Decoherence from Horizons: General Formulation and Rotating Black Holes

Recent work by Danielson, Satishchandran, and Wald (DSW) has shown that black holes -- and, in fact, Killing horizons more generally -- impart a fundamental rate of decoherence on all nearby quantum superpositions. The effect can be understood from measurement and causality: An observer (Bob) in the black hole should be able to disturb outside quantum superpositions by measuring their superposed gravitational fields, but since his actions cannot (by causality) have this effect, the superpositions must automatically disturb themselves. DSW calculated the rate of decoherence up to an unknown numerical factor for distant observers in Schwarzschild spacetime, Rindler observers in flat spacetime, and static observers in de Sitter spacetime. Working in electromagnetic and Klein-Gordon analogs, we flesh out and generalize their calculation to derive a general formula for the precise decoherence rate for Killing observers near bifurcate Killing horizons. We evaluate the rate in closed form for an observer at an arbitrary location on the symmetry axis of a Kerr black hole. This fixes the numerical factor in the distant-observer Schwarzschild result, while allowing new exploration of near-horizon and/or near-extremal behavior. In the electromagnetic case we find that the decoherence vanishes entirely in the extremal limit, due to the"Black hole Meissner effect"screening the Coulomb field from entering the black hole. This supports the causality picture: Since Bob is unable to measure the field of the outside superposition, no decoherence is necessary -- and indeed none occurs.


I. INTRODUCTION
Some of the greatest remaining conceptual challenges in theoretical physics involve the interplay among gravity, quantum mechanics, and measurement.Thought experiments involving black holes can be particularly sharp, pushing cherished physical ideas to their limits.In a recent paper [1] remarkable for its simplicity and generality, Danielson, Satishchandran, and Wald (DSW) showed that the mere presence of a black hole induces a fundamental rate of decoherence on all nearby quantum superpositions.In other words, it is not just difficult but in fact impossible to avoid entanglement with degrees of freedom inside the black hole.
As surprising as this decoherence may seem, there is actually a very simple argument for why it should occur. 1Suppose that Alice prepares a quantum superposition outside the black hole.The superposed matter creates a superposed gravitational field, which penetrates into the black hole, where an observer, Bob, could choose to measure it.By doing so, he has gained information about Alice's state and must therefore disturb it; however, by causality, his actions can have no effect whatsoever on her state.Since a hypothetical Bob can in principle take an action that should disturb Alice's state, the in-escapable conclusion is that her state must in fact disturb itself : the DSW decoherence.
Though powerful, this simple argument still leaves one wanting for deeper understanding-some identifiable mechanism or at least a connection to more familiar physics.DSW attribute the effect to "emission of soft gravitons" in light of a low-frequency approximation to the entangling graviton flux through the horizon.But this is hardly emission in the conventional sense, given that the relevant quantum superpositions are held stationary as they steadily decohere.A hint of deeper meaning might be found in the role of the surface gravity in the DSW calculation, suggesting that perhaps the decoherence can be understood in thermal terms.
Exploring and validating these interpretationscausality, soft radiation, thermal properties-will require detailed, general calculations of the decoherence.Thus far, the effect has been considered for distant observers in the Schwarzschild spacetime [1], Rindler observers in flat spacetime [4], and static observers in de Sitter spacetime [4].These cases do not allow one to vary the surface gravity independently of other physical parameters and do not contain an extremal limit.Furthermore, in all cases DSW only obtained the overall scaling of the rate, leaving a numerical factor undetermined.
In this paper we will derive a general formula for the precise decoherence rate for Killing observers near a bifurcate Killing horizon and apply it to an observer at an arbitrary location along the symmetry axis of a rotating black hole.We use the same FIG. 1.The DSW decoherence.Alice creates a spatial superposition, holds it for Killing time T , and then closes it.The superposed Coulomb fields are approximately stationary on the horizon for a time of order T , shown as a double-arrow.At large T , the coherence decreases exponentially like e −ΓT , making complete decoherence unavoidable.We find the precise rate to be Γ = 1  2 Cκ, where κ is the horizon surface gravity and C is the "decohering flux" from the superposed Coulomb fields.basic setup as DSW, while fleshing out and/or generalizing some key elements of the calculation.Our results confirm their findings (fixing unknown coefficients) and generalize the results to a context where the black hole mass, black hole temperature, and observer distance can be varied independently.However, our calculations are unfortunately limited to the electromagnetic (EM) and Klein-Gordon (KG) analogs.We leave the more challenging case of gravitational decoherence to future work. 2   We now describe the setup and the main results, focusing on the EM case.As in DSW, we consider a localized (but macroscopic) charged object that is placed into in a spatial superposition, held for a (Killing) time T , and then recombined (Fig. 1).The charged object is treated semiclassically in that each branch of the superposition couples to the quantized EM field as a fixed classical source.The sources are assumed to be on Killing orbits outside the horizon, and the photon field begins in the Unruh state asso- 2 In the the gravitational case the stress-energy of Alice's laboratory must be taken into account.
ciated with the Killing horizon. 3ach branch of the superposition has a different semi-classical source, so the two branches induce two different evolutions of the state |ψ⟩ of the photon field, denoted |ψ L ⟩ and |ψ R ⟩ for "left" and "right".If |ψ L ⟩ and |ψ R ⟩ become orthogonal after the experiment, then the experiment is maximally entangled with the photon field (a kind of "environment") and Alice's superposition has fully decohered.Working in the limit that the superposition is maintained for a long time T , we calculate the final overlap of these states to be4 where κ is the horizon surface gravity, T 0 is a nonuniversal timescale (dependent on the details of the transition), and C is a constant we call the "decohering flux".
To present the formula for C, let F R µν and F L µν represent the stationary solutions associated with the right and left branches (respectively) of the superposition.The difference between these fields contains "which-path information" about the superposition, and the important part turns out to be the "radial fields", i.e., the pullback of F and its dual * F to a horizon cross-section C. We define where x A (A = 1, 2) are coordinates on C, which has spatial metric q AB , area form ϵ AB , area element dS = √ qd 2 x A , derivative operator ∇ A , and Lapla- In terms of these definitions, the decohering flux is given by5 where (V A ) 2 denotes q AB V A V B .The KG result takes the same form (1) with the simpler expression C = π −1 ∆ϕ 2 dS (82) for the decohering flux.
We are able to evaluate the decohering flux C explicitly for a superposition on the symmetry axis of the Kerr spacetime.In the EM case we consider a spatial superposition in the radial direction, while in the KG case we consider a superposition of different values of charge (which is possible because the KG charge can be time-dependent).The KG and EM results are given in Eqs.(113) and Eq.(150) below, respectively, and are plotted in Fig. 2.
The extremal limit is particularly interesting.From (1) we see that if κ → 0 with C ̸ = 0, the exponential decoherence goes over to a power-law behavior, whereas if both vanish then the decoherence vanishes entirely.We see in Fig. 2 that indeed both vanish (C = κ = 0) for maximally spinning black holes (a = M ) in the electromagnetic case, meaning the effect goes away completely.This is a consequence of the "black hole Meissner effect" [5,6] that maximally spinning black holes screen out any external fields.The physical picture of "Bob in a black hole" thus holds up to detailed calculation: Since there is no Coulomb field for him to measure, there is no causal need for decoherence, and indeed the decoherence duly vanishes.
The remainder of the paper is organized as follows.The first four sections treat the KG case in detail.In Sec.II we consider field quantization in the presence of a persistent semiclassical source and introduce the notion of particles as quanta of the retarded-minus-advanced field.In Sec.III we describe the superposition experiment and show that the decoherence is related to the "expected number of entangling particles", fleshing out the key ideas of DSW and extending them to the retarded-minusadvanced context.In Sec.IV we express the decoherence as an integral on the past horizon.This produces the main formal results, namely the decoherence rates and formulas for the decohering flux.We calculate the decohering flux in the Kerr spacetime in Sec.V. We consider the EM case in Sec.VI, establishing some useful properties of a horizon-adapted gauge and repeating the steps of the KG calculation.We use units with G = c = ℏ = 1 and ϵ 0 = 1/(4π), and our metric signature is (−, +, +, +). 2. The decohering flux C as a function of spin for an observer at r0 = 3M on the symmetry axis of a Kerr black hole, normalized by its value C0 at a = 0.In the extremal limit, the EM flux vanishes as a result of the Black hole Meissner effect (Fig. 3).

