Topology Change of Black Holes
Daisuke Ida
and Masaru Siino
1Department of Physics, Gakushuin University, Tokyo 171-8588, Japan.
2Department of Physics, Tokyo Institute of Technology, Tokyo 152-8550, Japan.
(April 1, 2007)
Abstract.
The topological structure of the event horizon has been investigated in terms of the
Morse theory. The elementary process of topological evolution can be understood
as a handle attachment. It has been found that there are certain constraints on
the nature of black hole topological evolution: (i) There are n kinds of handle
attachments in (n+1)-dimensional black hole space-times. (ii) Handles are further
classified as either of black or white type, and only black handles appear in real
black hole space-times. (iii) The spatial section of an exterior of the black hole
region is always connected. As a corollary, it is shown that the formation of a
black hole with an Sn−2 × S1 horizon from that with an Sn−1 horizon must be
non-axisymmetric in asymptotically flat space-times.
1. Introduction
Black holes in space-times of greater than or equal to five dimensions have rich
topological structure. According to the well-known results of Hawking concerning
the topology of black holes in four-dimensional space-time, the apparent horizon
or the spatial section of the stationary event horizon is necessarily diffeomorphic
to a 2-sphere. [1, 2] This follows from the fact that the total curvature, which is
the integral of the intrinsic scalar curvature over the horizon, is positive under the
dominant energy condition and from the Gauss-Bonnet theorem. Alternative and
improved proofs of Hawking’s theorem have been given by several authors. [3, 4, 5, 6]
However in higher dimensional space-times, an apparent horizon or the spatial
section of the stationary event horizon may not be a topological sphere, [7, 8, 9, 10]
because the Gauss-Bonnet theorem does not hold in such cases. Nevertheless,
the positivity of the total curvature of the horizon still holds. This puts certain
topological restrictions on the black hole topology, though they are rather weak. For
example, the apparent horizon in five-dimensional space-time can consist of finitely
many connected sums of copies of S3/Γ and copies of S2 × S1. In fact, exact
solutions representing a black hole space-time possessing a horizon of nonspherical
topology have recently been found in five-dimensional general relativity. When such
black holes with nontrivial topologies are regarded as being formed in the course
of gravitational collapse, questions regarding the evolution of the topology of black
holes naturally arise. Our purpose here is to understand the time evolution of the
topology of event horizons in a general setting. The relation between the crease set,
where the event horizon is nondifferentiable, and the topology of the event horizon
is studied in Refs. [11, 12, 13] for four-dimensional space-times. In the present
work, we carry out a systematic investigation and find useful rules to determine
admissible processes of topological evolution for time slicing of a black hole.
Our approach is to utilize the Morse theory [14, 15] in differential topology. The
Morse theory is useful for the purpose of understanding the topology of smooth
http://arxiv.org/abs/0704.0100v3
manifolds. The basic tool used in this approach is a smooth function on a differ-
entiable manifold. The event horizon, however, is not a differentiable manifold but
has a wedge-like structure at the past endpoints of the null geodesic generators
of the horizon. For this reason, we first smooth the wedge. Then, the smooth
time function which is assumed to exist plays the role of the Morse function on the
smoothed event horizon. According to the Morse theory, the topological evolution
of the event horizon can then be decomposed into elementary processes called “han-
dle attachments.” In such a process, starting with a spherical horizon, one adds
several handles, each characterized by the index of the critical points of the Morse
function, which is an integer ranging from 0 to n (the dimension of the smoothed
horizon as a differentiable manifold).
The purpose of the present article is to show that there are several constraints
on the handle attachments for real black hole space-times.
2. The Morse theory for event horizons
Let M be an (n+1)-dimensional asymptotically flat space-time. We require the
existence of a global time function t : M → R that is smooth and has an everywhere
time-like and future-pointing gradient. The event horizon H is defined as the
boundary of the causal past of the future null infinity H = ∂J−(I +). [2] We treat
the event horizon defined with respect to a single asymptotic end, unless otherwise
stated. In other words, the future null infinity, I +, is assumed to be connected.
The black hole region B is defined as the interior region of H , specifically, as B =
M \ J−(I +), and the exterior region E of the black hole region is its complement,
E = int(J−(I +)). We refer to the intersection of the black hole region and the
time slice Σ(t0) = {t = t0} as the black hole B(t0) = B ∩ Σ(t0) at time t = t0.
The exterior region at time t = t0 is, accordingly, written E (t0) = E ∩ Σ(t0).
One of most basic properties of the event horizon is that it is generated by null
geodesics without future endpoints. In general, the event horizon is not smoothly
imbedded into the space-time manifold M , but it has a wedge-like structure at
the past endpoints of the null geodesic generators, where distinct null geodesic
generators intersect. We call the set of past endpoints of null geodesic generators
of H , from which two or more null geodesic generators emanate, the crease set
S. [11, 12] When no crease set S exists between the time slices t = t1 and t = t2, the
null geodesic generators of H naturally define a diffeomorphism ∂B(t1) ≈ ∂B(t2).
