which relates the Kruskal-Szekeres-like coordinates K and L to the radial Boyer-Lindquist coordinate r. Evaluating this equation at r = r + , we obtain the two solutions K = ±L. But since we consider black hole geometries, only the positive solution K = L describing future event horizons is of interest.

Applying this transformation leads to a Kinnersley tetrad in Kruskal-Szekeres-like coordinates that is regular at the event horizon

l ′ = ρ 2 0 √ 2 ρ 2 Ω ∂ K + ∂ L - a (L -K) α + β ρ 2 0+ ρ 2 0 r + r + r -r - ∂ φ+ n ′ = ρ 2 0 √ 2 ρ 2 Ω ∂ K -∂ L - a (L + K) α + β ρ 2 0+ ρ 2 0 r + r + r -r - ∂ φ+ m ′ = 1 √ 2 ς ia sin (θ) α + L ∂ K + K ∂ L -∂ θ + iρ 2 + ρ 2 0+ sin (θ) ∂ φ+ m ′ = - 1 √ 2 ς ia sin (θ) α + L ∂ K + K ∂ L + ∂ θ + iρ 2 + ρ 2 0+ sin (θ) ∂ φ+ (C3)
and to a regular dual co-tetrad

l ′ = Ω √ 2 ⎛ ⎝ ρ 2 + ρ 2 0+ + a 2 sin 2 (θ) K (L -K) ρ 2 0+ ρ 2 0 β r + r + r -r - dK - ρ 2 + ρ 2 0+ + a 2 sin 2 (θ) L (L -K) ρ 2 0+ ρ 2 0 β r + r + r -r - dL -a sin 2 (θ) α + (L -K) d φ+ ⎞ ⎠ n ′ = Ω √ 2 ⎛ ⎝ ρ 2 + ρ 2 0+ - a 2 sin 2 (θ) K (L + K) ρ 2 0+ ρ 2 0 β r + r + r -r - dK + ρ 2 + ρ 2 0+ + a 2 sin 2 (θ) L (L + K) ρ 2 0+ ρ 2 0 β r + r + r -r - dL -a sin 2 (θ) α + (L + K) d φ+ ⎞ ⎠ m ′ = - 1 √ 2 ς ia sin (θ) α + β ρ 2 0+ r + r + r -r - [L dK -K dL] -ρ 2 dθ + iρ 2 0 sin (θ) d φ+ m ′ = 1 √ 2 ς ia sin (θ) α + β ρ 2 0+ r + r + r -r - [L dK -K dL] + ρ 2 dθ + iρ 2 0 sin (θ) d φ+ . (C4)
We compute the spin coefficients by inserting the dual co-tetrad (C4) into the first Maurer-Cartan equation of structure in the Newman-Penrose formalism (A3), giving rise to an algebraic system with solution

κ ′ = σ ′ = λ ′ = ν ′ = 0 , ′ = - L -K √ 2β ρ ς , µ ′ = - L + K √ 2β ρ ς , τ ′ = - ia sin (θ) √ 2 ρ 2 , ′ = α + (L -K) 2 √ 2 ∂ r Ω , π ′ = ia sin (θ) √ 2 ς 2 , γ ′ = α + (L + K) 2 √ 2 2ia cos (θ) Ω ρ 2 + ∂ r Ω , α ′ = 1 2 √ 2 ς ⎛ ⎝ ia sin (θ) α + + 2r ρ 2 + ρ 2 0 cot (θ) ρ 2 ⎞ ⎠ , β ′ = - 1 2 √ 2 ς ia α + sin (θ) + ρ 2 0 cot (θ) ρ 2 .
(C5)

