Particle motion near high-spin black holes

Particle motion near high-spin black holes

Daniel Kapec kapec@ias.edu School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA    Alexandru Lupsasca lupsasca@fas.harvard.edu Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA Society of Fellows, Harvard University, Cambridge, MA 02138, USA
Abstract

General relativity predicts that the Kerr black hole develops qualitatively new and surprising features in the limit of maximal spin. Most strikingly, the region of spacetime near the event horizon stretches into an infinitely long throat and displays an emergent conformal symmetry. Understanding dynamics in this NHEK (Near-Horizon Extreme Kerr) geometry is necessary for connecting theory to upcoming astronomical observations of high-spin black holes. We review essential properties of NHEK and its relationship to the rapidly rotating Kerr black hole. We then completely solve the geodesic equation in the NHEK region and describe how the resulting trajectories transform under the action of its enhanced symmetries. In the process, we derive explicit expressions for the angular integrals appearing in the Kerr geodesic equation and obtain a useful formula, valid at arbitrary spin, for a particle’s polar angle in terms of its radial motion. These results will aid in the analytic computation of astrophysical observables relevant to ongoing and future experiments.

Contents

I Introduction

Extremal black holes have served as a rich source of novel ideas and techniques in quantum gravity and field theory for several decades Strominger1996 (); Maldacena1997 (); Guica2009 (); Maldacena2016a (); Maldacena2016b (). These fundamental advances have led to a mathematical description of numerous interesting quantum-mechanical and gravitational systems, but have yet to connect directly with experiment. However, with the advent of a new generation of powerful astronomical detectors such as LIGO and the Event Horizon Telescope Abbott2016 (); EHT2019 (), a subclass of astrophysically realistic near-extremal black holes stands poised to bridge this gap between formal theoretical investigation and successful experimental verification. The near-extremal Kerr black hole exhibits a number of striking phenomena showcasing strong-field general relativity, and a confirmation of even the most basic, qualitative prediction derived from the emergent symmetries of its near-horizon region would mark a huge success for both theory and experiment. If high-spin black holes do exist and come within observational reach, they will provide a window into a region of our Universe that is qualitatively similar to the extensively studied Anti-de Sitter (AdS) spacetime, which plays an outsize role in the modern holographic perspective on quantum gravity.

The traditional approach to the modeling of astrophysical black holes is based on extensive numerical simulation across large swaths of parameter space. While this method of analysis is perfectly adequate in the general setting, it must confront new complications that arise in the specific regime of high spin. As a black hole rotates faster, it develops an increasingly deep throat that nonetheless remains confined within a small coordinate distance from the event horizon. As a result, resolving near-horizon physics requires a spacetime mesh of increasingly fine resolution as the spin grows. Meanwhile, the overall size of the grid must remain large in order to accurately capture the asymptotically flat region far from the black hole. Eventually, this large separation of scales can incur a prohibitive computational cost. Fortunately, the same phenomenon that renders the problem numerically intractable also enables the application of a complementary analytic method.

The analytic approach proceeds from the key observation that near extremality, the Kerr spacetime separates into two distinct regions. While the extreme Kerr metric resolves physics away from the horizon, the throat region is instead described by the dramatically simpler Near-Horizon Extreme Kerr (NHEK) geometry, which possesses additional symmetries and can be viewed as a spacetime in its own right. The enhanced symmetry of the NHEK metric fixes the behavior of many near-horizon processes and turns complex dynamical questions into considerably simpler kinematic ones Lupsasca2014 (); Zhang2014 (); Lupsasca2015 (); Compere2015 (); Gralla2016a (); Chen2017 (); Gralla2017b (); Gralla2018 (); Hadar2019 (). Once a problem has been solved in the NHEK geometry, it can be matched onto the far region in order to produce astrophysical predictions. Indeed, there is a growing body of work that seeks to exploit the symmetries of the NHEK region to constrain physical observables targeted by ongoing and upcoming experiments Porfyriadis2014 (); Hadar2014 (); Hadar2015 (); Hadar2017 (); Compere2018 (); Gralla2015 (); Gralla2016b (); Burko2016 (); Gralla2016c (); Compere2017 (); Gralla2017a (); Porfyriadis2017 (); Lupsasca2018 (); Gates2018 (). Many, if not all, of these analyses require approximate or exact solutions to the geodesic equation in NHEK and Kerr. While Kerr geodesic motion is an old and well-studied subject, most treatments reduce the problem to first-order form and then simply seek numerical integration of the equations. On the other hand, NHEK geodesic motion has received far less attention. Previous work has focused primarily on equatorial geodesics and those geodesics obtainable from the equatorial case through a symmetry transformation Porfyriadis2014 (); Hadar2014 (); Hadar2015 (); Hadar2017 (); Compere2018 (). In this paper, we present drastically simplified analytic expressions for the angular geodesic integrals in Kerr, and solve outright the full geodesic equation in NHEK. Our results will directly aid in the calculation of gravitational wave and electromagnetic signals from non-equatorial sources in the NHEK region, which are expected to be relevant for both current and future experiments.

The outline of this paper is as follows. In Sec. II, we revisit the problem of geodesic motion in the Kerr spacetime for arbitrary values of the black hole spin, and derive new, improved expressions for the motion in the poloidal plane. In Sec. III, we present a pedagogical review of the NHEK geometry and its origin as the near-horizon scaling limit of a (near-)extreme Kerr black hole. We then completely solve the NHEK geodesic equation in Sec. IV, working first in the global strip, then in the Poincaré patch, and finally in the near-NHEK patch. We classify all categories of geodesic motion in each coordinate system and obtain explicit formulas for the motion as a function of coordinate time in each case. We conclude with a description of how NHEK geodesics transform under the action of the isometry group. Appendix A gathers mathematical definitions needed in the main body of the text. Appendix B describes the dimensional reduction of the NHEK geometry to AdS with a constant electric field, and the projection of NHEK geodesics to the trajectories of charged particles in AdS subject to the Lorentz force exerted by the background electromagnetic field.