Hence, the topological evolution of a black hole can take place only when the time
slice intersects the crease set S. Of course, the event horizon itself is a gauge-
independent object. Nevertheless, we often understand the dynamics of space-time
by scanning it along time slices. Thus, the topological evolution of a black hole
depends on the time function.
It is expected that Morse theory [14] provides useful techniques to analyze such
a process of topological evolution. Because the Morse theory is concerned with
functions on smooth manifolds, we first regularize H around the crease set S. The
event horizon is not necessarily smooth, even on H \ S, in the case that the future
null infinity I + has a pathological structure. [16] Here it is assumed that H is
smooth on H \ S. Then, small deformations of H near the crease set S will make
H a smooth hypersurface H̃ in M , while B̃(t0) remains deformed in such a manner
that ∂B̃(t0) = H̃ ∪ Σ(t0) holds and B̃(t0) remains homeomorphic to the original
black hole for all t0 ∈ R. This deformation is assumed to be such that the time
Figure 1. An example in which no smoothing procedure makes
t| eH a Morse function on H̃ . Here, the intersection of the crease set
S of the event horizon and t = t0 hypersurface has an accumulation
point.
function t| eH , which is the restriction of t on H̃, gives a Morse function on H̃ that
has only nondegenerate critical points, where the gradient of t| eH defined on H̃
becomes zero and where also the Hessian matrix (∂i∂jt| eH) of t| eH is nondegenerate.
Though this assumption should hold for a wide class of systems, it does not always
hold. Figure 1 gives an example for which no smoothing procedure makes the
induced time function t| eH a Morse function on H̃ , because the intersection of the
crease set S of the event horizon and the t = t0 hypersurface has an accumulation
point. It is highly nontrivial to determine whether such a smoothing procedure is
generically possible. It is, however, not easy nor the primary purpose of this article
to assertain the realm of validity of the assumption, and therefore we make this
assumption without inquiring into its validity.
According to the Morse Lemma, there is a local coordinate system {x1, · · · , xn}
on H̃ in the neighborhood of the critical point p ∈ H̃ such that the restriction t| eH
of the time function t on H̃ takes the form
t| eH(x
1, · · · , xn) = t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2.
The integer λ, ranging from 0 to n, is called the index of the critical point p. The
topology of the black hole B̃(t) changes when Σ(t) pass through critical points,
or equivalently, when the time function t takes critical values. This implies that
critical points appears only near the crease set S.
The gradient-like vector field for t| eH is defined to be the tangent vector field X
on H̃ such that Xt| eH > 0 holds on H̃ , except for critical points, and has the form
X = −2x1
− · · · − 2xλ
+ 2xλ+1
∂xλ+1
+ · · ·+ 2xn
near the critical point of index λ, in terms of the standard coordinate system
appearing in the Morse Lemma. We choose a gradient-like vector field X such that
it coincides with the future-directed tangent vector field of null geodesic generators
of H , except in a small neighborhood of the crease set S (Fig. 2). The effect of
a critical point p of index λ is equivalent to the attachment of a λ-handle. [14, 15]
The handlebody is just a topological n-disk Dn ≈ In (I = [0, 1]), but it is regarded
as the product space Dn ≈ Dλ×Dn−λ (Fig. 3). The λ-handle attachment to an n-
dimensional manifold N with a boundary consists of the set hλ = (Dλ ×Dn−λ, f),
where the attaching map f induces the imbedding of ∂Dλ × Dn−λ ⊂ ∂Dn into
∂N (Fig. 4). The new manifold obtained through the λ-handle attachment to N is
Figure 2. The smoothing procedure of the event horizon H . The
gradient-like vector field on H̃ can be constructed through a slight
deformation of the null geodesic generators of H . Here, the effect
of the crease set S has been replaced by that of the critical points
p1, p2 and p3.
Figure 3. The local structure around the critical point p of index
λ. It can be seen that H̃t(p)+ǫ is homeomorphic to H̃t(p)−ǫ with a
λ-handle attached.
given by
N ∪ hλ = N ∪ (Dλ ×Dn−λ)/(x ∼ f(x)), (x ∈ ∂Dλ ×Dn−λ).
Let us denote by H̃t0 the t ≤ t0 part of H̃. Then, H̃t(p)+ǫ (ǫ > 0) just above the
critical point p of index λ is homeomorphic (in fact diffeomorphic, taking account
of the smoothing procedure) to that just below p, H̃t(p)−ǫ attached with a λ-handle,
H̃t(p)+ǫ ≈ H̃t(p)−ǫ ∪ h
if there are no other critical points satisfying t(p)−ǫ ≤ t ≤ t(p)+ǫ. The handlebody
itself is denoted by hλ as well.
Let us consider several examples. The 0-handle attachment does not need an
attaching map f . It simply corresponds to the emergence of the (n − 1)-sphere
Sn−1 ≈ ∂Dn as a black hole horizon ∂B(t). A typical example is the creation of a
black hole (Fig. 5): A black hole always emerges as 0-handle attachment. The other
Figure 4. The attachment of a 1-handle and a 2-handle to a
3-manifold N creates a new 3-manifold N ∪ h1 ∪ h2.
