Creases, corners and caustics: non-smooth
structures on horizons
Harvey Reall
DAMTP, Cambridge University

Maxime Gadioux and HSR, 2303.15512

Introduction
Every point of an event horizon H belongs to a null geodesic thatlies within H. These geodesics are the generators of H.
A generator cannot have a future endpoint, i.e., it cannot leave H
to the future.
Generators can have past endpoints:

Horizon non-smoothness

We assume that spacetime is smooth.
Theorem: H is an achronal continuous hypersurface.
(Achronal: no two points of H are timelike separated.)

H is not smooth except in very special cases e.g. atime-independent black hole.
What is the nature of the non-smoothness of H?

There exist examples of spacetimes for which H is
non-differentiable on a dense set (Chrusciel & Galloway 96) but
these are “very artificial”
Theorem (Beem & Krolak 97):
I H is differentiable at p iff p lies on exactly one generator
I A point lying on more than one generator is an endpoint(converse untrue)
Let Hend be the set of past endpoints of horizon generators.
Siino & Koike 04: classification of points of Hend assuming aparticular definition of genericity

In explicit examples of gravitational collapse or black hole mergers,
Hend consists of a 2d spacelike crease set where pairs of generators
enter H, together with its boundary, which is a line of caustic
points (where “infinitesimally nearby generators intersect”)(Hughes et al 94, Shapiro et al 95, Lehner et al 99, Husa & Winicour ’99, Hamerly & Chen 10, Cohen et al 11,Emparan & Martinez 16, Bohn et al 16, Emparan et al 17)

In 2+1 dimensions:Asymmetric gravitational collapse in 3+1 dimensions:
BLACK HOLE FUSION IN THE EXTREME MASS RATIO LIMIT

PHYS. REV.black
D 97, 044004
Non-axisymmetrichole(2018)
merger (Emparan et al 17):
FIG. 13. Sketch of how the crease-forming generators map theasymptotic ðL; PÞ plane to the crease strip (here widened forbetter visibility) on the equatorial plane in an orthogonal merger.
This is like Fig. 11 but modified for clarity, with the same color
coding of Fig. 2. The crease strip winds around the Kerr horizonan infinite number of times. The gray part of the plane, withP ≠ 0, corresponds to generators that meet pairwise (withopposite values of P) at crossover points of the crease set.
The horizontal blue lines along the P ¼ 0 axis on the plane map
to the edges of the crease set; these are caustic points. The vertical
red lines L ¼ 0 map to the center of the strip. The mapping isreflection symmetric about the equatorial plane.

FIG. 15. Generators oα ¼ π=4, projected on thwith Lα ¼ 0. Those wit
signs have the same colowhich corresponds to t
represents an intermedi

their properties. We hπ=4, and 3π=8.
The numerical inte
matic in any case, andthe identification of csets in the parameter s
they are not easily see
the generic case theWhat features of these spacetimes lead to this simple structure forHend ?
What other structures are possible?
Assumptions
I Spacetime is globally hyperbolic
I H is smooth at late time: there exists a Cauchy surface Σ tothe future of Hend such that H? ≡ Σ ∩ H is smooth(No assumptions about equations of motion.)
We show that Hend is the past null cut locus of H? .

Null cut locus
A null geodesic emitted orthogonally to H? cannot be deformed toa timelike curve from H? locally. A null cut point is the first point
along a such a null geodesic beyond which it can be deformed intoa timelike curve. The null cut locus of H? is the set of all null cutpoints.

In Riemannian geometry a cut locus can be very complicated (e.g.
fractal). But it can be decomposed into parts with simpler
structure (Itoh & Tanaka 1998). We obtained a Lorentziananalogue of this decomposition.

Any point in a null cut locus is either a caustic or lies on at least 2generators (Beem & Ehrlich 81, Kemp 84, Kupeli 85). So we can classify points ofHend as follows:
I caustic points
I non-caustic points

normal crease points: lie on exactly 2 generators
normal corner points: lie on exactly 3 generators
I points on ≥ 4 generators
II

We prove:I (a) Normal crease points form a 2d spacelike crease

submanifoldI (b) Normal corner points form a 1d spacelike cornersubmanifold
I (c) All other points form a set of (Hausdorff) dimension ≤ 1Creases and corners
Normal crease points form a 2d spacelike crease submanifold.
Normal corner points form a 1d spacelike corner submanifold
Consider Σ ∩ H for some Cauchy surface Σ. Creases are lines atwhich two smooth sections of horizon meet. Corners are points at
which three smooth sections of horizon meet.