Ii Geodesics in Kerr

In this section, we review the standard treatment of geodesic motion in the background of a rotating black hole Bardeen1973 (); Chandrasekhar1983 (); ONeill1995 (). We begin in Sec. II.1 by re-deriving the Kerr geodesic equation in its first-order formulation. Then, we classify the different possible qualitative behaviors of the polar motion in Sec. II.2, before explicitly evaluating all the angular path integrals appearing in the geodesic equation in Sec. II.3. We take great care to unpack these integrals into sums of manifestly real and positive elliptic integrals, each of which is represented in Legendre canonical form. This results in compact expressions that are appreciably simpler than those previously given in the literature Vazquez2004 (); Kraniotis2005 (); Dexter2009 (); Fujita2009 (); Kraniotis2011 (); Hackmann2010 (); Hackmann2014 (), which either did not explicitly unpack the path integrals or did not reduce them to manifestly real and positive expressions. Our formulas then allow us to obtain in Sec. II.4 a simple expression for the polar angle as a function of the radial motion. Readers solely interested in the end results may skip directly to Sec. II.5 for a succinct summary.

ii.1 The Kerr geodesic equation in first-order form

The Kerr metric describes astrophysically realistic rotating black holes of mass and angular momentum . In Boyer-Lindquist coordinates and natural units where , the Kerr line element is

(1a)
(1b)

This metric admits two Killing vectors and generating time-translation symmetry and axisymmetry, respectively. In addition to these isometries, the Kerr metric also admits an irreducible symmetric Killing tensor111A Killing tensor satisfies . The antisymmetric tensor satisfies the Killing-Yano equation .

(2)

The motion of a free particle of mass and four-momentum is described by the geodesic equation,

(3)

In the Kerr geometry (1), geodesic motion is completely characterized by three conserved quantities,

(4)

denoting the total energy, angular momentum parallel to the axis of symmetry, and Carter constant, respectively. The first two quantities are the conserved charges associated with the Killing vectors and , respectively, whereas the conservation of the third quantity follows from the existence of the Killing tensor (2). While the Carter constant has the advantage of being manifestly positive, it is often useful to work instead with the so-called Carter integral

(5)

By inverting the above relations for , we find that a particle following a geodesic in the Kerr geometry (1) has an instantaneous four-momentum of the form

(6)

where the two choices of sign and depend on the radial and polar directions of travel, respectively. Here, we also introduced radial and polar potentials

(7a)
(7b)

One can then raise to obtain the equations for the geodesic trajectory,

(8a)
(8b)
(8c)
(8d)

The parameter is the affine parameter for massless particles (), and is related to the proper time by for massive particles. This system is completely integrable because it admits as many constants of motion as momentum variables, and can be integrated by quadratures. To do so, first note that

(9)

Integration along the geodesic from to yields

(10)

where the slash notation indicates that these integrals are to be evaluated along the geodesic, with turning points in the radial or polar motion occurring whenever the corresponding potential or vanishes. By definition, the signs and in front of and are always the same as that of and , respectively, so these integrals grow secularly (rather than canceling out) over multiple oscillations.

If the particle is located at when and at when , then this simplifies to

(11)

Likewise,

(12a)
(12b)

Aside from , the first integrand only contains -dependent terms and the second integrand only contains -dependent terms. Thus, we naturally replace using Eq. (8a) in the first integral and using Eq. (8b) in the second, resulting in

(13)

After repeating the same procedure for and shuffling constant pieces from one integral into the other, we find that

(14a)
(14b)
(14c)

To summarize, a geodesic labeled by connects spacetime points and if

(15a)
(15b)
(15c)

Generically, Kerr geodesics undergo multiple librations (polar oscillations) and rotations about the axis of symmetry. They may also undergo radial oscillations when they are bound () Wilkins1972 (). Kerr geodesics are therefore characterized by integers denoting the number of turning points in the radial motion, the number of turning points in the polar motion, and the winding number about the axis of symmetry, respectively.

ii.2 Qualitative description of the polar motion

We now wish to compute the angular integrals that appear in the Kerr geodesic equation (15),

(16)

and then solve for the final angle in the part of the equation, which is of the form with

(17)

To do so, it is convenient to rewrite the angular potential as

(18)

There are three qualitatively different cases that we will consider in turn:

  1. corresponds to null geodesics with , or unbound timelike geodesics with .

  2. corresponds to marginally bound timelike geodesics with .

  3. corresponds to bound timelike geodesics with .

Here, the bound/unbound nomenclature refers to the radial motion—the polar motion is of course always bounded.

The positivity condition implies that a geodesic can reach a pole at or if and only if . We will assume the genericity condition , in which case the polar motion is strictly restricted to oscillations bounded by turning points .222The special case needs to be treated separately, as one must account for the possibility that geodesics may climb over the black hole and pass through the rotation axis. This oscillatory motion can be of two qualitatively different types:

  • Oscillatory motion about the equatorial plane between and with .

  • “Vortical” motion between turning points or , corresponding to geodesics that never cross the equatorial plane and are instead confined to a cone lying either entirely above or entirely below the equatorial plane.

