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Photon Emission Near Extreme Kerr Black Holes

Ongoing astronomical efforts extract physical properties of black holes from electromagnetic emissions in their near-vicinity. This requires finding the null geodesics which extend from the near-horizon region out to a distant observatory. In general these can only be found numerically. In this paper, for the interesting special case of extremally spinning Kerr black holes, we use an emergent near-horizon conformal symmetry to find near-superradiant geodesics analytically.


Introduction
In the coming decades, LIGO [1], the Event Horizon Telescope [2] and a number of other experiments [3][4][5] are expected to advance observational black hole astrophysics to a qualitatively new level of precision. The phenomenology of black holes is based on the Kerr solution which describes neutral rotating black holes in vacuum general relativity. A large number of observed black holes, especially supermassive ones in active galactic nuclei, are thought to be rotating very fast [6,7]. Therefore extreme Kerr, which is rotating at the maximally allowed theoretical limit, is of particular observational relevance.
From a purely theoretical perspective extreme Kerr is also of particular interest for several reasons. An enhanced conformal symmetry group of the geometry emerges as the horizon is approached [8,9] which greatly constrains and simplifies the gravitational dynamics in the near-horizon region. This is known as the Kerr/CFT (Conformal Field Theory) correspondence, because the same symmetry group appears in two-dimensional CFTs. The conformal symmetries imply a variety of qualitatively new and potentially observable phenomena [10][11][12][13][14][15][16][17][18][19][20]. At the quantum level, they have conjectural implications for black hole information [9,21,22]. More generally, the extreme Kerr black hole is a critical point in the Kerr family of astrophysical black holes and hence worthy of special attention.
Recently the Kerr/CFT correspondence has been used to compute explicit analytical gravitational waveforms for extreme-mass-ratio-inspirals in the near-horizon region of nearextreme Kerr black holes [10,11,15,16]. It has also been used to analytically solve the equations of force-free electrodynamics which are expected to govern the near-horizon magnetosphere [12,14,17,18].
In this paper we lay the groundwork for an exploration of the consequences of Kerr/CFT on electromagnetic emissions from the near-horizon region of an extreme Kerr black hole. The intense frame-dragging in this region leads to the expectation that such emissions peak near the superradiant bound, at which the ratio of the angular momentum to the energy is the Schwarzschild radius. We solve analytically, to leading order in the deviation from the superradiant bound, the equations for null geodesics that connect emission points in the near-horizon region of an extreme Kerr to observation points in the region far from the black hole. 1 Using geometric optics these are trajectories of photons which start near the horizon and end at the telescope. Previously such geodesics were known only numerically. These analytic results may be useful for the study of a variety of problems related to observations of electromagnetic radiation which originates from or passes through the nearhorizon region of extreme black holes. For example, we may use them to compute the profile of iron line emissions from the near-horizon of an extreme Kerr, which have been extensively analyzed in [23][24][25][26][27]. We also hope to apply the equations derived herein to other potential observables such as extreme black holes silhouettes or orbiting hot spots. The rest of the paper is organized as follows. In section 2 we set up the problem and give our conventions. In section 3 we solve for the radial null geodesic motion in the r − θ plane and in section 4 we do the same for the motion in the r − φ and r − t planes. Section 5 summarizes the results and discusses future directions.

Geodesic equations
The Kerr metric in Boyer-Lindquist coordinates x a ∼ (t,r, θ,φ) is given by: This describes neutral rotating black holes of mass M and angular momentum J = aM. The general form of the geodesic equation for x a (τ ) with affine parameter τ is where µ = 0 for null geodesics and µ = 1 for time-like geodesics. In the Kerr space-time there are three additional conserved quantities along every geodesic: The Carter constant Q here is constructed from the Killing tensor (square of Killing-Yano), E andL are the energy and angular momentum while Q measures the kinetic energy out of the equatorial plane. These conservation laws enable integration of the geodesic equation into the form [28] The integrals are understood to be path integrals along the trajectory. At every point of the trajectory the conditionsR ≥ 0 ,Θ ≥ 0 must hold but the signs of dr ′ , dθ ′ and R , Θ change at corresponding turning points in such a way that dr ′ / R and dθ ′ / Θ are always positive. In general (2.5-2.7) can only be evaluated numerically.
In this paper we consider null geodesics, in the extreme Kerr space-time, which are near the so-called superradiant bound: In this case we will find the radial integrals in (2.5-2.7) are analytically soluble to leading order in λ. A particle or photon on a λ < 0 geodesic cannot cross the horizon because, where it do so, the resulting black hole would violate the cosmic censorship bound. Hence, as already noted by Bardeen [29], λ = 0 is a special point in the parameter space of extreme Kerr geodesics. One expects many photons emitted from the near-horizon region to have small λ. It is therefore gratifying that this case is analytically soluble.
It is convenient to introduce the dimensionless Bardeen-Horowitz coordinates, The near-horizon region is then r ≪ 1, and the metric takes the form [8] where Γ(θ) = (1 + cos 2 θ)/2 , Λ(θ) = 2 sin θ/(1 + cos 2 θ) and the subleading corrections are suppressed by powers of r. The leading term displayed on the right hand side solves the Einstein equation on its own. It is referred to as the Near-Horizon Extreme Kerr or 'NHEK' metric and has an enhanced symmetry group [8,9]. We are interested in trajectories of photons emitted near the horizon and observed by a distant telescope. These are geodesics which start at (t n , r n , θ n , φ n ) with r n ≪ 1 in the NHEK region and end at (t f , r f , θ f , φ f ) with r f ≫ 1 in the far asymptotically flat region.
In the null caseÊ may be scaled out of the geodesic equation, whose solutions are labeled by λ and the convenient dimensionless shifted Carter constant where the last inequality expresses positivity of the kinetic energy in a local frame. We will show below that, given (2.12), a geodesic that originates in NHEK and reaches out to the far asymptotically flat region must have a positive q 2 . We will take q > 0 and hold it fixed while expanding in small λ. Motion in the equatorial plane has q = √ 3. In terms of the coordinates (2.13) and parameters (2.12) and (2.15) the geodesic equations (2.5-2.7) become: where R = r 4 + 4r 3 + (q 2 + 8λ − 4λ 2 )r 2 + 8λr + 4λ 2 , The integrals in (2.16-2.18) are of elliptic type and are generally treated numerically. In this paper we will perform the radial integrals analytically to leading order in small λ using the method of matched asymptotic expansions (MAE) for all geodesics extending from the near to the far region. We will proceed in the small λ regime by dividing the spacetime into two regions: We now proceed to solve the equations in the near and far regions and match the solutions in the overlap region.