FIG. 3.
If a charge q is placed in a spatial superposition of proper separation d, the "which path information" is contained in the difference of the Coulomb fields, which is field of a dipole p = qd.Here we show the electric field lines (level sets of electric flux through a surface of revolution) for a dipole at r0 = 3M on the symmetry axis of a non-rotating (left) and maximally rotating (right) Kerr black hole.(The black hole region r < M + √ M 2 − a 2 is shown in black.)In the extremal case, the field does not penetrate inside (the black hole Meissner effect), and correspondingly there is no horizon-induced decoherence.

II. KLEIN-GORDON FIELD WITH A CLASSICAL SOURCE
In this section we review the Fock quantization of the Klein-Gordon field in the style of [7], first in the free case and then in the presence of a semiclassical source.This establishes notation and physical interpretation for our analysis of the superposition experiment.We work in a globally hyperbolic spacetime in coordinates (t, x), where t is constant on a family of Cauchy surfaces and x refers to a collection of spatial coordinates.We consider the case of three spatial coordinates, but the modifications for other dimensions are trivial.
The free KG equation is where □ = g µν ∇ µ ∇ ν and ∇ is the metriccompatible covariant derivative.Given two solutions ϕ 1 and ϕ 2 , the KG product is defined as where an overbar denotes complex conjugation and n µ is the future-directed unit normal to a spacelike Cauchy surface with induced metric h µν and volume element √ hd 3 x.The KG product is independent of the choice of surface, meaning in particular that its value is independent of t.
We consider a complete set of mode solutions ϕ i (t, x) normalized as Here the index i represents a collection of continuous or discrete indices, with δ ij representing a product of Kronecker and/or Dirac deltas as appropriate.If a solution is expanded as then its "positive-frequency part" is The map K depends on the choice of mode functions.The KG product of a real solution (with itself) is always zero, but the KG product of its positivefrequency part is given by The quantum theory in the Heisenberg picture is obtained by promoting ϕ to a self-adjoint operator satisfying the field equations, □ φ = 0, (11) expressing the general solution in terms of a complete set of mode functions imposing ladder-operator commutation relations on the coefficients âi , and building a Fock space in the usual way.The product â † i âi is interpreted as the number operator for ϕ i -particles.The notion of a particle thus depends on the choice of mode function.This procedure is equivalent to canonical quantization [8].Now suppose that there is a fixed classical source ρ(t, x), so that the Heisenberg-picture field now satisfies The form (12) is no longer a solution of the equations of motion; instead, we must add a particular solution to (14).This choice adds another freedom in the construction of the Fock space, beyond the choice of mode function.The choice of mode function determines the notion of particle, while the two choices together-mode functions and particular solutionjointly determine the state of the field in the "vacuum" annihilated by âi .For the semiclassical source to be the "only influence" on the state of the quantum field, the natural condition is the absence of incoming radiation, corresponding to the (classical) retarded solution ϕ ret ,6 where 1 is the identity.We again impose the commutation relations (13).The operators âi define an "in" Fock space F in with an associated "in" vacuum, which has no in-particles.However, this notion of particle is unnatural at late times, since a field configuration with outgoing radiation would be deemed to have no particles.Instead, at late times the natural expression of φ is where the use of the advanced field ϕ adv guarantees that the b-vacuum has no outgoing radiation, as desired if one wishes to have a quantized description of outgoing radiation.These operators define an "out" Fock space F out with an associated "out" vacuum, bi |out⟩ = 0, (18) which has no out-particles.Notice that we use the same mode functions ϕ i at early and late times.This in effect neglects any particle creation due to the spacetime itself.Such effects can easily be added on with a Bogoliubov transformation.
In general the source ρ will create particles in the sense that ⟨in| Nout |in⟩ ̸ = 0, where Nout = b † i bi .We can express the particle creation by noticing that where α i are the coefficients of the retarded-minusadvanced solution, Since the retarded-minus-advanced solution is a homogeneous solution to the Klein-Gordon equation, these coefficients α i are constants independent of time (and space).It then follows that the expected number of particles created in each mode i is given by In light of Eqs. ( 9) and ( 10), we see that the total expected particle number is given by That is, the expected number of particles is the KG product of the positive-frequency part of the retarded-minus-advanced solution.
In the next section we will consider the response of the KG field to a quantum superposition of semiclassical sources that agree at early times.For this purpose it is useful introduce the Schrödinger picture defined relative to some early time t 0 .We denote the Schrödinger-picture operator by φ0 and the initial Schrödinger-picture state by |ψ 0 ⟩, The Schrödinger-picture state evolves as where the quantum-mechanical propagator Û satisfies with φ given in Eq. (15).