Appendix D: Isolated Horizon Conditions

We impose the isolated horizon conditions on the regular Kinnersley tetrad (C3) by requiring, on the one hand, that the real-valued null vector l ′ , evaluated at r = r + , constitutes the equivalence class of expansion-free null normals (1) and, in addition, that it coincides with the generator of the black hole's Killing horizon. On the other hand, we want the associated complex null vectors m ′ and m ′ to be tangential to the horizon 2-spheres, spanning their intrinsic geometry [START_REF] Ashtekar | Isolated horizons: Hamiltonian evolution and the first law[END_REF][START_REF] Ashtekar | Geometry of generic isolated horizons[END_REF]. The first condition on l ′ is stated as follows. Let ν be a null normal to ∆, i.e.,

⟨ν, V ⟩ = 0 ∀ V ∈ T ∆ and ⟨ν, ν⟩ = 0 , (D1)
where ⟨⋅ , ⋅⟩ ∶= g (⋅ , ⋅) ∶ T M × T M → R is the canonical scalar product with respect to the metric (B5). From (D1), we find that the null normal takes the form

ν r=r+ = ν t r=r+ α + L (∂ K + ∂ L )
with ν t being the time-like component in Boyer-Lindquist coordinates. Evaluating the vector l ′ given in (C3) at r = r + , we can immediately show that it is already of this form and, therefore, normal to the horizon. Secondly, we adjust the local isolated horizon surface gravity to the surface gravity of a Kerr black hole, which is defined at space-like infinity, by requiring that the null normal l ′ r=r+ corresponds to the generator of the black hole's Killing horizon

χ ∶= ∂ t + a ρ 2 0+ ∂ ϕ = α + L (∂ K + ∂ L ) .
We impose this condition by using a class III local Lorentz transformation (A4)

l ′′ r=r+ = ξ l ′ r=r+ = χ with transformation parameters ξ = 2 β + ρ + L ρ 2 0+
and ψ = 0 .

Applying this transformation to (C3) yields

l ′′ r=r+ = α + L (∂ K + ∂ L ) n ′′ r=r+ = ρ 4 0+ 2α + Lρ 4 + Ω 2 + (∂ K -∂ L ) - ar + α + ρ 2 0+ ρ 2 + ∂ φ+ m ′′ r=r+ = 1 √ 2 ς + ia sin (θ) α + L [∂ K + ∂ L ] -∂ θ + iρ 2 + ρ 2 0+ sin (θ) ∂ φ+ m ′′ r=r+ = - 1 √ 2 ς + ia sin (θ) α + L [∂ K + ∂ L ] + ∂ θ + iρ 2 + ρ 2 0+ sin (θ) ∂ φ+ .
Next, we formulate the conditions for the complex pair (m ′′ , m ′′ ) r=r+ . Let W be a vector field that is tangential to the horizon 2-spheres, that is,

W = W θ ∂ θ + W φ+ ∂ φ+ .
In order to bring m ′′ r=r+ and m ′′ r=r+ into this form, we use a class I local Lorentz transformation (A4) that eliminates the ∂ K L -components

m ′′′ r=r+ = m ′′ r=r+ + a l ′′ r=r+ = m ′′′ θ r=r+ ∂ θ + m ′′′ φ+ r=r+ ∂ φ+ m ′′′ r=r+ = m ′′ r=r+ + a l ′′ r=r+ = m ′′′ θ r=r+ ∂ θ + m ′′′ φ+ r=r+ ∂ φ+ ,
where the complex parameter a reads a = -ia sin (θ) √ 2 ς + .

With this additional transformation, we obtain a Newman-Penrose frame that is adapted to the boundary conditions of Kerr isolated horizons

l ′′′ r=r+ = α + L (∂ K + ∂ L ) n ′′′ r=r+ = ρ 4 0+ 2α + Lρ 4 + Ω 2 + ⎛ ⎝ 1 - a 2 sin 2 (θ)L 2 β + ρ 4 0+ ∂ K -1 + a 2 sin 2 (θ)L 2 β + ρ 4 0+ ∂ L ⎞ ⎠ - a ρ 2 0+ 1 + r + α + ρ 2 + ∂ φ+ m ′′′ r=r+ = 1 √ 2 ς + -∂ θ + iρ 2 + ρ 2 0+ sin (θ) ∂ φ+ m ′′′ r=r+ = - 1 √ 2 ς + ∂ θ + iρ 2 + ρ 2 0+ sin (θ) ∂ φ+ . (D2)
Its dual becomes