The Kerr geometry also admits planar geodesics at any fixed polar angle . These arise in the special limit where the turning points of the angular motion coalesce at . In that case, the geodesic equation (15) degenerates and a separate treatment is necessary. In the null case, the planar geodesics are the well-known principal null congruences, which endow the Kerr geometry with many of its special properties.333The Kerr principal null congruences are shear-free. By the Goldberg-Sachs theorem, this implies that the Kerr spacetime is algebraically special of Petrov Type D. This property guarantees the existence of the Killing-Yano tensor Stephani1978 (), from which many of the special properties of the Kerr geometry—including the separability of the wave equation and integrability of geodesic motion—are derived. We presently exclude this fine-tuned situation, which has been extensively studied in the literature Chandrasekhar1983 (). To study the two generic types, it is useful to define signs

(19a)
(19b)

where and denote the polar momentum evaluated at the endpoints and of the geodesic, respectively. By working through all the possible configurations, one can check that the angular path integral unpacks as follows:

Type A: (20a)
Type B: (20b)

where for both types, we presented two equivalent representations that differ only in the choice of turning point taken as a reference for the integrals. It will turn out that the type of oscillation is picked out by the sign of :

  1. corresponds to Type A oscillations. These are allowed for all signs of .

  2. corresponds to Type B (vortical) oscillations. These are only allowed for .444This is because Eq. (8b) implies that . Thus is only possible if , which requires that .

  3. corresponds to a singular limit of Type B motion in which the cone of oscillation touches the equatorial plane, where the integrals develop a nonintegrable singularity. Such geodesics are also only allowed for .

From now on, we will work with the variable , in terms of which

(21)

Here we defined

(22)

For future convenience, we also introduce the quantities

(23)

ii.3 Computation of the angular integrals

ii.3.1 Case 1:

In this case, we must necessarily have and . Hence, the oscillation is of Type A with turning points at . The only integrals we need are

(24a)
(24b)
(24c)

where, in order to ensure that each integral is real and positive, we used the substitution

(25)

Thus, in the case (where necessarily ), we obtain

( required):
(26b)
(26c)
(26d)

ii.3.2 Case 2:

In this case, we must necessarily have and . Hence, the oscillation is of Type A with turning points at . The only integrals we need are

(27a)
(27b)
(27c)

where we defined and, in order to ensure that each integral is real and positive, we used the substitution

(28)

Thus, in the case (where necessarily ), we obtain

( required):
(29b)
(29c)
(29d)

ii.3.3 Case 3:

If , then and the oscillation is of Type A with turning points at . The only integrals we need are

(30a)
(30b)
(30c)

where, in order to ensure that each integral is real and positive, we used the substitution

(31)

Thus, in the , case, we obtain

, :
(32b)
(32c)
(32d)

If , then and the oscillation is of Type B with turning points at and , where the upper/lower sign corresponds to vortical oscillation within a cone lying entirely above/below the equatorial plane. If we use as the reference turning point (, in both hemispheres, we integrate from the turning point closest to/farthest from the equator), then the only integrals we need are

(33a)
(33b)
(33c)

where we used the substitution

(34)

in order to ensure that each integral is real and positive. Thus, in the , case, we obtain

, :
(35b)
(35c)
(35d)

To be clear, in these equations, the choice of upper/lower sign leads to equivalent representations of the integrals corresponding to different choices of reference turning point.

If , then and the oscillation appears to be of Type B with turning points at the equator and , where the upper/lower sign corresponds to vortical oscillation within a cone lying entirely above/below the equatorial plane. However, in practice, the geodesic motion can only turn at , as it is barred from reaching the equator at , which corresponds to a nonintegrable singularity of the angular integrals. Hence, the complete motion can undergo at most one libration, , it can only have or . Therefore, in this special situation,

(36)

Thus, using as the reference turning point (which we must, in order to avoid the singularity at the equator), the only integrals we need are

(37a)
(37b)
(37c)

where we did not need any substitution to obtain simpler trigonometric representations of the integrals, which are all real and positive. In conclusion, in the , case, we obtain

, :
(38b)
(38c)
(38d)

ii.4 Solution to the equation

The part of the Kerr geodesic equation (15) is of the form , with

(39)

We want to solve this equation for . We will proceed by considering each case in turn, starting with the simplest.

In the case (where necessarily ), Eq. (26b) tells us that

(40)

Using the fact that is an odd function, this can be rewritten

(41)

from which it follows that

(42)

This expression can be further simplified by noting that

(43)

Since the function satisfies the periodicity condition , it follows that

(44)

from which we conclude that is in fact independent of the number of turning points along the trajectory:

( required):
(46)

We now turn to the remaining cases (with ), which are slightly more complicated but can nonetheless be treated using a similar approach. As a preliminary, note that because is an odd function,

(47)

In the case with , Eqs. (29b) and (32b) tell us that

(48)

where . Using Eq. (47), this can be rewritten

(49)

where we defined

(50)

and used the fact that is odd in its first argument. The inverse function of the elliptic integral of the first kind is the Jacobi elliptic function , which satisfies . Using this identity, it immediately follows from Eq. (49) that

(51)

This expression can be further simplified by noting that

(52)

Since the function satisfies the periodicity condition , it follows that

(53)

from which we again conclude that is independent of the number of turning points along the trajectory:

, with :
(55)

In the case (where necessarily ), Eq. (35b) tells us that

(56)

where either choice of sign is equally valid. This can be rewritten as

(57)

where we introduced

(58)

Next, we use the fact that the Jacobi elliptic function satisfies , and therefore

(59)

Applying this identity to yields

(60)

The last step follows from Eq. (57), together with the observation that

(61)

which follows from

(62)

Using Eq. (47) and the fact that is even in its first argument, we find that

(63)

where in the last step, we used the fact that for Type B vortical geodesics. This expression can be further simplified by noting that

(64)

Since the function satisfies the periodicity condition , it follows that

(65)

from which we again conclude that is independent of the number of turning points along the trajectory:

, with arbitrary:
(67)

While this expression for (valid for ) superficially differs from that given in Eq. (55) (valid for ), the two expressions can be brought into the same form. For , the reciprocal-modulus theorem Fettis1970 ()

(68)

determines the real and imaginary parts of in terms of and The difference in sign results from the branch cut in extending from to along the positive real axis. For real , choosing the primary branch for fixes

(69)

With this choice, identical manipulations demonstrate that

(70a)
(70b)

We now apply these identities with and . Since , we apply (70a) for the upper choice of sign and (70b) for the lower choice of sign. Combined with Eq. (61), we find that

(71)

After multiplying by , this becomes

(72)

In order to proceed, we also need the identity555This can be derived, for instance, from the two standard identities and (Eqs. (16.8.9) and (16.20.1) of Ref. Abramowitz1972 ()) which imply that . The result is then obtained by using Eq. (69) together with the imaginary periodicity condition whenever it is needed.