III. THE SUPERPOSITION EXPERIMENT
Let H matter denote the Hilbert space of the matter degrees of freedom manipulated by our intrepid experimentalist.She prepares a superposition state with the following properties.First, the constituent states are perfectly distinguishable, Second, their charge densities can be treated semiclassically, Here ρ is the charge operator and ρ I is its expectation value in the (normalized) state |Ψ I ⟩.Third, the two semiclassical charge densities are identical at sufficiently early and late times, We name the states left and right since we imagine a spatial superposition, following DSW.Such a superposition can be achieved if matter with an embedded spin is sent through a Stern-Gerlach apparatus, held in spatial superposition for a time T , and then sent through a reversing Stern-Gerlach apparatus [2].However, in the KG case the charge of the particle is not conserved, and we will actually consider the simpler situation where the "left" and "right" states are two different time-evolutions of the charge, without any spatial separation.These distinctions are unimportant until Sec.V below.Now consider the evolution of the system including the Klein-Gordon coupling.By the assumption of no backreaction, each state |Ψ I (t)⟩ in the superposition induces a corresponding evolution |ψ I (t)⟩ in the field via the corresponding semiclassical source ρ I .Because the sources agree at early times, the corresponding retarded solutions ϕ ret I agree at the initial time t 0 .Using a fixed set of mode functions ϕ i , the Shrödinger-picture field φ [see (23) and (15)] is therefore independent of the choice of right or left and can be identified with the single Shrödinger-picture field operator in the superposition experiment.Similarly, the time-independent annihilation operators âi define the shared initial state of the field via This construction results in an initial tensor product state for the joint evolution, The interpretation is that Alice has carefully isolated her apparatus.As she conducts the superposition experiment, the photon field reacts differently do the right and left branches, and the system as a whole evolves as If the left and right field states become orthogonal, then the superposition has decohered.
To express the evolution of the field states we add an index to the propagator Û , Whereas in Sec.II Û was the propagator for the problem with a fixed semiclassical source, here ÛI is the evolution operator for a particular part of a particular state.More precisely, if the matter states |Ψ⟩ evolve by Û, then The Heisenberg-picture operators of Sec.II depend on the source and are not associated with any picture (Heisenberg or otherwise) in the superposition problem.Instead, these are viewed as useful definitions to obtain information about ÛI .From Eq. ( 15), we have and from (23) we have Our goal is to calculate where in light of Eq. ( 34), we define To find this operator we note that since the LHS of (36) is independent of I, we have where we have suppressed the (t, x) dependence of the fields and the (t, t 0 ) dependence of the propagators.From (35) we have which combined with (40) gives At late times t → ∞, the LHS is D † φL D. Since the advanced fields agree at late times, we may add ϕ adv L − ϕ adv R to the RHS in this limit.We thus have where we introduce Eq. ( 43) holds at all times, although D was defined using a late-time limit.At late times, Eq. ( 43) reduces to D † φL D = φR , showing that D displaces the late-time left field to the late-time right field.
The form (43) is convenient because the presence of the retarded-minus-advanced fields makes ∆ϕ a source-free solution for all times.Its mode coefficients are therefore time-independent, and we can think of D as a time-independent displacement operator.According to the general theory of coherent states [9], every operator that obeys (43) may be represented as where ∼ = indicates equality up to phase and α I i are the time-independent mode coefficients of (ϕ ret I − ϕ adv I ).Using (31), one can then calculate the overlap (37) as Recall from ( 21) that i |α i | 2 can be interpreted as the expected number of particles produced by the semiclassical source.Here it is the left-right difference α R i − α L i that appears, which by linearity can be interpreted as the expected number of particles produced by the difference of the left and right semiclassical sources.Since these particles evidently determine the extent to which the system decoheres, it is natural to call them entangling particles following DSW.We therefore define ⟨N e ⟩, the "expected number of entangling particles", as which is also the KG product of the positivefrequency part of the left-right difference of the retarded minus advanced solutions (44), Putting everything together, Eqs.(37), (46), and (47) become This reproduces the physical picture of Ref. [2] and DSW: if the experiment emits entangling particles, the superposition decoheres.Notice that nearcomplete decoherence occurs with the emission of just a handful of particles.
In computing ⟨N e ⟩ using (48), the KG product may be computed on any Cauchy slice, since ∆ϕ is a homogeneous solution.However, if we use a slice at late times, where the advanced solutions agree (ϕ adv L = ϕ adv R ), then only the retarded solutions appear.That is, the expected number of entangling particles is the KG product of the positive-frequency part of the difference between the left and right retarded solutions, evaluated at late times.This is the form of the result obtained in DSW.

FIG. 4. Bifurcate Killing horizon
In calculations we will find it convenient to instead push the Cauchy surface to early times, where (for the Unruh state) the non-vanishing mode functions are purely positive frequency.In this case it is the retarded solutions that agree (ϕ ret L = ϕ ret R ), and so only the advanced solutions appear.That is, we may also compute ⟨N e ⟩ as the KG product of the positive-frequency part of the difference between the left and right advanced solutions, evaluated at early times.