Figure 5. The emergence of a black hole through a 0-handle attachment.
Figure 6. The emergence of a bubble in the black hole region by
0-handle attachment, which does not occur in the real black hole
space-times.
Figure 7. The collision of a pair of black holes, creating a single
black hole, is realized through 1-handle attachment.
possiblity is the creation of a bubble that is subset of J−(I +) in a black hole region
(Fig. 6). One might think that this corresponds to wormhole creation between the
internal and external regions of the event horizon. Although in the framework of the
standard Morse theory on H̃ , these two examples are indistinguishable, we below
see that the latter process is in fact impossible.
Next, we consider 1-handle attachment. A typical example is the collision of two
black holes. A 1-handle serves as a bridge connecting black holes, or it corresponds
to taking the connected sum of each component of multiple black holes (Fig. 7).
Figure 8. The bifurcation of one black hole into two is represeted
by an (n − 1)-handle attachment. This, however, never occurs in
real black hole space-times.
Figure 9. The structure of λ-handle. The core Dλ × {0} corre-
sponds to the stable submanifold with respect to the flow gener-
ated by the gradient-like vector field, and the co-core {0} ×Dn−λ
corresponds to the unstable submanifold.
The time reversal of the collision of black holes consists of the bifurcation of one
black hole into two. This would be realized through an (n− 1)-handle attachment,
if such a process were possible (Fig. 8). It is, however, well known that such a
process is forbidden. [2] In general, the time reversal of the λ-handle attachment
corresponds to (n− λ)-handle attachment.
Before discussing general cases, let us consider the structure of a handlebody.
Recall that a λ-handle consists of the product space Dλ × Dn−λ. The subset
Dλ × {0} ⊂ Dλ × Dn−λ is called the core of the handlebody, and {0} ×Dn−λ ⊂
Dλ ×Dn−λ is called the co-core. The core and co-core intersect transversely at a
point. This point can be regarded as a critical point p.
Let us refer to the subset Ws(p) of H̃
(1) Ws(p) = {q ∈ M | lim
expq tX = p}
which consists of points that converge to p along the flow generated by the gradient-
like vector field X , as the stable manifold with respect to the critical point p. The
stable manifold Ws(p) is homeomorphic to R
λ if the index of p is given by λ. [17]
Similarly, let us refer to the subset Wu(p) ⊂ H̃ consisting of points which converge
to p along the flow generated by (−X) as the unstable manifold with respect to p.
For the unstable manifold, Wu(p) ≈ R
n−λ holds. The portions of the stable and
unstable manifolds in the handlebody can be regarded as corresponding to the core
and co-core, respectively.
The effect of smoothing the event horizonH to H̃ is to deform the null vector field
generating H into a gradient-like vector field X . The primary difference between
the null geodesic generators and the flow generated by X is that the former does
not have future endpoints, but the latter can. Thus, there are admissible and
inadmissible processes for the smoothed manifold H̃. An admissible process is
given by H̃ , which is obtained from an in priciple realizable event horizon, while an
inadmissible one is constructed from a spurious event horizon, i.e., one that consists
of the null hypersurface containing null geodesic generators with a future endpoint.
3. The structure of the critical points
The spatial topology of a black hole changes only when the time function takes a
critical value. The time evolution of the black hole topology can be understood by
considering its local structure around critical points. To determine whether a given
topological change is admissible or inadmissible, it is not sufficient to consider only
the intrinsic structure of the event horizon. Rather, it is required to take account
of its imbedding structure relative to the space-time.
In a time slice, any point separate from H̃ belongs to either of the black hole
or the exterior of the black hole region. It is useful to consider the local behavior
of the black hole region or the exterior region near the critical point p. Let us call
the exterior E of the black hole region simply the exterior region, for brevity. The
exterior region is slightly deformed by the smoothing procedure. The deformed
exterior region is denoted by Ẽ , and the deformed exterior region at the time t by
Ẽ (t) = Ẽ ∩ Σ(t) = Σ(t) \ B̃(t).(2)
The 0-handle is placed at some t ≥ t(p). Such an attachment describes the emer-
gence of the black hole region at the critical point p and its expansion with time.
The emergence of a bubble, which consists of a part of J−(I +), in the background
of the black hole region would also be described by a 0-handle attachment. This,
however, never occurs, as we explain below in detail. Hence, a 0-handle attachment
always describes the creation of a black hole homeomorphic to the n-disk.
An n-handle attachment corresponds to the time reversal of a 0-handle attache-
ment. This process, however, never occurs in real black hole space-time. An n-
handle is defined for t ≤ t(p), which means that it terminates at the critical point
p. The crease set is isolated into critical points during the course of the smoothing
procedure. The gradient-like vector field, which can be regarded as being tangent
to the generator of the deformed event horizon H̃ , may have several inward (con-
verging) directions at the critical point due to this smoothing procedure, while the
original null generator of the event horizon does not have an inward direction at
the crease set. In the case of the n-handle, all the directions become inward at the
critical point. This implies that the null generators of the event horizon H must
have future endpoints at the critical point, which is, of course, impossible. It is
thus seen that an n-handle attachment never occurs in real black hole space-times.