(73)

This relation holds for either choice of sign, which we take to be . Substituting and combining with Eq. (67) yields

(74)

Finally, using Eq. (72), this reduces to

(75)

Therefore Eqs. (55) and (67) can be combined into the single expression

, with :
(77)

In the case (where necessarily ), Eq. (38b) tells us that

(78)

which can be rewritten

(79)

Since is an odd function, it follows that

(80)

Therefore, the overall signs disappear upon squaring, leaving

(81)

Using the fact that is an even function, and that for vortical geodesics, we finally conclude:

, :
(83)

ii.5 Summary of results

Here, we collect the simplified expressions for the angular integrals , and , as well as expressions for the final angle obtained by solving the part of the geodesic equation (15). Our conventions are such that all of the terms appearing in these expressions are real and positive.

with :
(84b)
(84c)
(84d)
(84e)
with : with
(85b)
(85c)
(85d)
(85e)
(with required): with arbitrary
(86b)
(86c)
(86d)
(86e)
(with required):
(87b)
(87c)
(87d)
(87e)

Finally, we note that Eqs. (85e) and (86e) can be conveniently combined into a single expression:

and with :
(89)

Iii Near-horizon geometry of (near-)extreme Kerr

The Kerr family of metrics has two adjustable parameters corresponding to the mass and angular momentum of the black hole. Geometries satisfying the Kerr bound have smooth event horizons concealing a ring singularity, while solutions that violate this bound exhibit naked singularities visible from infinity. Black holes that (nearly) saturate the Kerr bound are termed (near-)extremal, and there is strong evidence to suggest that no physical process can drive a (sub-)extremal black hole over the Kerr bound Sorce2017 (). However, one would expect accretion of matter onto an astrophysical black hole to push it towards extremality, and indeed the vast majority of measured supermassive black holes spins are close to maximal Brenneman2013 (); Reynolds2019 (). The limiting behavior of the Kerr metric in the extremal limit is therefore of both theoretical and astronomical interest.

This section is dedicated to a pedagogical review of the qualitatively new and surprising features that the Kerr black hole develops in the high-spin regime. We begin in Sec. III.1 by presenting a timelike equatorial orbit that seemingly lies on a null hypersurface. This apparent paradox is resolved in Sec. III.2 by the presence of an infinitely deep throat-like region bunched up near the event horizon: the Near-Horizon Extreme Kerr (NHEK) geometry. This motivates the more systematic investigation of near-horizon scaling limits that we conduct in Sec. III.3. These different limits are then related by an emergent conformal symmetry of the throat in Sec. III.4. Finally, in Sec. III.5, we introduce global coordinates and describe the causal structure of the throat geometry using its Carter-Penrose diagram.

iii.1 Peculiar features of the extremal limit

The limit of the Kerr geometry poses a series of puzzles whose resolution requires a careful analysis of the near-horizon geometry of the extreme black hole. A particle orbiting a Kerr black hole on a prograde, circular, equatorial geodesic at radius has four-velocity Bardeen1972 ()

(90)

The energy and angular momentum of this geodesic is given by

(91)

This orbit is stable provided that the orbital radius exceeds the marginally stable radius of the Innermost Stable Circular Orbit (ISCO), given by

(92a)
(92b)

In the context of black hole astrophysics, these orbits provide a simple model for accretion onto a black hole: to a very good approximation, a thin disk of slowly accreting matter consists of particles following the geodesics (90) Novikov1973 (); Page1974 (). In reality, their trajectories also have a small inward radial component, but it can be neglected down to the ISCO radius, which delineates the innermost edge of the disk where accretion terminates. Beyond this edge, the particles quickly plunge into the black hole and their radial momentum can no longer be ignored. Instead, their motion is described by infalling geodesics with the conserved quantities of the marginally stable orbit Cunningham1975 (); Penna2012 ().

Consider a Kerr black hole with spin parameter . When the deviation from extremality is small, , the black hole has a small Hawking temperature of order ,

(93)

where denotes the radius of the (outer/inner) event horizon,

(94)

Thus, the extremal limit is equivalent to a low-temperature limit . A detailed investigation of this limit raises several puzzles:

  1. The first puzzle pertains to the fate of the ISCO in the extremal limit. According to Eq. (92),

    (95)

    Comparing with Eq. (94), it would appear that the ISCO moves onto the event horizon in the extremal limit:

    (96)

    However, for any sub-extremal black hole, the ISCO is a timelike geodesic, while the event horizon is ruled by null geodesics. Clearly, the extreme Kerr metric fails to accurately portray the spacetime geometry in the ISCO region correctly.

  2. Indeed, although the ISCO and extremal horizon appear to coincide, the proper radial distance (as measured on a Boyer-Lindquist time-slice) between the two actually diverges logarithmically in this limit:

    (97)

    Thus, even though generic timelike (null) geodesics fall into the horizon in finite proper (affine) time, the near-horizon region acquires an infinite proper three-volume in the extremal limit.