l ′′′ r=r+ = α + L ρ 4 + Ω 2 + ρ 4 0+ (dK -dL) n ′′′ r=r+ = 1 2α + L ⎛ ⎝ 1 + a 2 sin 2 (θ) L 2 β + ρ 4 0+ dK + 1 - a 2 sin 2 (θ) L 2 β + ρ 4 0+ dL ⎞ ⎠ m ′′′ r=r+ = 1 √ 2 ς + ⎛ ⎝ ia sin (θ) Lρ 4 + Ω 2 + ρ 4 0+ α + + r + ρ 2 + (dL -dK) + ρ 2 + dθ -iρ 2 0+ sin (θ) d φ+ ⎞ ⎠ m ′′′ r=r+ = 1 √ 2 ς + ⎛ ⎝ - ia sin (θ) Lρ 4 + Ω 2 + ρ 4 0+ α + + r + ρ 2 + (dL -dK) + ρ 2 + dθ + iρ 2 0+ sin (θ) d φ+ ⎞ ⎠ . (D3)
The corresponding spin coefficients are determined by transforming (C5) according to (A5) and (A6). We find

κ ′′′ r=r+ = σ ′′′ r=r+ = ′′′ r=r+ = 0 , π ′′′ r=r+ = ia sin (θ) √ 2 ς + α + + 1 ς + , λ ′′′ r=r+ = - a 2 sin 2 (θ) ς 2 + α + 2 + 1 ς + , µ ′′′ r=r+ = 1 ρ 2 + -r + + a 2 sin 2 (θ) α + 2 , γ ′′′ r=r+ = 1 2ρ 2 + ⎛ ⎝ -ς + + r + a 2 sin 2 (θ) ρ 2 + - α + 2 β + ρ 4 0+ L 2 + a 2 sin 2 (θ) + ρ 2 0+ 4r 2 + α + ⎞ ⎠ , ν ′′′ r=r+ = ia sin (θ) √ 2 ς + ρ 2 + ⎛ ⎝ -2 ς + + ia 3 cos (θ) sin 2 (θ) ρ 2 + -α + ρ 2 + + β + ρ 4 0+ 2L 2 + a 2 sin 2 (θ) 2 + ρ 2 0+ 4r 2 + α + ⎞ ⎠ , τ ′′′ r=r+ = - ia sin (θ) √ 2 ρ 2 + , ′′′ r=r+ = α + 2 , α ′′′ r=r+ = 1 √ 2 ς + ⎛ ⎝ cot (θ) 2 + ia sin (θ) α + 2 + 1 ς + ⎞ ⎠ , β ′′′ r=r+ = - 1 2 √ 2 ς + cot (θ) + ia α + sin (θ) . (D4)
It can be easily verified that the expansion of the real-valued Newman-Penrose vector l ′′′ r=r+ given in (D2), which constitutes the equivalence class of null normals (1), vanishes on the horizon, as in the Newman-Penrose formalism the associated expansion scalar is expressed in terms of the real part of the spin coefficient ′′′ [START_REF] Ashtekar | Isolated horizons: Hamiltonian evolution and the first law[END_REF]. Thus, using ′′′ r=r+ as stated in (D4), we directly see that the expansion vanishes on the horizon

θ (l ′′′ ) r=r+ = -2 Re( ′′′ r=r+ ) = 0 .
Moreover, since

κ ′′′ r=r+ = σ ′′′ r=r+ = Im( ′′′ r=r+ ) = 0 ,
we know that l ′′′ r=r+ is geodesic as well as shear-and twist-free. Finally, we remark that the validity of Eq.( 2) can be shown by direct computation and that in the family of Kerr spacetimes, the only equivalence class (1) satisfying the isolated horizon condition (v) is the one containing constant multiples of l ′′′ r=r+ [START_REF] Ashtekar | Isolated horizons: Hamiltonian evolution and the first law[END_REF].