The remaining cases are λ-handle attachments for 1 ≤ λ ≤ n − 1. In these
cases, the λ-handle lies on either side of the critical point p both in the future
[t > t(p)] and in the past [t < t(p)]. Then, we consider the case in which the handle
exists during the sufficiently small time interval t ∈ [t(p) − δ, t(p) + δ] (δ > 0), to
understand the topological change of the black hole region at the critical point p.
Figure 10. The neighborhood U of p is separated by hλ into the
future region, U+, and the past region, U−.
First, we introduce a coordinate system {t, xi} (i = 1, · · · , n) in the neighborhood
U of p, where t is a given function of time, and {xi} is the extension over U of
the cannonical coordinate appearing in the Morse Lemma such that each curve
(x1, · · · , xn) = [const] is timelike in U . We assume that U is the solid cylinder
given by t ∈ [t(p)−δ, t(p)+δ],
(xi)2 ≤ δ. In this coordinate system, the λ-handle
hλ is given by the saddle surface
t = t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2
in U , which is an acausal set if the constant δ is taken sufficiently small, since hλ
is tangent to the space-like hypersurface t = t(p) at p. Therefore, hλ separates
U into two open subsets, the future and past regions U+ and U− of U , where
U+ and U− are the subsets lying chronological future and past, respectively, of
hλ: U± = I±(hλ) ∩ U . Explicitly, the future and past regions U± are the regions
satisfying
t ≷ t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2
in U , respectively (Fig. 10).
Because the λ-handle is a subset of the black hole boundary H̃ , one of U± is
contained in the black hole region, B̃, and the other in the exterior region, Ẽ .
However, the future region U+ of U is always included in the black hole region, i.e.
U+ ⊂ B̃, and hence we have U− ⊂ Ẽ , since the horizon is the boundary of the past
set, J−(I +). Therefore, the black hole region B̃(t(p) − ǫ) ∩ U in U at the time
t = t(p)− ǫ just before the critical time is given by
(x1)2 + · · ·+ (xλ)2 > (xλ+1)2 + · · ·+ (xn)2 + ǫ,
which is homotopic to the (λ − 1)-sphere Sλ−1. (For λ = 1, S0 simply consists of
two points.) Similarly, B̃(t(p) + ǫ) ∩ U just after the critical time is given by
(x1)2 + · · ·+ (xλ)2 + ǫ > (xλ+1)2 + · · ·+ (xn)2,
which is homotopic to the n-disk. In this way, the black hole region restricted to
the small neighborhood of the critical point p is initially homotopic to a sphere.
Then, the internal region of the sphere is filled up at the critical time t = t(p) and
eventually becomes homotopically trivial. The exterior region, Ẽ (t) ∩ U , in U is
initially homotopic to an n-disk for t = t(p) − ǫ. Then, its (n − λ)-dimensional
direction is penetrated by the black hole region at t = t(p), and thus it becomes
homotopic to an (n− λ− 1)-sphere Sn−λ−1 for t = t(p) + ǫ. If the spurious event
horizon is also taken into account, the future region U+ might be a subset of Ẽ , and
therefore the past region U− might be a subset of B̃. Then, the black hole region in
the λ-handle might be homotopic to an n-disk initially and become homotopic to an
(n−λ−1)-sphere finally, and vice versa for the exterior region. Let us refer to such
a topological change of the black hole region B̃(t)∩U from a region homotopic to a
sphere to a region homotopic to a disk as a black handle attachment, and that from
a region homotopic to a disk to a region homotopic to the sphere as a white handle
attachment. The above observation shows that only a black handle attachment
occurs if a sufficiently small neighborhood of the critical point is considered. For
example, a collision of black holes corresponds to a black 1-handle attachment, while
the bifurcation of a black hole corresponds to a white (n − 1)-handle attachment
in the sense that the homotopy type of the exterior region Ẽ (t) ∩ U changes from
that of Sn−2 to that of Dn. This local argument also elucidates te reason that a
black hole collision is admissible while a black hole bifurcation, which is its time
reversal, is inadmissible. We also note that the effect of time reversal is to convert
a black λ-handle attachment into a white (n− λ)-handle attachment.
It is appropriate to refer to the 0-handle attachment corresponding to the cre-
ation of a black hole as a black 0-handle attachment. Then, the proposition above
also applies to a 0-handle attachment.
4. Connectedness of the exterior region
There also exist processes that are unrealizable due to global conditions. Let
us, for a moment, consider the event horizon in maximally extended Schwarzschild
space-time. Though we are interested in the event horizon defined with respect to
a specific asymptotic end, for the purpose of explanation, we examine the event
horizon defined with respect to a pair of asymptotic ends in Schwarzschild space-
time (Fig. 11).