These observations were noted early on by Bardeen, Press and Teukolsky Bardeen1972 () and later revisited in Refs. Jacobson2011 (); Gralla2016a (). Taken together, these peculiarities indicate that the extremal Kerr metric grossly misrepresents the spacetime geometry near the event horizon of the extremal black hole. While it is true that the Boyer-Lindquist coordinates become singular at the horizon, we stress that these problems are not a coordinate artifact: they still arise even in coordinates that are smooth across the horizon. The existence of the infinite throat region is a coordinate-invariant statement, and describing it requires a careful resolution of the near-horizon geometry. This was accomplished by Bardeen and Horowitz Bardeen1999 () by introducing a horizon-scaling limit tailored to this task, to which we now turn.

iii.2 The extreme Kerr throat

In Sec. III.1, we saw that as a black hole spins up and approaches the limiting extremal geometry with , a deep throat of divergent proper depth develops outside of its event horizon. Moreover, from the perspective of the far observer, particles on the ISCO co-rotate with the black hole horizon in this limit. This motivates a coordinate transformation from Boyer-Lindquist coordinates to Bardeen-Horowitz coordinates given by

(98)

where denotes the angular velocity of the extremal black hole horizon,

(99)

These coordinates are adapted to a local near-horizon observer co-rotating with the black hole, since

(100)

Local finite-energy excitations near the horizon of a black hole have large gravitational redshift relative to an observer at infinity. For black holes far from extremality, this region of spacetime is small and contains no stable orbits. However, for extremal black holes, the stable orbits extend down the throat, and the high-redshift emissions from sources in this region are phenomenologically interesting. In order to resolve the degeneracy arising from the infinite redshift while zooming into the horizon, we perform an infinite dilation onto the horizon, implemented by the rescaling

(101)

If the black hole is precisely extremal (), this scaling procedure has a finite limit and yields the NHEK geometry, with non-degenerate line element Bardeen1999 ()

(102a)
(102b)

Since the NHEK geometry arises as a non-singular scaling limit of the extreme Kerr solution, it manifestly solves the vacuum Einstein equations and can be studied as a spacetime in its own right. Moreover, since in the limit , the resulting metric is -independent, further coordinate rescalings leave the NHEK line element (102) invariant: physically, the throat-like region is sufficiently deep that it becomes self-similar in the extremal limit.

Therefore, the region of spacetime in the throat displays an emergent scaling symmetry, which is generated at the infinitesimal level by the dilation Killing vector . Surprisingly, yet another, no-less constraining symmetry—invariance under special conformal transformations generated by —also emerges in this limit. Together, these symmetries generate the global conformal group , with commutation relations

(103)

Hence, in the high-spin regime, the symmetry of the Kerr metric (1) generated by the Killing vectors and (associated with stationarity and axisymmetry, respectively) is enlarged within the near-horizon region to an isometry group generated by

(104)

In fact, although the Killing tensor (2) in extreme Kerr is associated to a non-geometrically-realized symmetry, in the near-horizon limit, this irreducible Killing tensor descends to a reducible Killing tensor in NHEK AlZahrani2011 (). More precisely, it is given (up to a mass term) by the Casimir of :

(105)

Thus, the Kerr metric’s hidden symmetries become explicit in the emergent throat region, where it decomposes into

(106)

Finally, we note that the induced metric on a hypersurface of fixed is that of warped three-dimensional Anti de-Sitter space (WAdS) with warp factor Song2009 (). This warp factor goes to unity, , at the critical angle , which corresponds to the so-called “light surface” of the extreme Kerr black hole Aman2012 (). The NHEK metric becomes precisely that of AdS on this surface, which seems to play a priviledged role in the propagation of light out of the throat Gralla2017a ().

iii.3 Near-horizon scaling limits for near-extreme Kerr

In Sec. III.1, we saw that the extreme Kerr metric fails to resolve near-horizon physics. Then, we argued in Sec. III.2 that this failure is caused by the emergence, in a certain scaling limit, of a near-horizon region of the Kerr spacetime that resembles an infinite gravitational potential well. This throat-like region is sufficiently deep that in the extremal limit, it becomes self-similiar and enjoys an enhanced isometry group: the global conformal group .

The scaling limit to the NHEK region is unique for a precisely extremal black hole. But, according to the classical laws of black hole thermodynamics, such a black hole is unphysical since it has zero temperature, and no adiabatic process can turn a non-extremal black hole extremal Israel1986 (). Thus, it is more realistic to consider black holes with a small deviation from extremality and a correspondingly small temperature. However, for such a near-extremal black hole, there exist infinitely many bands of near-horizon radii that become infinitely separated from each other in the extremal limit Gralla2015 (). More precisely, if one considers two Boyer-Lindquist radii and that scale to the horizon at different rates as extremality is approached, so that

(107)

then the proper radial separations along a Boyer-Lindquist time-slice have the limiting form

(108)

Only radii that scale to the horizon at the same rate have finite radial separation in the extremal limit. This indicates that there are in fact infinitely many physically distinct near-horizon limits, each of which resolves the throat physics at different scales. The relevant scaling limits straightforwardly generalize Eqs. (98) and (101) to

(109)

The limit with (which physically amounts to zooming into the near-horizon region at the same rate that the black hole is dialed into extremality) and held fixed yields the so-called near-NHEK geometry Bredberg2010 ()

(110)

which is the finite-temperature analogue of the NHEK metric (102). It also has an isometry group generated by

(111)

This region lies deepest in the throat and resolves the horizon at , along with all other radii that also scale like in the limit. Examples of physically interesting radii that scale into near-NHEK include the photon orbit at and the (prograde) marginally bound orbit at , also known as the Innermost Bound Circular Orbit (IBCO) radius Bardeen1972 (),

(112a)
(112b)

The proper radial separation between two prograde, equatorial, circular geodesics in near-NHEK is given by

(113)

This expression matches the limiting radial separation of equatorial geodesics calculated in the Kerr geometry (108), provided that one identifies the near-NHEK radius with the radius in Kerr scaling as . Note that this distance is not scale-invariant due to the presence of the horizon: physically, the presence of a small temperature breaks the scaling symmetry exhibited by the NHEK distance. Mathematically, this can also be seen from Eq. (109), where the presence of the temperature precludes the scaling limit from being a coordinate limit: unlike the NHEK scaling (101), the dilation into near-NHEK also acts on the parameter , which is why it is not forced to become an isometry in the limit.