ω (0)(1) = -2 Re(γ) l -2 Re( ) n + (α + β) m + (α + β) m ω (0)(2) = ω (0)(3) = -τ l -κ n + m + σ m ω (1)(2) = ω (1)(3) = ν l + π n -µ m -λ m ω (2)(3) = 2 i Im(γ) l + 2 i Im( ) n -(α -β) m + (α -β) m . (E1)
We note that it is sufficient to compute the pullback of these quantities to the isolated horizon 2-spheres as the entire horizon can be covered by the action of the Lie derivative along its local null normal vector field. Accordingly, inserting the spin coefficients (D4) and the dual Kinnersley frame (D3) into (E1), we obtain for the spin connection on the isolated horizon 2-spheres

⇐ ω ′′′ (0)(1) = a sin (θ) ρ 2 + ⎛ ⎝ a cos (θ) dθ + ρ 2 0+ sin (θ) α + + r + ρ 2 + d φ+ ⎞ ⎠ ⇐ ω ′′′ (0)(2) = ⇐ ω ′′′ (0)(3) = 0 ⇐ ω ′′′ (1)(2) = ⇐ ω ′′′ (1)(3) = 1 √ 2 ς + ρ 2 + ⎛ ⎝ ρ 2 + r + + a 2 sin 2 (θ) ς + dθ + iρ 2 0+ sin (θ) -r + + a 2 sin 2 (θ) α + + 1 ς + d φ+ ⎞ ⎠ ⇐ ω ′′′ (2)(3) = i ρ 2 + -r + a sin (θ) dθ + ρ 4 0+ cos (θ) ρ 2 + d φ+ , (E2)
in which the double arrow underneath denotes the horizon 2-sphere pullback. Since the Ashtekar and Ashtekar-Barbero connection variables are usually defined on space-like hypersurfaces of foliations of globally hyperbolic spacetimes, we now express the spin connection (E2) via an orthonormal frame that is fixed in the time gauge (see, e.g., [START_REF] Doná | Introductory lectures to loop quantum gravity[END_REF] and references therein). This frame is related to the Newman-Penrose frame via

e (0) = l ′′′ + n ′′′ √ 2 , e (1) = l ′′′ -n ′′′ √ 2 , e (2) = m ′′′ + m ′′′ √ 2
, and e

(3) = i (m ′′′ -m ′′′ ) √ 2 , ( E3
)
and the metric is given by the Minkowski metric with signature (1, 3) (E5)

η(a)(b) = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 ⎞ ⎟ ⎟ ⎟ ⎠ . (E4) Substituting (D2) into (E3) yields e (0) r=r+ = α + L √ 2 ⎛ ⎝ 1 + 1 2ρ 2 + β + ρ 4 0+ L 2 -a 2 sin 2 (θ) ∂ K + 1 - 1 2ρ 2 + β + ρ 4 0+ L 2 + a 2 sin 2 (θ) ∂ L ⎞ ⎠ - a √ 2ρ 2 0+ 1 + r + α + ρ 2 + ∂ φ+ e (1) r=r+ = α + L √ 2 ⎛ ⎝ 1 - 1 2ρ 2 + β + ρ 4 0+ L 2 -a 2 sin 2 (θ) ∂ K + 1 + 1 2ρ 2 + β + ρ 4 0+ L 2 + a 2 sin 2 (θ) ∂ L ⎞ ⎠ + a √ 2ρ 2 0+ 1 + r + α + ρ 2 + ∂ φ+ e (
As we work only on the horizon 2-spheres, the orthonormal frame is in the time gauge when ⟨e (0) r=r+ , W ⟩ = 0 for all W = W θ ∂ θ + W φ+ ∂ φ+ ∈ T S 2 . This condition leads to a time-like basis vector of the general form where the transformation matrix is given by (see (E3) and Appendix A)

Λ (a) (b) = 1 √ 2 ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 0 0 -1 1 0 0 0 0 -1 -1 0 0 i -i ⎞ ⎟ ⎟ ⎟ ⎠
and the metrics η and η by (A1) and (E4), respectively, results in the expressions (3).