Let I +1 and I
2 be the pair of future null infinities of the maximally extended
Schwarzschild space-time. The event horizon here is defined by H = ∂J−(I +1 ∪
2 ), which is nondifferentiable at the bifurcate horizon F = ∂J
−(I +1 )∩∂J
−(I +2 ).
Let t be a global time function and χ be a global radial coordinate function such
that each two-surface t, χ = [const] is invariant under the SO(3) isometry. These
coordinates are chosen such that the bifurcation surface F is located at t = χ = 0
and the event horizon H is determined by t = |χ| around F . The smoothed event
horizon H̃ is also taken to be invariant under the SO(3) isometry. Due to the
symmetry of the configuration, the time function t has critical points of degenerate
type. In fact, any point on bifurcate horizon F is critical. Here, we are not interested
in such a nongeneric situation. Instead, we consider a slightly different time slicing
determined by the new time function
t′ = t+ ǫ sin2
where ǫ > 0 is a sufficiently small positive constant and ϑ, which satisfies 0 ≤ ϑ ≤ π,
is the usual polar coordinate of the 2-sphere. Then, there appears only a pair of
isolated critical points at the north pole (ϑ = 0) and the south pole (ϑ = π) on
the bifurcate horizon F , and the time function t′ becomes the Morse function on
H̃. At the time t′ = 0, the black hole appears at the north pole. This is the
0-handle attachment. The black hole formed there grows into a geometrically thick
spherical shell with a hole at the south pole, which is nevertheless a topological
3-disk. At the time t′ = ǫ, the puncture at the south pole is filled, and the black
hole region becomes topologically S2 × [0, 1]. The deformed event horizon H̃ splits
into a disjoint union of a pair of 2-spheres. This is the 2-handle attachment.
This kind of 2-handle attachment occurs because the event horizon is defined
with respect to the two asymptotic ends, which is in general inadmissible if the
future null infinity is connected, as we assume from this point. To understand the
above statement, it should be noted that there is no process through which the
several connected components of the exterior region Ẽ (t) = Ẽ ∩ Σ(t) at time t
merge together at a later time because such a handle attachment is not admissible.
It is also seen that no connected component of Ẽ (t) disappears, because possi-
ble n-handle attachments are inadmissible. These facts imply that the number of
connected components of the exterior region Ẽ (t) cannot decrease with the time
function t. On the other hand, there is only one connected component of the exte-
rior region Ẽ (t) for sufficiently large t, because of the connectedness of I +. This
observation shows that the exterior region Ẽ (t) remains connected in any process.
The only possible process through which the number of connected components
of the exterior region Ẽ (t) changes is an (n− 1)-handle attachment, as constructed
above in the Schwarzschild space-time. This is because the subset Dλ × ∂Dn−λ of
H2 H1
t' = t' (p)
B (t'(p))
II 12
Figure 11. The figure on the left is a conformal diagram of the
maximally extended Schwarzschild space-time. The structure of
the event horizon defined with respect to the two asymptotic ends
is depicted on the right, with one dimension omitted. The shaded
region represents the black hole region at the critical time t = t(p).
This corresponds to the 2-handle attachment, where the exterior
region is separated into a pair of connected components.
the boundary of the λ-handle
∂hλ ≈ (∂Dλ ×Dn−λ) ∪ (Dλ × ∂Dn−λ),
namely the part of ∂hλ which is the complement of the preimage of the attaching
f : ∂hλ ⊃ ∂Dλ ×Dn−λ → H̃t,
is disconnected only when λ = n − 1. In this case, the homotopy type of the
exterior region Ẽ (t) changes from that of an n-disk to that of S0, namely two points.
Note, however, that this does not imply that the exterior region Ẽ (t) is always
separated into two disconnected parts through the (n− 1)-handle attachment. For
example, a transition from the black ring horizon ≈ Sn−2 × S1 to the spherical
black hole horizon ≈ Sn−1 is realized through a black (n− 1)-handle attachment,
which pinches the longitude {a point}×S1 ⊂ Sn−2 ×S1 into a point. The exterior
region Ẽ (t) remains connected all the while. Thus, there are both admissible and
inadmissible processes for (n−1)-handle attachments. An (n−1)-handle attachment
is inadmissible if it separates the exterior region Ẽ (t).
5. Concluding remarks
The arguments given in this paper are summerized by the following rules. As-
sume that (i) an (n+ 1)-dimensional space-time M is asymptotically flat and the
future null infinity I + is connected, or the event horizon H = ∂J−(I +) is defined
with respect to a single asymptotic end, (ii) the space-time M admits a smooth
global time function t, (iii) the event horizon H can be deformed so that the black
hole B̃(t) deformed accordingly at each time t is smooth and homeomorphic to orig-
inal one B(t) at each time t and the time function t becomes the Morse function
on H̃. Then, the topological evolution of the event horizon can be regarded as a
λ-handle attachment (0 ≤ λ ≤ n) subject to the following rules:
(1) The n-handle attachment is inadmissible.
(2) Only the black λ-handle attachment (0 ≤ λ ≤ n − 1), where the black
hole region in the neighborhood of the critical point varies from the region
homotopic to the sphere Sλ−1 (regarded as the empty set for λ = 0) to the
n-disk Dn, is admissible.