IV. ENTANGLING PARTICLES ON A BIFURCATE KILLING HORIZON IN THE UNRUH STATE
We now restrict to spacetimes with a bifurcate Killing horizon (Ref.[10] and App.B), in which a future horizon H + and a past horizon H − intersect at a bifurcation surface B. We label the horizon generators of H ± by their coordinates x A on B and use an affine parameter U ∈ (−∞, ∞) on the past horizon and V ∈ (−∞, ∞) on the future horizon, such that U = V = 0 is B. Thus H + is described by coordinates (V, x A ) and H − is described by coordinates (U, x A ).In black hole spacetimes with the conventional definitions of U and V , the regions of H ± bordering the exterior are U < 0 on H − and V > 0 on H + .We will denote these portions as H E ± .We denote the horizon surface gravity by κ and introduce Killing time coordinates u and v by In particular, These coordinates u and v extend into the conventional exterior region of a black hole spacetime.Some of these properties are illustrated in Fig. 4.
A. KG product in terms of affine-time Fourier transform The formula (6) for the KG product holds on a spacelike surface.We will be interested in the limit where a portion of this surface approaches the past horizon H − .One can check that the limiting product is where dS is the volume element on the bifurcation surface B, while ℓ µ = dx µ /dλ is tangent to the horizon generators x µ (λ) with λ some parameter that is constant on Consider now a set of modes ϕ i used for quantization.Suppose in particular that the subset of such modes that are non-vanishing on the past horizon are purely positive-frequency with respect to affine time U when evaluated there.(This situation arises for black holes formed from gravitational collapse; we will describe it as working in the "Unruh state" [11]).Describing each non-vanishing mode by an affine frequency Ω > 0 and a set of (continuous or discrete) spatial indices L, we have where Y L is some complete set of modes for functions f (x A ) which are orthonormal with respect to dS,7 The prefactor (4πΩ) −1/2 in (53) guarantees that the modes are properly normalized in the sense of (7).
The mode coefficient c ΩL is then evaluated to be Noting that c ΩL is proportional to the affine-time Fourier transform of each spatial mode of the field, we can express the Fourier transform as a sum over modes, where our Fourier conventions are Integrating over the spatial directions, we have using the orthonormality of the harmonics.The KG product of the positive-frequency part of a field ϕ is thus The range of the Ω-integral follows from the fact that the mode functions are purely positive frequency (defined only for Ω > 0).
The future-horizon analog of Eq. ( 60) was used by DSW in the electromagnetic and gravitational cases in the Schwarzschild spacetime [1].The futurehorizon version holds for the Hartle-Hawking state, but DSW argue that the associated decoherence results also apply to the Unruh state, citing lowfrequency equivalence between the two.
We will find it helpful to introduce another form of Eq. (60).Using the definition (57) of the Fourier transform, we can introduce two integrals over affine time, This last integral can be performed using the iden- which is easily checked by contour integration.Here H(ω) is the Heaviside function, equal to 1 for ω > 0 and 0 for ω < 0. Inverting this Fourier transform, we find which is the needed integral.Eq. (60) then becomes This formula also arises in the rigorous approach to QFT used in [10] -see equation 4.13 therein.

B. KG product in terms of Killing-time Fourier transform
Up to now we have considered the entirety of H − , including both the "exterior" U < 0 and "interior" U > 0. However, our application of the formula involves fields that are non-zero only in the exterior part U < 0, denoted H − E .For such fields we may reduce the integration range of U 1 and U 2 to negative values only, and we may change variables to the Killing time u related by U = −e −κu (50).After some algebra we obtain where ϕ is evaluated on H − E and regarded as a function of (u, x A ).We write H − E on the LHS to remind the reader that this integral ranges over only the exterior portion of the past horizon.However, the full horizon is still in some sense involved through the K map, which refers to positive frequency with respect to U ∈ (−∞, ∞).
Next we reintroduce the Fourier transform, this time with respect to Killing time, We may now invert for ϕ| H − E (u, x A ) and plug in to both instances of ϕ in Eq. (65).After changing variables to u = u 1 and ∆ = u 2 − u 1 , we see that the u integral reduces to a delta function in frequency, leaving The last integral may be performed by contour integration, giving which may be compared with (60).
From Eqs. ( 48), (68), and (66) and (44) the contribution from H − E to the expected number of entangling photons is given by where ∆ φ is the Killing-time Fourier transform of ∆ϕ (44) evaluated on the horizon, The left and right retarded solutions agree at early times, so they do not contribute to ∆ϕ when evaluated on Eq. ( 69) has a thermal interpretation.If the calculation were repeated using Boulware modes (positive frequency with respect to u on H − E ) to quantize the field in the exterior, then Eq. (69) would appear without the coth factor.The difference between the Unruh case under consideration and the Boulware case [12][13][14][15] is thus an integral multiplied by coth(πω/κ) − 1, which is a thermal factor 2/(exp[ω/T H ]−1) where T H = κ/(2π) is the horizon temperature.

C. Estimate of entangling particle number
For the superposition experiment in a spacetime with sufficient decay properties, the left and right fields will agree at early and late times, meaning that ∆ϕ will vanish at early and late times.If the superposition is maintained for a sufficiently long time, then the left and right fields will be roughly constant over most of the time the superposition is maintained, and ∆ϕ is similarly constant over a corresponding lapse of time.We will denote this constant by ∆ϕ(x A ). Again assuming the relevant decay properties, it can be computed from solutions where the sources persist forever, where ϕ stat R and ϕ stat L are stationary (invariant under the horizon-generating Killing field8 ) solutions with stationary sources ρ stat R and ρ stat L corresponding to the superposition.We write evaluation on a horizon cross-section C (as opposed to the past horizon H − E ) to emphasize that these are stationary solutions.We include a minus sign in the definition (72) to match the minus sign in (71), which originates from the use of retarded minus advanced fields in the difference field (44).This is purely a cosmetic issue since only the square of ∆ϕ will appear in the decohering flux, Eq. (82) below.
We may thus think of ∆ϕ as transitioning from zero to ∆ϕ then back to zero.We will work with Killing time u and denote the Killing timescales as follows: T b : The largest "background" timescale (associated with propagation on the background spacetime) T t : The "transition" timescale (associated with opening or closing the superposition) T : The timescale for which the superposition is maintained Finally we define which represents the timescale for the field on the horizon to transition.
Our goal is to estimate the integral (69) in the regime To do so we introduce an intermediate frequency ω c satisfying For example, if 1/T ∼ ϵ then we can take ω c ∼ ϵ 1/2 as ϵ → 0. We split the integral (69) into two pieces, with In the first integral (77) we have ωT ≪ 1 on the entire range of integration, so that variations in ∆ϕ on scales ≲ T are not visible in the Fourier transform.In this regime we may approximate ∆ φ as the Fourier transform of a sharp top hat function of height ∆ϕ(x A ), Using this approximation in (77), we find where is the constant we call the decohering flux.In passing from (80) to (81) we made the change of variables λ = ωT .
The analysis now bifurcates according to whether κ vanishes or not.For κ ̸ = 0 we may replace coth x by its small-x behavior 1/x in the large-T limit, noting that ω c T → ∞ with the intermediate scaling (75) and using the integral Alternatively, in the κ → 0 limit we may replace coth x by its large-x value of 1, where in the second step we drop constant terms, including log ω c .In both cases the arbitrary scale ω c disappears from the leading behavior at large T .
Since our calculation assumes a bifurcate Killing horizon, strictly speaking it applies only for non-zero κ.The expression "κ = 0" in (86) (and elsewhere below) is shorthand for the assumption κT ≪ 1, which establishes the hierarchy T ≪ T ≪ 1/κ.One might worry that the Aretakis instability [16][17][18] will make this hierarchy difficult to achieve, since growth occurs on timescales of order κ −1 for near-extremal black holes [19].However, the field itself decays during this time, albeit at a slow, power-law rate.Now consider the second integral (78).For analyzing this integral it is helpful to write ∆ϕ as where g 1 (u, x A ) and g 2 (u, x A ) are smooth, Tindependent functions that transition from 0 to ∆ϕ(x A )/2 in a region of size ∆u ∼ T near u = 0.These functions describe the details of the transition periods, including (for example) oscillations from quasi-normal mode ringing.The Fourier transform of (87) is where g1 (ω, θ, ϕ) and g2 (ω, θ, ϕ) are the Fourier transforms of g 1 and g 2 , respectively.The magnitude squared is thus In the second integral (78) we have ωT ≫ 1 everywhere on the domain of integration and by (89) the T -dependence of |∆ φ| 2 is only in the form of a rapid oscillation.These oscillations integrate to zero, leaving Although the integrand is now independent of T , the integral can still depend on T through the lowerlimit ω c , which has an intermediate scaling, such as ω c ∼ 1/ √ T .However, we have already seen that ω c disappears from the leading, large-T approximation for the other integral I 1 (linearly diverging in the non-extremal case, and logarithmically diverging in the extremal case).Since the full integral I 1 + I 2 is manifestly independent of ω c , it follows that I 2 is also independent of ω c at these orders in the large-T expansion.But since I 2 can only have T -dependence through ω c , we conclude that I 2 makes no contribution at these orders in the large-T expansion.
The leading large-T behavior of ⟨N e ⟩ (76) thus comes entirely from the low-frequency integral I 1 .Collecting the results from Eqs. ( 84) and (86), we have holding for T ≫ T .The log of a dimensionful quantity appears because we have dropped subleading constant contributions; these depend on the details of the transitions.
Recall from (49) that the exponential of −⟨N e ⟩/2 gives the final value of the of the inner product |⟨ψ L |ψ R ⟩| of the left and right states of the experiment.If there are no significant contributions to the entangling photon flux from other boundaries, then Eq. (91) provides the decoherence rate.We see that in the non-extremal case the dependence on the separation time T is exponential, which we can interpret as saying that the state decoheres exponentially.On the other hand, in the extremal case we have a slower, power-law decoherence where we now include an undetermined constant T 0 .This constant is non-universal and depends on the details of the transition.By contrast, the decohering flux C can be determined from the stationary field equations via Eq.(82).It remains to calculate the decohering flux in situations of physical interest.