The r-θ motion
In this section we solve the radial integral to leading order in λ via MAE. Given (2.12) we have that R ≈ r 4 + 4r 3 + q 2 r 2 + 8λr + 4λ 2 .

(3.4)
We first check the conditions under which small λ geodesics in the near and far regions connect. Turning points in the radial motion occur at the zeroes of R(r). In the far region these are located at If q 2 is positive, this has no turning points with positive r. On the other hand, if q 2 is negative, there is a turning point at positive r, and the geodesic bounces off the black hole before it penetrates the near region. We therefore take ignoring the measure zero case q = 0. In the near region the zeroes are at Since 0 < q 2 < 4 to leading order in λ, the square root is always a positive number less than 2. Hence there is a positive root if and only if λ is negative. In that case, a geodesic emanating from the horizon will turn around before it reaches the far region at r ∼ √ λ. This is due to the fact that photons on such trajectories exceed the superradiant bound.
On the other hand, there are two turning points, and if the geodesic originates in the near region but outside both turning points, r > r n± , it can reach the far region. Hence λ can have either sign but geodesics with negative λ cannot get all the way to the horizon.
To leading order in λ the near and far radial integrals can now be performed analytically: where C n , C f are integration constants. In the overlap region (2.25) we have: I n (r) = 1 q ln r + ln 2q 2 + q C n + 4λ q 2 r + . . . , (3.10) Matching I n = I f in the overlap region we find: The integral (3.1) is given by I = I f (r f ) − I n (r n ) and using (3.8), (3.9), and (3.12) we find: where, 4 The r-φ and r-t motion Given (2.12) we have that Φ ≈ r 3 + 4r 2 + 3r + 4λ , (4.1) T ≈ r 4 + 4r 3 + 7r 2 + 4r + 4λ , and we can perform the radial integrals for the r-φ and r-t motion in (2.17) and (2.18), via MAE as follows.

Conclusion
In this paper we studied null geodesics in the extreme Kerr space-time. We used the method of matched asymptotic expansions to integrate, to leading order in the deviation from the superradiant bound, all radial integrals in Carter's integral geodesic equations. The key equations derived in this paper are (3.13), (4.16), and (4.17).
In the special case of motion confined to the equatorial plane we derived the algebraic equations (4.18,4.19) which relate an endpoint of a null geodesic near the horizon to its endpoint in the region far from the black hole and the associated angular momentum constant along the geodesic.
For geodesics which move in the θ direction we note that positivity of the shifted Carter constant q 2 (derived in (3.6) for all geodesics which connect between the near and far regions) restricts the values of θ that these geodesics may possibly explore. This is because Θ ≥ 0 then implies 3+cos 2 θ−4 cot 2 θ ≥ 0 which means that θ must lie between θ 0 = arccos 2 √ 3 − 3 ≈ 47 • and π − θ 0 ≈ 133 • . This fact has been derived previously from an analysis of the NHEK geodesics alone in [30]. Solving the null geodesic equations studied in this paper for a fixed observer far from the black hole and using the geometrical optics methods of [31,32] one may obtain various observables related to propagation of electromagnetic radiation from the near-horizon region of extreme black holes. These include broadened Fe Kα emission lines and images of hot orbiting spots. We hope that the results obtained in this paper together with an analytical treatment of the θ integrals will lead to a derivation of analytical formulas for such observables.
Finally, it is worth mentioning that while in our expansion in small λ we have been keeping q 2 fixed and finite, all of the equations and statements derived in this paper are also true for the case of small q 2 provided that it remains q 2 ≫ √ λ. In this case the only thing that needs to be modified are the definitions of the near region (2.23) to r ≪ min (1, q 2 ) and the overlap region (2.25) to √ λ ≪ r ≪ min (1, q 2 ).