(3) The (n − 1)-handle attachment which separates the spatial section of the
exterior region of the black hole is inadmissible.
The first rule simply states that no connected component of a black hole disap-
pears. It also implies that if a bubble of the exterior region forms within the black
hole region, it does not vanish.
The second rule is concerned with the imbedding structure of the event horizon
relative to the space-time manifold. The neighborhood of the critical point is sep-
arated into two regions by the event horizon. One changes homotopically from a
sphere to a disk and the other from a disk to a sphere. We call it a black handle at-
tachment when the former corresponds to the black hole region and a white handle
attachment otherwise. Then, the second rule states that a white handle attach-
ment never occurs. The reverse process, in which a black hole region homotopically
changes from a disk to a sphere, is ruled out. A white 0-handle attachment, which
Figure 12. Black ring formation from a spherical black hole must
be non-axisymmetric in real black hole space-times.
describes the emergence of the exterior region, is also forbidden. This gives an-
other reason for the well-known result that a black hole cannot bifurcate, because
it corresponds to a white (n− 1)-handle attachment.
The second rule applies to more general situations. For example, let us consider
the topological evolution of the event horizon from Sn−1 to Sn−2 × S1 in (n+ 1)-
dimensional space-time (n ≥ 3). When it is realized with a single critical point, it
corresponds to a 1-handle attachment. Here, one might expect two possibilities if
the second rule is not considered. One possibility is that the 1-handle is attached in
the exterior region of the black hole. This is locally equivalent to the merging of a
pair of black holes, where these two black holes are connected elsewhere irrelevant.
The other possibility is that it is attached from the inside such that the 1-handle
pierces the black hole region. In asymptotically flat space-times, only the latter
includes axisymmetric configurations such that a spherical black hole is pinched
out along the symmetric axis; here the axisymmetric configuration is such that
the space-time possesses the SO(n − 1) isometry and the time slicing respects
this symmetry. However, this latter possibility corresponds to a white 1-handle
attachment, which is impossible, and only the former, which corresponds to a black
1-handle attachment, is possible. In particular, a transition from a spherical event
horizon (≈ Sn−1) to a black ring horizon (≈ Sn−2×S1) in asymptotically flat space-
times is always non-axisymmetric in the sense that such a configuration cannot
possess SO(n− 1) symmetry (Fig. 12).
While the apparent horizon must be diffeomorphic to a two-sphere in four-
dimensional space-times under the dominant energy condition, a torus event hori-
zon may appear, even under the dominant energy condition, via a black 1-handle
attachment to the spherical horizon. More generally, an event horizon with an
arbitrary number of genura may be formed by several black 1-handle attachments.
The third rule is not directly determined by the local structure of the critical
point. It states that the exterior region E (t) = E ∩ Σ(t) at each time is always
connected under the assumption that I + is connected. Thus, the possibility that
there forms a bubble of the exterior region inside the black hole horizon is ruled
out. It should, however, be noted that such a process is possible if I + consists
of several connected components. This may also be related to the topological
censorship theorem. [19] The topological censorship theorem states that all causal
curves from I − to I + are homotopic under the null energy condition. This also
forbids the formation of a bubble of the exterior region inside the black hole, because
otherwise there would be two nonhomotopic causal curves from I − to I +, one
passing inside the horizon and the other outside. Our argument, however, does not
depend on energy conditions.
References
[1] S. W. Hawking, Commun. Math. Phys. 25 (1972), 152.
[2] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-times (London, Cam-
bridge University Press, 1973).
[3] D. Gannon, Gen. Relat. Gravit. 7 (1974), 219.
[4] P. T. Chrusciel and R. M. Wald, Class. Quantum Grav. 11 (1994), L147; gr-qc/9410004.
[5] T. Jacobson and S. Venkataramani, Class. Quantum Grav. 12 (1995), 1055; gr-qc/9410023.
[6] S. F. Browdy and G. J. Galloway, J. Math. Phys. 36 (1995), 4952.
[7] M. l. Cai and G. J. Galloway, Class. Quantum Grav. 18 (2001), 2707; hep-th/0102149.
[8] C. Helfgott, Y. Oz and Y. Yanay, JHEP 0602 (2006), 025; hep-th/0509013.
[9] G. J. Galloway and R. Schoen, Commun. Math. Phys. 266 (2006), 571; gr-qc/0509107.
[10] G. J. Galloway, gr-qc/0608118.
[11] M. Siino, Phys. Rev. D 58 (1998), 104016; gr-qc/9701003.
[12] M. Siino and T. Koike, arXiv:gr-qc/0405056.
[13] S. L. Shapiro, S. A. Teukolsky and J. Winicour, Phys. Rev. D 52 (1995), 6982.
[14] J. Milnor, Lectures on h-cobordism theorem (Princeton, Princeton University Press, 1965).
[15] I. Tamura, The differential topology (Tokyo, Iwanami Shoten Publishers, 1978).