Application: Bousso entropy bound (99)
A lightsheet is a non-expanding null hypersurface ending at caustic
set. Consider entropy S crossing lightsheet emanating orthogonallyfrom a 2d spacelike surface Σ of area A. Conjecture: S ≤ A/4G ~.
Proofs: (a) for matter possessing a local entropy current obeying
reasonable conditions (Flanagan, Marolf & Wald 99); (b) for quantum
fields (Bousso et al 14)
Terminate lightsheet
at null cut locus of
Σ? (Tavakol & Ellis 99).
Our results
apply to a generalnull cut locus.
Combining with theabove proofs givesS ≤ (A−2Acrease )/4G ~.

Perestroikas
Let τ be a time function and Στ denote a Cauchy surface ofconstant τΣτ ∩ H is the “horizon at time τ ”. This will have somearrangement of creases, corners and caustics.
As τ varies, this arrangement may undergo a qualitative change ata critical value of τ . We call this a perestroika (restructuring).
A crease perestroika occurs at a time τ for which Στ is tangent tothe crease submanifold.
Near the point of tangency, H is (part of) the union of two
intersecting null hypersurfaces. By introducing Riemannian normal
coordinates around this point we can determine the exact local
behaviour of H.
There are three qualitatively different possibilities. Shift τ so thatperestroika occurs at τ = 0.

Flying saucer

This perestroika describes the nucleation of a component of H in
generic gravitational collapse

Length of crease and angle at crease scale as√τ , area scales as τCollapse of hole in horizon
In examples of gravitational collapse or a black hole merger, somechoices of time function give a brief period where horizon has
toroidal topology (Hughes et al 94, Siino 97, Cohen et al 11, Bohn et al16). The “hole in the torus” collapses superluminally. The collapse
is described by a perestroika:

Length of crease and angle at crease scale as√−τ .

Black hole merger

This perestroika describes the merger of two (locally) disconnected
sections of horizon e.g. two merging black holes.

Angle at creases scales asp|τ |Genericity/stability
Which features of Hend are stable under small perturbations?
e.g. spherically symmetric collapse: Hend is a single (caustic)
point. If we perturb spacetime then non-trivial crease submanifold
is present, so original structure of Hend is unstable/non-generic.
Siino & Koike 04: classification of points of Hend assuming aparticular mathematical notion of genericity
I Non-caustic points of double, triple, quadruple

self-intersection of H
I Lines of caustic points “of type A3 ”

But: how to relate this notion of genericity to genericity w.r.t.
perturbations of metric?

Generic caustic point: A3
A3 caustic points form spacelike lines. A horizon cross-section
generically has isolated A3 caustic points. If we extend generators
beyond their past endpoints we obtain the swallowtail:Why can’t an A2 caustic occur on H? Would violate achronality!

A3 perestroikas
Occur when Στ is tangent to A3 line.

Crease contribution to black hole entropy
Old idea: some/all of black hole entropy is entanglement entropyof quantum fields across horizon (Bombelli et al 86, Srednicki 93,
Susskind & Uglum 94). Flat space entanglement entropy exhibits
novel features in the presence of a crease (Casini & Huerta 06, Hirata& Takayanagi 06, Klebanov et al 12, Myers & Singh 12)

Suggests that a crease might contribute to black hole entropy as
Z1√
f (Ω)dlG ~ crease
where Ω is angle at crease and f ∝ 1/Ω as Ω → 0. This issubleading compared to Bekenstein-Hawking entropy A/4G ~: how
to test this idea?
Consider “hole in the horizon” perestroika: this term remains finite
and non-zero as τ → 0−, so discontinuous at τ = 0. Consistency
with second law implies that residue of f at Ω = 0 must be
non-positive.

Gauss-Bonnet term in entropy
A “Gauss-Bonnet” term in gravitational action is topological in 4d
but contributes to black hole entropy (Jacobson & Myers 93, Iyer & Wald 94)
Z√
SGB = γd 2 x µR[µ]HOn smooth horizon SGB = 4πγχ where χ is Euler number of H.
For non-smooth horizon, “regulate” SGB , defining via a limit ofsmooth surfaces to obtain same result. SGB is discontinuous inblack hole formation or merger, so only γ = 0 is consistent with
2nd law (Sarkar & Wall 11)
But: does SGB actually need regulating? No: integral iswell-defined for creases, corners and A3 caustics. No longer
topological, continuous in black hole formation/merger.
Still find γ = 0 if no “higher order” terms in entropy but γ
unconstrained if such (EFT) terms are present.

Summary
We’ve studied the endpoint set of the generators of a horizon thatis smooth at late time
Structure of endpoint set: crease submanifold, corner submanifold,
1d set (similarly for a general null cut locus). Examples of corners?
Applications of crease submanifold?
Perestroikas describe qualitative changes in the structure of thehorizon as a function of time: crease perestroikas describe blackhole nucleation and black hole merger; caustic perestroikas describe
decay of crease via (dis)appearance of pairs of caustic points
There are open issues concerning the correct notion of genericity
for caustics and the classification of caustics in curved spacetime
Creases may contribute to black hole entropy. 4d Gauss-Bonnet
term may not be dead.