V. OBSERVER ON THE SYMMETRY AXIS OF A ROTATING BLACK HOLE
The Kerr metric in Boyer-Lindquist coordinates reads where ∆ = r 2 +a 2 −2M r and Σ = r 2 +a 2 cos 2 θ.The horizon radius r + , angular velocity Ω H , and surface gravity κ are given by The Boyer-Lindquist coordinates are not regular on the horizon.To describe the past horizon H − we introduce where dr * /dr = (r 2 +a 2 )/∆ and dr # /dr = a/∆.See Eqs.(5.62) and (5.63) in Ref. [20] for explicit expressions for r * and r # .The coordinates (u, r, θ, ψ − ) are regular on the past horizon, but they are not constant on its generators, which instead rotate with angular velocity Ω H .We therefore introduce an additional angle which remains constant on the generators of the past horizon.We will refer to the set (u, r, θ, φ) as "horizon-adapted coordinates".These satisfy the construction described in Sec.IV with x A = (θ, φ).
The induced metric on the horizon is where the + indicates evaluation at r = r + .The area element is The horizon-generating Killing field takes the form The partial derivatives in the first equality refer to Boyer-Lindquist coordinates, whereas the partial derivative in the second equality refers to horizonadapted coordinates.
The state of a quantum field near a black hole formed from gravitational collapse is described at late times by the Unruh vacuum [21], which corresponds to choosing a set of modes that are positive frequency with respect to affine time U when evaluated on the past horizon H − (along with additional modes that vanish there).Using this same set of modes in the framework of Sec.III describes the superposition experiment near a black hole formed from gravitational collapse.Sec.IV showed that under these circumstances the decoherence rate is determined by the integral of the square of the stationary difference field on the horizon, ∆ϕ, as in Eq. (82).Our task is to compute ∆ϕ in cases of physical interest.
We will consider point sources described by a charge q(τ ) on a worldline x µ (τ ) (where τ is proper time), Notice that unlike in the electromagnetic case, the charge can be time-dependent.For a constant point charge q 0 held at radius r 0 on the symmetry axis of a Kerr black hole, we have where α = √ g tt is the "redshift factor", The stationary solution ϕ c = ϕ c (r, θ) satisfies The solution has been given in integral form in Ref. [22] as Eq. ( 10) with s = 0. (Our ϕ c is identified with −2αq 0 f in that reference.)Evaluating the integral gives where R is defined as Notice that the constant prefactor in the KG Coulomb solution (110) is the redshifted charge αq 0 .In particular, the charge inferred at infinity as the coefficient of the 1/r falloff is αq 0 , which is not equal to the locally defined charge q 0 .As an extreme example, when a charge q 0 is slowly lowered into the black hole, the field everywhere outside the black hole decreases in strength and ultimately vanishes when the charge enters the horizon.There is no obstacle to this physical process since there is no conservation law for the scalar charge.The disappearance of the field is a manifestation of the "no scalar hair" property of black holes [23].
These properties make the KG spatial superposition experiment behave differently from the EM case.To adhere most closely to the DSW electromagnetic setup, we would consider a point charge q 0 held in superposition at two slightly different radii r L and r R , which results in a dipolar difference source.However, if the dipole is defined in the Lorentzinvariant way as the charge q 0 times the proper distance d, then the field actually falls off like 1/r at infinity, not as 1/r 2 as it does in electromagnetism.In essence, the KG dipole still behaves like a monopole.
We therefore find it more convenient to simply work with a monopole superposition, which is possible since Alice can manipulate the value of the KG charge as a function of time.That is, we consider the conceptually simpler experiment in which the particle position is definite but the value of its charge is placed in superposition and later restored to the original value.We will still use the labels "left" and "right", but the two semiclassical sources share a single position r 0 , differing only in the timeevolutions q R (τ ) and q L (τ ) of their charges.The difference source ∆q = q R − q L evolves from zero to some value Q and back.The horizon difference field ∆ϕ (72) is given by the difference of Coulomb fields on the horizon, Using (104) we may compute the decohering flux as where we remind the reader that Q is the difference in charge between the branches of the superposition.
In general there are also contributions to the decoherence from null infinity.In App.A we show that the expected number of entangling particles on I − at large T is given by⟨N e ⟩ I − = 2C ∞ log T (A18), where C ∞ is the analogous decohering flux on null infinity (A19).From Eq. ( 110) we compute the difference of Coulomb fields at r → ∞ to be ∆ϕ ∞ = −αQ, and the integral (A19) gives This gives rise to a power-law contribution to large-T decoherence going as (T /T 0 ) −C∞ , similar to the extremal horizon result (93).
In the non-extremal case the decoherence from the horizon is exp[−(1/2)CκT ] by ( 92) and always dominates the power-law contribution from null infinity.On the other hand, in the extremal case the horizon's contribution is also a power law, and the dominant effect is the one with the larger decohering flux.Comparing ( 113) and ( 114), we find (as expected) that the horizon dominates when the observer is sufficiently near the horizon, while infinity dominates when the observer is sufficiently far away.Precise equality occurs at the critical value We began this paper by reviewing a heuristic argument involving Bob making measurements from behind a horizon, which suggests the presence of a decoherence effect associated with horizons.The precise calculation shows there is also a decoherence associated with null infinity (in the KG case), and it is interesting to consider whether a similar heuristic applies.While no single Bob can gather information at infinite distance, we can still imagine an army of Bobs stationed on a distant sphere in the distant future, i.e., a congruence of observers approaching null infinity.The question of whether the Bobs gather finite information in the limit depends on falloff conditions, and we do not attempt to analyze it directly.However, the non-zero decoherence result suggests that for the KG field, the Bobs can indeed gather which-path information from far away.It would be very interesting to make this correspondence precise.