[16] P. T. Chrusciel and G. J. Galloway, Commun. Math. Phys. 193 (1998), 449 gr-qc/9611032.
[17] S. Smale, Ann. of Math. 74 (1961), 199.
[18] R. Emparan and H. S. Reall, Phys. Rev. Lett. 88 (2002), 101101; hep-th/0110260.
[19] J. L. Friedman, K. Schleich and D. M. Witt, Phys. Rev. Lett. 71 (1993), 1486 [Errata; 75
(1995), 1872]; gr-qc/9305017.
http://arxiv.org/abs/gr-qc/9410004
http://arxiv.org/abs/gr-qc/9410023
http://arxiv.org/abs/hep-th/0102149
http://arxiv.org/abs/hep-th/0509013
http://arxiv.org/abs/gr-qc/0509107
http://arxiv.org/abs/gr-qc/0608118
http://arxiv.org/abs/gr-qc/9701003
http://arxiv.org/abs/gr-qc/0405056
http://arxiv.org/abs/gr-qc/9611032
http://arxiv.org/abs/hep-th/0110260
http://arxiv.org/abs/gr-qc/9305017
	1. Introduction
	2. The Morse theory for event horizons
	3. The structure of the critical points
	4. Connectedness of the exterior region
	5. Concluding remarks
	References

  The topological structure of the event horizon has been investigated in terms
of the Morse theory. The elementary process of topological evolution can be
understood as a handle attachment. It has been found that there are certain
constraints on the nature of black hole topological evolution: (i) There are n
kinds of handle attachments in (n+1)-dimensional black hole space-times. (ii)
Handles are further classified as either of black or white type, and only black
handles appear in real black hole space-times. (iii) The spatial section of an
exterior of the black hole region is always connected. As a corollary, it is
shown that the formation of a black hole with an S**(n-2) x S**1 horizon from
that with an S**(n-1) horizon must be non-axisymmetric in asymptotically flat
space-times.

Introduction
Black holes in space-times of greater than or equal to five dimensions have rich
topological structure. According to the well-known results of Hawking concerning
the topology of black holes in four-dimensional space-time, the apparent horizon
or the spatial section of the stationary event horizon is necessarily diffeomorphic
to a 2-sphere. [1, 2] This follows from the fact that the total curvature, which is
the integral of the intrinsic scalar curvature over the horizon, is positive under the
dominant energy condition and from the Gauss-Bonnet theorem. Alternative and
improved proofs of Hawking’s theorem have been given by several authors. [3, 4, 5, 6]
However in higher dimensional space-times, an apparent horizon or the spatial
section of the stationary event horizon may not be a topological sphere, [7, 8, 9, 10]
because the Gauss-Bonnet theorem does not hold in such cases. Nevertheless,
the positivity of the total curvature of the horizon still holds. This puts certain
topological restrictions on the black hole topology, though they are rather weak. For
example, the apparent horizon in five-dimensional space-time can consist of finitely
many connected sums of copies of S3/Γ and copies of S2 × S1. In fact, exact
solutions representing a black hole space-time possessing a horizon of nonspherical
topology have recently been found in five-dimensional general relativity. When such
black holes with nontrivial topologies are regarded as being formed in the course
of gravitational collapse, questions regarding the evolution of the topology of black
holes naturally arise. Our purpose here is to understand the time evolution of the
topology of event horizons in a general setting. The relation between the crease set,
where the event horizon is nondifferentiable, and the topology of the event horizon
is studied in Refs. [11, 12, 13] for four-dimensional space-times. In the present
work, we carry out a systematic investigation and find useful rules to determine
admissible processes of topological evolution for time slicing of a black hole.
Our approach is to utilize the Morse theory [14, 15] in differential topology. The
Morse theory is useful for the purpose of understanding the topology of smooth
http://arxiv.org/abs/0704.0100v3
manifolds. The basic tool used in this approach is a smooth function on a differ-
entiable manifold. The event horizon, however, is not a differentiable manifold but
has a wedge-like structure at the past endpoints of the null geodesic generators
of the horizon. For this reason, we first smooth the wedge. Then, the smooth
time function which is assumed to exist plays the role of the Morse function on the
smoothed event horizon. According to the Morse theory, the topological evolution
of the event horizon can then be decomposed into elementary processes called “han-
dle attachments.” In such a process, starting with a spherical horizon, one adds
several handles, each characterized by the index of the critical points of the Morse
function, which is an integer ranging from 0 to n (the dimension of the smoothed
horizon as a differentiable manifold).
The purpose of the present article is to show that there are several constraints
on the handle attachments for real black hole space-times.
2. The Morse theory for event horizons
Let M be an (n+1)-dimensional asymptotically flat space-time. We require the
existence of a global time function t : M → R that is smooth and has an everywhere
time-like and future-pointing gradient. The event horizon H is defined as the
boundary of the causal past of the future null infinity H = ∂J−(I +). [2] We treat
the event horizon defined with respect to a single asymptotic end, unless otherwise
stated. In other words, the future null infinity, I +, is assumed to be connected.