Appendix F: Ashtekar and Ashtekar-Barbero Formulations

In the Ashtekar formulation, instead of using a tetrad frame (e (a) ) and six so 1,3 (R)-valued spin connection 1-forms ω (a)(b) , with a, b ∈ {0, 1, 2, 3}, one employs a triad (e (i) ) and three self-dual sl 2 (C)-valued connection 1forms A

+ , i ∈ {1, 2, 3}, as configuration variables (for a detailed review see, e.g., [START_REF] Rovelli | Ashtekar formulation of general relativity and loop-space nonperturbative quantum gravity: a report[END_REF]). The self-dual connection, commonly known as Ashtekar connection, is defined by [START_REF] Ashtekar | New variables for classical and quantum gravity[END_REF] A (i) + ∶= P where τ is an eigenfunction of the ⋆-operator. Then, the spin connection can be split into a self-dual and an anti-self-dual part

ω = ω + + ω -,
in which both ω + and ω -contain the same information as ω itself. The Ashtekar connection (F1) can be generalized to an su 2 (C)-valued 1-form, the so-called Ashtekar-Barbero connection A (i) γ , replacing the imaginary unit in the projector (F2) by a complex-valued parameter γ, known as Barbero-Immirzi parameter. Different values for γ yield equivalent classical theories, however, their quantum theories are unitarily inequivalent, resulting in ambiguous physical predictions. In the framework of loop quantum gravity, one usually restricts γ to values in R {0} [START_REF] Barbero | Real Ashtekar variables for Lorentzian signature space-times[END_REF], which in turn leads to real-valued SU(2) connections. This restriction arises from the current level of development in functional analysis, where mathematical methods are worked out in detail only for real-valued connections.

  0 , which in the tetrad formulation reads de (a) = γ (a) (b)(c) e (b) ∧ e (c) with the Ricci rotation coefficients γ (a) (b)(c) e (b) = e (a)

γ

  (a)(b)(c) are the Ricci rotation coefficients, into the Newman-Penrose formalism employing (A2)

ρ 2 +

 2 ∂ θ -r + ρ 2 0+ sin (θ) ∂ φ+ .

e+ ρ 4 + ρ 6 0+ α + + r + ρ 2 +e

 462 (0) r=r+ = e (0)Kr=r+ ∂ K + e (0)L r=r+ ∂ L + aLΩ 2 (0)L r=r+ -e (0)K r=r+ ∂ φ+ .Direct computation shows that the time-like basis vector e (0) r=r+ of the orthonormal tetrad (E5) is already of this form. Transforming the spin connection (E2) into the orthonormal frame according to the transformation law for connections ⇐ ω (a)(b) = η(a)(c) η(h)(b) η (d)(e) Λ

  homomorphism reading out the self-dual part of the spin connection. Note that, since the so 1,3 (R) Lie algebra cannot be decomposed into a direct sum of self-dual and anti-self-dual algebras, one has to work with the complexified Lie algebra so 1,3 (C) for which such a decomposition exists, namely so 1,3 (C) = so 1,3 (C) + ⊕ so 1,3 (C) - with the self-dual and anti-self-dual algebras so 1,3 (C) + ∶= {τ ∈ so 1,3 (C) ⋆ τ = iτ } and so 1,3 (C) -∶= {τ ∈ so 1,3 (C) ⋆ τ = -iτ } ,

Acknowledgments

I am grateful to Alejandro Perez, Felix Finster, Simone Murro, Guillaume Idelon-Riton, Ernesto Frodden, and Daniele Pranzetti for useful discussions and comments, and to Katharina Proksch, Horst Fichtner, and Florian Schuppan for a careful reading of the first version of this paper. Furthermore, I thank the anonymous referees for helpful and constructive comments. This research was supported by a DFG Research Fellowship.