VI. ELECTROMAGNETIC DECOHERENCE
The EM case is highly analogous to the KG case, except for some additional subtleties related to the choice of gauge.We will first establish some some general properties of a horizon-adapted gauge, before proceeding to perform the calculation.

A. Horizon-adapted gauge
Following DSW we work in a gauge where A µ vanishes when contracted with the horizon generator.Since we work on the past horizon, the condition takes the form In the horizon-adapted coordinates of Sec.IV we may write or We will call this a horizon-adapted gauge.
In App.B we show that, in the horizon-adapted gauge, the pullback of Maxwell's equation to the horizon takes the form of a total derivative, ∂ ∂u Integrating then gives in terms of an integration constant f .Under the residual gauge freedom We can thus utilize the residual gauge freedom to make f (x A ) lie complement of the image of the Laplacian.By theorem 4.13a of Ref. [24], for compact manifolds (in our case, the horizon cross-section C) this complement is just the set of constant functions.We may thus set f to be a constant, After integration, we see that f 0 is proportional to the charge enclosed in the compact horizon, where dS is the natural area element on C and A = C dS is the cross-sectional area.If there is no charge in the compact horizon then we have simply For non-compact horizons with suitable falloff conditions, we expect that the complement of the image of the Laplacian will either be empty or can be removed with a similar physical assumption.
For simplicity we will assume that f 0 = 0 so that (123) holds.However, it is trivial to repeat the analysis of the paper with f 0 included.One simply assumes that the left and right branches of the superposition share the same f 0 , which corresponds physically to the statement that the superposition was created from a single initial state.(In the compacthorizon case, f 0 is interpreted as total charge.)The constant f 0 then cancels out of the right-left difference field and hence does not appear in the final decoherence rate.This cancellation occurs in our treatment of null infinity in App.A 3, where the charge cannot be assumed to vanish separately in the left and right branches.
A second equation analogous to (123) can be derived much more easily.From the definition of the field strength we have Contracting this equation with ϵ AB produces a magnetic analog of (123).Displaying these two equations together gives a pleasing pair, The right-hand-sides provide invariant notions of the radial electric and magnetic fields.
These equations suggest that both components of A A can be determined from the field strength F µν at each time u.We can make this explicit by introducing electric and magnetic Hertz potentials E(u) and B(u), Each potential is then sourced by its corresponding radial field, where ∇ 2 = q AB ∇ A ∇ B is the Laplacian on the horizon cross-section.We define the inverse Laplacian ∇ −2 so that the integral of ∇ −2 f vanishes for any f in the domain.(This is always possible by theorem 4.13e of Ref. [24].)We then invert as We expect similar expressions to hold in the noncompact case provided that there are suitable falloff conditions.

B. Entangling photons
With the above facts established, the KG analysis of Secs.II-V generalizes straightforwardly to EM fields.The modes satisfy the free Maxwell equation, and are normalized according to gauge-invariant, 9 surface-independent product In the presence of a source J µ we have We make the same assumptions of Sec.III with sources J µ R and J µ L instead of ρ R and ρ L , and use identical arguments to show the decoherence rate is given by e −⟨Ne⟩/2 (49) with 9 Under Aµ → Aµ + ∂µΛ, Eq. ( 133) changes by boundary terms, which are the integrals of F 1 Λ and F 2 Λ.We do not address these subtleties in the mode quantization argument, but the final results involve the inner product evaluated on solutions where such terms would vanish-see discussion below Eq. ( 136).
analogously to (48).Here ∆A is the right-left difference of the retarded-minus-advanced fields, where we suppress the tensor index and use notation analogous to (44).Note that the associated field strength ∆F is the retarded minus advanced solution arising from a source J µ R − J µ L of compact spacetime support.This ensures that ∆F has compact spatial support on any Cauchy surface, and therefore that the product (135) is gauge-invariant.
The steps of Sec.IV also generalize.On the horizon in the horizon-adapted gauge we have which closely parallels the corresponding KG expression (52).The ensuing steps proceed identically until we establish the analog of (69), where with ÃA is the Killing-time Fourier transform on the Horizon [analogous to (70)], In the KG case we proceeded to estimate the integral based on the assumption that ∆ϕ transitions from zero to a constant ∆ϕ and back, with the transitions occurring over a timescale T and the constant period occurring for T ≫ T (Fig. 1).In the EM case the same assumptions will be valid for the field strength F µν , but the behavior in a given gauge A A need not mimic that of the field strength.However, we have shown in Sec.VI A that the horizonadapted gauge can be constructed locally in time from the field strength (at least for compact horizons, and presumably in the non-compact case with suitable falloff), meaning that it will in fact share the properties described above.We may thus perform the estimate identically, arriving at the exact same results (92) and (93), where the decohering flux C is now given by analogously to (82).The quantity ∆A is defined analogously to (72), where (with tensor indices suppressed) A stat To derive the formula for C given in the introduction as Eq. ( 4), we use the solution (130) and (131) for the electric and magnetic potentials in the decomposition (A46) for the horizon vector potential.Plugging this decomposition into (142), we find that cross-terms are total derivatives.These may be dropped for a horizon of compact support like that of the Kerr spacetime, and also for noncompact horizons in spacetimes with suitable decay properties (such as Rindler) given the assumption of a localized source outside the horizon.