The limit with any physically corresponds to spinning up the black hole faster than one zooms into the horizon, and always produces the same NHEK metric (102). However, each geometry thus obtained corresponds to a physically distinct region of the throat: a given choice of resolves a band of Boyer-Lindquist radii that scale like .666For a precisely extremal black hole, this distinction is irrelevant since all the scales we discuss lie precisely at , so there is a single NHEK limit that resolves them. The proper radial separation in NHEK,

(114)

matches the corresponding limiting radial separation of equatorial geodesics calculated in the Kerr geometry (108). However, because the NHEK expression is scale-invariant, the identification of NHEK radii with Kerr radii is ambiguous: one identifies the NHEK radius with the radius in Kerr scaling as , up to an overall -independent factor. It is only after the throat is reattached to the asymptotically flat region, and the dilation symmetry is broken, that NHEK radii can be unambiguously identified with Kerr radii.

A physically interesting Kerr radius that scales into NHEK is the ISCO radius, which according to Eq. (92) has a near-horizon limit . The band of radii in Kerr with a finite radial separation from the ISCO in the extremal limit all scale like , and the limit (109) with produces precisely the NHEK metric (102). In this limit, the four-velocity of the ISCO becomes Gralla2016a ()

(115)

which is both timelike and finite.777However, plunging trajectories that were resolved by the far region will be null in this limit; see, , Fig. 1 of Ref. Gralla2016a (). Therefore, the “ NHEK” resolves the part of the throat in which ISCO-scale physics occurs. Again, note that while the near-horizon limit of Eq. (92) appears to identify the ISCO radius with the NHEK radius , the dilation invariance of the NHEK metric (102) indicates that there is in fact no meaningful way to assign a definite radius to the ISCO within NHEK. In fact, in contrast to near-NHEK, all circular geodesics in NHEK are marginally stable: from the near-horizon viewpoint, the ISCO is in some sense everywhere within the NHEK Gralla2015 ().

Far region
NHEK
near-NHEK
Figure 1: Throat geometry of a high-spin black hole. (Figure reproduced from Ref. Bardeen1972 () with additional labels.) “High spin” means with . As the black hole approaches extremality (), its near-horizon region develops an infinitely deep throat, which becomes a spacetime in its own right. Its geometry is described by the Near-Horizon Extreme Kerr (NHEK) metric (102) and exhibits an emergent scaling symmetry consisting of dilations of the throat (or equivalently, translations along the infinite length of the cylinder).

To summarize, any two radii that scale to the horizon at the same rate (“lie in the same band”) end up in the same near-horizon geometry at a finite proper radial distance from each other. For instance, the photon orbit and IBCO radii both lie in the horizon band, and hence scale to the same near-NHEK region. Accordingly, the proper radial distance between these scales tends to a finite limit:

(116a)
(116b)
(116c)

On the other hand, radii that lie in different bands, , that scale to the horizon at different rates, end up in different NHEKs that are infinitely far apart (separated by a divergent proper radial distance). For instance, the ISCO band is infinitely far from both the horizon band, as well as from the mouth of the throat, which we may for instance define as the spin-independent equatorial radius of the ergosphere, :

(117a)
(117b)
(117c)

These facts are summarized in Fig. 1, where the different bands appear stacked on top of one another, with cracks denoting the logarithmically divergent proper radial distance separating them. From this point of view, the precisely extremal, zero-temperature black hole is a degenerate limit in which all the throat geometries merge: near-NHEK disappears and the different NHEKs coalesce into one.

Finally, note that the expansion about extremality defined by Eq. (109) can be viewed both as a small-temperature expansion and an expansion in the divergent proper depth of the throat: indeed, at leading order,

(118)

Thus, subleading corrections due to deviations from extremality are exponentially suppressed in the characteristic length scale of the system, which diverges in the extremal limit. Similar behavior is of course observed near critical points in condensed matter systems—this analogy was further developed in Ref. Gralla2016a ().

When studying the extremal Kerr black hole, it is important to note that neither the far metric (extreme Kerr) nor the near metric (NHEK) is more fundamental than the other: away from extremality, the Kerr metric resolves physics in the entire spacetime, but near extremality, the spacetime decouples into two regions. Each of these two regions is described by its own metric, which fails in the other region: while NHEK resolves the near-horizon region, it fails to resolve the far region (for instance, it is not asymptotically flat), and the far metric does not resolve the throat region. As is usual for smooth extremal solutions in general relativity, the extreme Kerr geometry serves to interpolate between two separate vacuum solutions: flat space in the far region and NHEK in the near region. The two regions of spacetime are on equal footing.

iii.4 Emergent conformal symmetry

In many situations (including those of astrophysical interest), it is appropriate to treat the Kerr geometry as a fixed background while neglecting gravitational backreaction of the matter system (as well as gravitational excitations). When this approximation is valid, it suffices to work strictly with the NHEK metric and its exact isometries. In other applications, one is interested not only in the vacuum NHEK geometry, but in all spacetimes that approach NHEK asymptotically in some appropriate sense.888The precise choice of boundary conditions is delicate Guica2009 (); Dias2009 (); Amsel2009 (); Compere2012 (). Although these geometries all possess a long throat and approximate scale-invariance, generic members of this class of spacetimes have no exact isometries. It is the symmetries of the class of spacetimes, rather than the symmetries of a specific spacetime, that control gravitational dynamics in the throat.