Appendix A: The Newman-Penrose Formalism Let F M and F ⋆ M be a null frame bundle and its dual on M. On their sections, we introduce local tetrad and co-tetrad basis (e (a) ) and (e (a) ) with a ∈ {0, 1, 2, 3}. These are related to the basis vectors of the sections of the tangent and cotangent bundles T M and T ⋆ M via e In the Newman-Penrose formalism [START_REF] Newman | An approach to gravitational radiation by a method of spin coefficients[END_REF][START_REF] Newman | Note on the Bondi-Metzner-Sachs group[END_REF], the null tetrad consists of two real-valued vectors l = e (0) = e (1) and n = e (1) = e (0) , as well as the complex-conjugated vectors m = e (2) = -e (3) and m = e (3) = -e (2) . These fulfill the null conditions

and the cross-normalization conditions

which depend on the chosen signature convention. In this particular null frame, the metric is given by

Appendix B: Analytical Extension of Kerr Geometry -From Boyer-Lindquist to Kruskal-Szekeres-like Coordinates

The Kerr metric g in Boyer-Lindquist coordinates (t, r, θ, ϕ), with t ∈ R, r ∈ R >0 , θ ∈ [0, π], and ϕ ∈ [0, 2π), as well as with the signature convention (+, -, -, -), is given by [START_REF] Boyer | Maximal analytic extension of the Kerr metric[END_REF][START_REF] Kerr | Gravitational field of a spinning mass as an example of algebraically special metrics[END_REF]

where ∆ = ∆(r) ∶= (r -r + )(r -r -) = r 2 -2M r + a 2 is the horizon function, ρ 2 = ρ 2 (r, θ) ∶= r 2 + a 2 cos 2 (θ), and ρ 2 0 = ρ 2 0 (r) ∶= r 2 + a 2 . The parameter M denotes the mass, aM the angular momentum, and r ± ∶= M ± √ M 2 -a 2 the event horizon and the Cauchy horizon of the Kerr black hole. For the analysis of Kerr isolated horizons, we require a coordinate system that is regular at the event horizon. As Boyer-Lindquist coordinates are singular there, they are not suitable for the purpose at hand. Instead, we apply horizon-penetrating Kruskal-Szekeres-like coordinates (K, L, θ, φ+ ), with K ∈ R, L ∈ R >0 , θ ∈ [0, π], and φ+ ∈ [0, 2π), which read [START_REF]Maximal extension of Schwarzschild metric[END_REF] K ∶= e α+r⋆ sinh (α + t) , L ∶= e α+r⋆ cosh (α + t) , θ = θ , and

where

Appendix C: Regular Kinnersley Tetrad and Spin Coefficients in Kruskal-Szekeres-like Coordinates

The Kinnersley tetrad in Boyer-Lindquist coordinates yields [START_REF] Chandrasekhar | The mathematical theory of black holes[END_REF][START_REF] Kinnersley | Type D vacuum metrics[END_REF]

with the quantity ς = ς(r, θ) ∶= r + ia cos (θ) satisfying ς ς = ρ 2 . The corresponding dual Kinnersley co-tetrad is given by

Substituting the basis vectors and 1-forms (B3) and (B4) into (C1) and (C2), we obtain the Kinnersley tetrad and its dual in Kruskal-Szekeres-like coordinates. Although the underlying coordinate system is now regular up to the Cauchy horizon, the resulting expressions for the frame vector and 1-form l are still singular at the event horizon. As these singularities arise in multiplicative factors, they can be eliminated by rescaling, using a class III local Lorentz transformation (A4) with parameters ξ = L -K √ 2β ρ and ψ = 0 .