C. Calculation in Kerr
The charge-current of a point source in a curved spacetime is given by For a stationary charge on the symmetry axis of a Kerr black hole, we have The stationary, regular solution with charge e can be called the Coulomb field F c µν .A vector potential is given in closed form in Eq. ( 9) of Ref. [25].However, this potential is poorly behaved on the horizon and its well-behaved components are not in the horizonadapted gauge.While we could in principle proceed by finding an explicit gauge transformation to a regular, horizon-adapted gauge, we find it more convenient to compute the gauge-invariant field strength and reconstruct a a suitable A c A by solving Eqs. ( 125) and (126).In the axisymmetric case of present interest, these equations become holding on the horizon r = r + .The integrals can be done in closed form, and after some calculation we find imposing regularity on the pole to obtain a unique solution.
Eq. (148) refers to the Coulomb field of a single point charge e.For the superposition experiment, a charge e is held in a spatial superposition of two nearby radii r 0 ± ϵ/2.We require the difference of the two vector potentials, which by linearity is the vector potential due to the difference of the sources.Since the proper separation is √ g rr ϵ, the difference source is an effective dipole p = e √ g rr ϵ.In the limit ϵ → 0 (fixing p) the difference field is Using Eq. ( 148) we can obtain ∆A (143) and evaluate the decohering flux integral (142).We find This function is plotted in Fig. 2 as a function of black hole spin.
We can obtain simpler expressions in a few different limits.First we we consider the non-spinning case a = 0, We can further expand at large r 0 , C = 32 3π This expression may be compared to analogous results in Ref. [1], which quotes a decoherence timescale in Eq. ( 15).In light of Eq. ( 92) we define the decoherence timescale T D = 2/(Cκ).Using κ = 1/(4M ) and Eq. ( 152) for C, and noting that our p equals their qd, we confirm the scaling of Eq. ( 15) of Ref. [1].Noting our units ϵ 0 = 1/(4π), we find that their Eq.( 15) should contain a prefactor 3π 2 in order to correspond to our definition of T D .This is the precise version of their order of magnitude estimate.
Alternatively we can consider a near-horizon limit.Letting b = 2 2M (r 0 − 2M ) + O((r 0 − 2M ) 3/2 ) denote the proper distance to the horizon, we find This may be compared with analogous results for Rindler spacetime in Ref. [4], noting that the acceleration a of a near-horizon observer is related to proper distance b by a = 1/b + O(b).Since Ref. [4] quotes a rate in proper time, we also use κT = aτ + O(b) to convert our decoherence rate T D to a proper decoherence rate τ D = T D (κ/a) = 2/(Ca).This τ D may be compared to T D in Eq. (3.27) of Ref. [4]; we confirm the scaling and find that a prefactor of 12π 2 is required to agree with the precise rate.
It is also instructive to look at the large r 0 limit at finite spin, Finally, we consider extremal limits.Introducing ϵ = 1 − a 2 /M 2 and letting ϵ → 0 at fixed M and r 0 , we find The flux vanishes like ϵ 2 , which is linear in a, as seen in Fig. 2.
Alternatively, we can take the extremal limit while simultaneously approaching the horizon.Defining x 0 = (r 0 − r + )/r 0 and taking the extremal limit ϵ → 0 with x 0 ∼ ϵ gives a finite limit, If time is also rescaled like 1/ϵ, then this limit applied to the metric yields the the"nearNHEK" patch [26] of the NHEK metric [27].Here we have used the definitions of Ref. [28].
In the case of KG fields there was also a contribution to decoherence from null infinity.This occurs because the Coulomb fields are non-zero at infinity in the sense that the 1/r part of the difference field is non-zero.This occurs for the monopole case we studied as well as the KG dipole case, as discussed in Sec.V.However, in the EM case the dipole field decays like 1/r 2 and there is no contribution to decoherence from null infinity.This may be seen explicitly from the general expression (A57) derived in App. A. butions from null infinity.We consider the KG and EM cases separately.We find that there is a logarithmic contribution in the KG case, whereas the EM case is finite as T → ∞.

Klein-Gordon product
As already presented in the main text as Eq. ( 6), the KG product of two source-free solutions ϕ 1 and ϕ 2 on a timeslice Σ t is defined as where n µ is the future-directed unit normal to Σ t , and h is the determinant of the induced metric.It follows directly from the source-free Klein-Gordon equation that the inner product is independent of the surface Σ t .
We may approach the past horizon by letting r → r + at fixed {u, θ, φ}, where these coordinates were defined in Sec.V. Since t → −∞ in this limit, it represents a portion of the limiting Σ t .In this limit, a direct computation gives where dS is given in Eq. (104) and ∂ ∂u µ is the coordinate basis vector in the horizon-adapted coordinate system {u, r, θ, φ} (also equal to the horizongenerating Killing field).The contribution to the KG product in this limit is thus where the partial derivative refers to horizonadapted coordinates.
We may approach past null infinity by letting r → ∞ at fixed (v, θ, ϕ), where v is the advanced time v = t + r * .Since t → −∞ in this limit, it also represents a portion of the limiting Σ t .In this limit, a direct computation gives where dΩ = sin θdθdϕ is the 2-sphere element and ∂ ∂v µ is the coordinate basis vector in the advanced coordinate system {v, r, θ, ϕ}.The volume element diverges, but this will be canceled in the inner prod-uct by falloff of the fields,10 ϕ(v, r, y A ) = ϕ| I − (v, y A )r −1 + o(r −1 ).(A5) (We adopt the notation that evaluation of a scalar field at I − represents the 1/r part.)The contribution to the KG product in this limit is thus where the partial derivative refers to advanced coordinates.The fields in the integrand are understood as the r −1 parts defined in Eq. (A5).
Finally we consider the pieces of the limiting Σ t that are not captured by the horizon or null infinity.Colloquially, these are the "contributions from past timelike infinity".Although we do not formalize a proof, it is clear that these pieces do not contribute to the limiting KG product.Extensive numerical evidence (beginning with [30]) shows that solutions with compact sources decay polynomially in time at fixed spatial coordinate.Rigorous decay results have also been obtained in hyperboloidal slicings (Ref.[31], Corollary 3.1).The time-reverse of these results applies to decay of the advanced field at early times, such that the retarded-minus-advanced field we consider should decay as t → −∞ away from H − and I − .We therefore assert that there is no contribution from timelike infinity,