In attempting to compute this generalized symmetry group, one often finds an enhancement of the global conformal isometry group to an infinite-dimensional local conformal symmetry Compere2012 (). The details of calculations of this type depend delicately on the choice of boundary conditions. We will focus on a particular class of symmetry transformations that have been repeatedly utilized Porfyriadis2014 (); Hadar2014 (); Hadar2015 (); Hadar2017 (); Compere2018 () in calculating geodesics in NHEK and near-NHEK, and defer a complete asymptotic symmetry group analysis to future work. These large diffeomorphisms are the NHEK analogue of boundary reparameterizations of the AdS throat discussed in Ref. Maldacena2016b () and should be related to inequivalent ways of reattaching the Kerr throat region to the exterior geometry.

Starting with the NHEK line element (102),

(119)

we consider a coordinate transformation of the form999We thank Abhishek Pathak for help in deriving this transformation from its AdS analogue Roberts2012 ().

(120)

The resulting line element is given by

(121)

where we introduced the Schwarzian derivative

(122)

These metrics are the NHEK analogues of the AdS Bañados metrics Banados1999 (); Compere2016 (). Note that at the boundary , this coordinate change implements a time reparameterization , and that as a result, subleading components of the metric transform like the expectation value of a stress-tensor component in CFT.

Infinitesimally, the conformal transformation (120) is implemented by the action of the vector field

(123)

This can be decomposed into modes

(124)

which obey the Witt algebra at the boundary,

(125)

The isometry group of NHEK is generated by the vector fields

(126)

whose corresponding finite diffeomorphisms are given by the Möbius transformations with vanishing Schwarzian:101010More precisely: dilations by are obtained by setting and ; time-translations by are obtained by setting and ; special conformal transformations by a parameter are obtained by setting and .

(127)

The rest of the symmetry transformations with nonvanishing are spontaneously broken. Of particular interest here is the exponential map

(128)

for which the metric becomes

(129)

Near the boundary, this diffeomorphism acts as the exponential map on the boundary time. It is the analogue of the usual conformal transformation from the plane to the cylinder, which puts a CFT at finite temperature. In fact, the metric (129) is actually near-NHEK, as can be seen by performing a further (small) diffeomorphism

(130)

which puts it in the form of Eq. (110). By composing these transformations, one can directly map near-NHEK,

(131)

into NHEK via the coordinate change

(132)

This transformation also maps the NHEK Killing vectors (104) and near-NHEK Killing vectors (111) into each other. It is important to note that this map is not surjective: its range covers only a subset of the NHEK Poincaré patch. Within that image, the inverse transformation is

(133)

Since the map (132) is a diffeomorphism between near-NHEK and a subset of the Poincaré patch in NHEK (rather than its entirety), the near-NHEK and NHEK patches are locally (but not globally) diffeomorphic. Of course, since both near-NHEK and NHEK have horizons, they are not geodesically complete spacetimes. As we will discuss in the next section, they have the same maximal extension: global NHEK.

iii.5 The global strip and the Poincaré patch

To obtain the maximal extension of the NHEK spacetime, we pass from the Poincaré coordinates with a coordinate singularity at to global coordinates that can be smoothly continued past this surface. The transformation from Poincaré NHEK to a patch of global NHEK is given by

(134)

Different branches of the arctangent map Poincaré NHEK to diffeomorphic patches of global NHEK which differ by translations in global time. For this reason, all geodesic motion in global NHEK is oscillatory with period . The inverse map from the patch of global NHEK to Poincaré NHEK is given by

(135)

Under this coordinate transformation, the NHEK metric (102) becomes

(136)

and the NHEK Killing vector fields (104) are mapped into

(137a)
(137b)
(137c)

It will be convenient for us to complexify this algebra by introducing new (complex) generators

(138)

which obey the same commutation relations:

(139)

These two sets of generators are related by

(140)

The generator of global time translations is the analogue of the Hamiltonian of a CFT defined on the cylinder.

The causal structure of the global NHEK geometry is best understood by introducing a compactified radius111111The inverse transformation on the domain is given by .

(141)

in terms of which the global NHEK line element (136) becomes

(142)

As explained in detail in App. B, the part of the metric describes the global strip of AdS parameterized by and . Since this two-dimensional metric is conformally flat, with lines of manifestly null, it is straightforward to obtain its Carter-Penrose diagram, depicted in Fig. 2.

Under the embedding (135) of Poincaré NHEK into global NHEK, we see that the Poincaré coordinates cover the patch with (in particular, the future/past horizon is located at ):

(143)

Hence, the lines of constant and constant are respectively given by

(144)

and in Fig. 3, they are plotted at constant intervals of , where and .

Under the embedding of near-NHEK into global NHEK obtained by composing Eqs. (132) and (135), we see that the Poincaré coordinates cover the patch with :

(145)

Hence, the lines of constant and constant are respectively given by

(146)

and in Fig. 3 they are plotted at constant intervals of , where and .

Note that the finite transformations—Eq. (120) with a Möbius transformation—leave the NHEK metric (102) invariant, but not its geodesics: they are mapped into each other under the action of the global conformal group. Equivalently, we can study geodesics in global coordinates and obtain different geodesics in the Poincaré patch by using the embedding (134) composed with the transformations. This strategy can be used to map circular orbits to (slow or fast) plunges and was used to great effect in Refs. Hadar2014 (); Hadar2015 (); Hadar2017 (); Compere2018 (). In fact, most preexisting analyses of geodesics in NHEK focused on equatorial circular geodesics or plunging geodesics obtainable through the above mapping from the ISCO geodesic (115).