Decoherence from Null Infinity
We now consider the superposition experiment setup of the main body.In Secs.IV and V we evaluated the contribution to decoherence from the past horizon, first in a general setup and then for an observer on the symmetry axis of the Kerr spacetime.We now repeat this analysis for the contribution from null infinity.The steps are highly analogous.
For the Unruh state, the non-vanishing mode functions are positive frequency with respect to v, Following [10], we assume there exists a oneparameter isometry whose fixed points form an orientable, spacelike, co-dimension 2 (hence 2dimensional) surface B. Let ℓ µ and n µ be two (continuously chosen, future directed) null normals B satisfying ℓ • n = −1.(For brevity we denote inner products by v µ w µ = v•w.)We extend n µ off of B via the affinely-parameterized null geodesic equation, The null surface generated by n µ is called the past horizon H − .The portion of H − in the causal past of B is the "exterior" past horizon H − E , while the part in the future is the "interior" H − I .An analogous construction can be made for the future horizon H + , but we will not use it in this appendix.In what follows, all equations are understood to be evaluated on H − E .Let ξ µ denote the generator (Killing field) of the isometry.As explained in Sec. 2 in Ref. [10], ξ µ is tangent to H ± .On H − E it is related by a positive function f , We introduce time coordinates u and U on H − E by We choose U such that the bifurcation surface B is at U = 0, while U = −1 coincides with u = 0.It is shown in Sec. 2 in Ref. [10] that where κ is the "surface gravity".The constancy of κ on horizon can be seen using the same proof in Sec.12.5 of Ref. [34].It follows immediately that The null geodesic congruence n µ is also normal to the horizon.It follows that the geodesic congruence is expansion-less, shear free, and twist-less, i.e., v µ 1 v ν 2 ∇ µ n ν = 0 (B7) for all v µ 1 , v µ 2 tangent to the horizon H − E .This result again can be directly seen from an anologous argument leading to Eq. (12.5.20) in Ref. [34].
If x A is a coordinate system on the bifurcation surface B, we obtain a coordinate system {U, x A } (hence {u, x A }) on H − E by letting x A be constant on the null geodesic generators.If we adjoin a fourth cooordinate r that is constant on H − E , then on H − E we have Since ξ µ is a Killing field, we have ∂ u g AB = 0 and hence This is equivalent to the statement that in the preferred coordinates (U, r, x A ), the components m µ are constant on horizon generators, In particular, m is tangent to horizon cross-sections; its only non-zero components are m A .
The inner products m • n, m • m and m • m are preserved by the Lie transport.For example, The second equality follows from Killing's equations ∇ µ ξ ν + ∇ ν ξ µ = 0 together with Eq. (B2). 15The third equality holds because m µ n µ = 0 and m µ m µ = 0 on H − E .Thus the conditions m • n = 1, m • m = 0, and m • m = 1 hold everywhere on H − E .To form a complete tetrad basis on the horizon, we choose ℓ µ to be the future-directed null vector that is uniquely specified by ℓ • n = −1 and m • ℓ = 0.This defines a complex null tetrad on the entire H − E .For future use we list the standard relations obeyed by a null tetrad.As shown above, the vectors are orthonormal, Since the dot products of tetrad members are constant, for any two tetrad members X, Y ∈ {ℓ, n, m, m}, we have This is equivalent to the anti-symmetry of the Ricci rotation coefficients.In particular, X µ ∇ ν X µ = 0.The metric is given by The volume elements for the 4-space ϵ and the crosssection ϵ (2) are given by We label the tetrad components of tensors with a lowered index; for example, ℓ µ n ν T µ ν = T ℓn .All tetrad components are seen as lower indices; no raising and lowering of tetrad indices appears in this work.Derivatives are written similarly; for example, ∇ ℓ = ℓ µ ∇ µ .When a derivative acts on a tensor with tetrad indices, the tetrad contraction is taken first.For example, for a tensor T µ1µ2 ν1ν2 , we might write ∇ m T µ1µ2 ℓn = m λ ∇ λ (T µ1µ2 ν1ν2 ℓ ν1 n ν2 ) .(B20) 15 Eq.(B2) holds only on the horizon, but in Eq. (B12) the indices are contracted with vectors m and n which are tangent to the horizon.

Cross-sections C
The spatial two-surfaces obtained by Lie transport of the bifurcation surface will be called cross-sections C. The induced metric on a cross-section may be written and satisfies q µλ q λ ν = q µν .(B22) Since q AB = g AB , by (B9) the components are constant on horizon generators, It follows from orthonormality of m that q AB = m A m B + m A m B is equal to the matrix inverse of q AB .The derivative compatible with q AB is denoted ∇ A .For any one-form v µ , the components v A satisfy Using the Leibniz rule we have in the notation of (B20).The two terms in Eq. (B26) can be simplified as follows: where c.c. denotes the complex conjugate of the preceeding terms on that line.We have used the antisymmetry of F µν , the vanishing of the projection of ∇ µ n ν onto the cross-section C [Eq. (B7)] (in particular, m µ m ν ∇ µ n ν = m µ m ν ∇ µ n ν = 0), antisymmetry of the Ricci rotation coefficients [Eq.(B16)], and the geodesic equation [Eq.(B1)].
FIG.2.The decohering flux C as a function of spin for an observer at r0 = 3M on the symmetry axis of a Kerr black hole, normalized by its value C0 at a = 0.In the extremal limit, the EM flux vanishes as a result of the Black hole Meissner effect (Fig.3).
in horizon-adapted gauge in the presence of stationary sources J stat R and J stat L (respectively) corresponding to the superposition.

) 2 .
Null tetrad on H − E We may obtain a null tetrad on B by selecting a normalized complex-null vector m (m • m = m • m = 0, m • m = 1) that is orthogonal to ℓ and n.We extend m µ to all of H − E by Lie transport along the null generators, L n m µ = [n, m] µ = 0. (B10)

4 .
Maxwell's equation on H − E We now consider the component of Maxwell's equations tangent to the horizon generators, n ν ∇ µ F µν = 0. (B25)