Figure 2: Carter-Penrose diagrams for the equatorial plane of near-extreme Kerr [left], extreme Kerr [middle] and NHEK [right]. (The diagrams depend on because these geometries are not spherically symmetric and, away from the equatorial plane, it is possible to continue past the singularity.) In the diagrams for (near-)extreme Kerr, denotes future/past null infinity ( with ), is spacelike infinity (), is the outer/inner horizon, and the singularity lies at . The thin lines correspond to hypersurfaces of constant , which are timelike outside the black hole [red] and spacelike inside [gray].
Figure 3: Carter-Penrose diagrams for the equatorial plane of global NHEK [blue] and the Poincaré patch for NHEK [red] and near-NHEK [green]. The global NHEK strip is covered by the global coordinates with and . The Poincaré patch for NHEK [red] can be embedded at any vertical position along the strip; here, it is depicted as the red patch at , corresponding to the mapping (134). Likewise, near-NHEK can also be placed anywhere on the strip; here, it is depicted as the green patch embedded at , , corresponding to the composition of the mappings (132) and (134). In both the NHEK and near-NHEK patches, slices of constant Poincaré time and radius are depicted by blue and red lines, respectively. For this choice of NHEK patch, the future/past horizon is located at (or, equivalently, with ).

Iv Geodesics in NHEK

In this section, we analyze geodesic motion in global NHEK, the Poincaré patch, and near-NHEK. For each of these spacetimes, we follow the procedure outlined for Kerr in Sec. II and recast the geodesic equation in first-order form. We then obtain explicit expressions for all the path integrals appearing in the equation. Finally, inverting the expression for the time-lapse allows us to solve for the radial motion as a function of coordinate time and thereby derive a complete and explicit parameterization of all NHEK geodesics.

We begin by analyzing the geodesic equation in global NHEK (Sec. IV.1), as it is a geodesically complete spacetime, unlike Poincaré NHEK (Sec. IV.2) and near-NHEK (Sec. IV.3). As a byproduct of our analysis, we elucidate the action of the NHEK isometry group on the space of geodesics. We find that the group orbits are classified by the Casimir and angular momentum , which completely determine the polar motion. Any two NHEK geodesics with the same polar motion can be mapped into each other by the action of using the explicit transformations presented at the end of App. B.

iv.1 Geodesics in global NHEK

Recall from Sec. III.5 that in global coordinates, the NHEK line element is

(147)

and the generators of are

(148)

The Casimir of is the (manifestly reducible) symmetric Killing tensor

(149)

It is related to , the NHEK limit (105) of the irreducible Killing tensor on Kerr (2), by

(150)

The motion of a free particle of mass and four-momentum is described by the geodesic equation,

(151)

Geodesic motion in global NHEK is completely characterized by the three conserved quantities121212Observe that , where or are the conserved quantities associated with the generators of . Since these are not independent of each other, we use the Casimir, which is in involution with all the AlZahrani2011 (). This is exactly analogous to exploiting the conservation of and , rather than , in a problem with symmetry.

(152a)
(152b)

denoting the global energy, angular momentum parallel to the axis of symmetry, and “Casimir” constant, respectively. Here, we introduced

(153)

When connecting NHEK geodesics to the far region in Kerr, it is more convenient to work with the Carter constant

(154)

which is directly related to its Kerr analogue . In this paper, however, we restrict our attention to motion within NHEK, which is more easily characterized by the Casimir .

By inverting the above relations for , we find that a particle following a geodesic in the global NHEK geometry (147) has an instantaneous four-momentum of the form

(155)

where the two choices of sign and depend on the radial and polar directions of travel, respectively. Here, we also introduced radial and polar potentials

(156a)
(156b)

One can then raise to obtain the equations for the geodesic trajectory,

(157a)
(157b)
(157c)
(157d)

The parameter is the affine parameter for massless particles (), and is related to the proper time by for massive particles.

Following the same procedure as in Kerr, we find from Eqs. (157) that a geodesic labeled by connects spacetime points and if

(158a)
(158b)
(158c)

We may rewrite these conditions as

(159)

where we have defined the integrals

(160a)
(160b)

iv.1.1 Qualitative description of geodesic motion

Before solving this geodesic equation outright, it is useful to determine the qualitative behavior of the geodesics projected onto the poloidal plane. The analysis of the polar motion follows directly from our earlier discussion in Sec. II for Kerr: indeed, it suffices to note that takes the form under the identification

(161)

Therefore, the NHEK angular integral takes the same form as the Kerr angular integral with

(162)

From Eq. (161), we see that NHEK geodesics (unlike their Kerr counterparts) are necessarily Type A (non-vortical) because cannot be negative, since positivity of the angular potential requires that

(163)

Moreover, it is straightforward to check that is only allowed for purely equatorial geodesics, a special case requiring a separate (and simpler) treatment. Generically, we therefore have the strict inequality , and the angular motion is necessarily of Type A with bounds given by Eq. (84) (when ) or Eq. (85) (when ).

Having completed the analysis of the polar motion, we now turn to the radial motion. Its allowed range is heavily constrained by the requirement that the radial potential remain positive at every point along the trajectory. As in Kerr, real zeroes of the radial potential correspond to turning points in the radial motion. In global NHEK, the radial motion can be of two qualitatively different types:

  • Oscillatory motion between turning points , corresponding to bound particles that are confined to NHEK and traverse the totality of the global strip, . The global-time-lapse between successive turning points is : each period of motion lies in a single Poincaré patch of the global strip.

  • Single-bounce motion (from a boundary at to a turning point and back), corresponding to unbound particles that probe a single Poincaré patch of the global strip, . The continuation of the geodesic beyond the intersection with the boundary depends on boundary conditions.131313In Anti-de Sitter space, one typically imposes reflective boundary conditions in order to describe a closed system. In the present case, such a choice of boundary conditions would lead to periodic multiple-bounce trajectories. We will not explicitly consider such solutions as they are unlikely to be relevant to astrophysical black holes, but it is trivial to construct them by gluing together a sequence of single bounces.

The type of motion is determined by the properties of the roots of the radial potential . For generic values of the geodesic parameters, the (possibly complex) roots of are given by

(164)

As we vary the geodesic parameters, these roots move around in the complex -plane. When one or more of the roots approaches or pinches the contour of integration, the radial motion of the allowed geodesics is then constrained to lie exclusively on one side of the root.

Positivity of energy in the local frame of the particle,