The black hole region B is defined as the interior region of H , specifically, as B =
M \ J−(I +), and the exterior region E of the black hole region is its complement,
E = int(J−(I +)). We refer to the intersection of the black hole region and the
time slice Σ(t0) = {t = t0} as the black hole B(t0) = B ∩ Σ(t0) at time t = t0.
The exterior region at time t = t0 is, accordingly, written E (t0) = E ∩ Σ(t0).
One of most basic properties of the event horizon is that it is generated by null
geodesics without future endpoints. In general, the event horizon is not smoothly
imbedded into the space-time manifold M , but it has a wedge-like structure at
the past endpoints of the null geodesic generators, where distinct null geodesic
generators intersect. We call the set of past endpoints of null geodesic generators
of H , from which two or more null geodesic generators emanate, the crease set
S. [11, 12] When no crease set S exists between the time slices t = t1 and t = t2, the
null geodesic generators of H naturally define a diffeomorphism ∂B(t1) ≈ ∂B(t2).
Hence, the topological evolution of a black hole can take place only when the time
slice intersects the crease set S. Of course, the event horizon itself is a gauge-
independent object. Nevertheless, we often understand the dynamics of space-time
by scanning it along time slices. Thus, the topological evolution of a black hole
depends on the time function.
It is expected that Morse theory [14] provides useful techniques to analyze such
a process of topological evolution. Because the Morse theory is concerned with
functions on smooth manifolds, we first regularize H around the crease set S. The
event horizon is not necessarily smooth, even on H \ S, in the case that the future
null infinity I + has a pathological structure. [16] Here it is assumed that H is
smooth on H \ S. Then, small deformations of H near the crease set S will make
H a smooth hypersurface H̃ in M , while B̃(t0) remains deformed in such a manner
that ∂B̃(t0) = H̃ ∪ Σ(t0) holds and B̃(t0) remains homeomorphic to the original
black hole for all t0 ∈ R. This deformation is assumed to be such that the time
Figure 1. An example in which no smoothing procedure makes
t| eH a Morse function on H̃ . Here, the intersection of the crease set
S of the event horizon and t = t0 hypersurface has an accumulation
point.
function t| eH , which is the restriction of t on H̃, gives a Morse function on H̃ that
has only nondegenerate critical points, where the gradient of t| eH defined on H̃
becomes zero and where also the Hessian matrix (∂i∂jt| eH) of t| eH is nondegenerate.
Though this assumption should hold for a wide class of systems, it does not always
hold. Figure 1 gives an example for which no smoothing procedure makes the
induced time function t| eH a Morse function on H̃ , because the intersection of the
crease set S of the event horizon and the t = t0 hypersurface has an accumulation
point. It is highly nontrivial to determine whether such a smoothing procedure is
generically possible. It is, however, not easy nor the primary purpose of this article
to assertain the realm of validity of the assumption, and therefore we make this
assumption without inquiring into its validity.
According to the Morse Lemma, there is a local coordinate system {x1, · · · , xn}
on H̃ in the neighborhood of the critical point p ∈ H̃ such that the restriction t| eH
of the time function t on H̃ takes the form
t| eH(x
1, · · · , xn) = t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2.
The integer λ, ranging from 0 to n, is called the index of the critical point p. The
topology of the black hole B̃(t) changes when Σ(t) pass through critical points,
or equivalently, when the time function t takes critical values. This implies that
critical points appears only near the crease set S.
The gradient-like vector field for t| eH is defined to be the tangent vector field X
on H̃ such that Xt| eH > 0 holds on H̃ , except for critical points, and has the form
X = −2x1
− · · · − 2xλ
+ 2xλ+1
∂xλ+1
+ · · ·+ 2xn
near the critical point of index λ, in terms of the standard coordinate system
appearing in the Morse Lemma. We choose a gradient-like vector field X such that
it coincides with the future-directed tangent vector field of null geodesic generators
of H , except in a small neighborhood of the crease set S (Fig. 2). The effect of
a critical point p of index λ is equivalent to the attachment of a λ-handle. [14, 15]
The handlebody is just a topological n-disk Dn ≈ In (I = [0, 1]), but it is regarded
as the product space Dn ≈ Dλ×Dn−λ (Fig. 3). The λ-handle attachment to an n-
dimensional manifold N with a boundary consists of the set hλ = (Dλ ×Dn−λ, f),
where the attaching map f induces the imbedding of ∂Dλ × Dn−λ ⊂ ∂Dn into
∂N (Fig. 4). The new manifold obtained through the λ-handle attachment to N is
Figure 2. The smoothing procedure of the event horizon H . The
gradient-like vector field on H̃ can be constructed through a slight
deformation of the null geodesic generators of H . Here, the effect
of the crease set S has been replaced by that of the critical points
p1, p2 and p3.
Figure 3. The local structure around the critical point p of index
λ. It can be seen that H̃t(p)+ǫ is homeomorphic to H̃t(p)−ǫ with a
λ-handle attached.
given by
N ∪ hλ = N ∪ (Dλ ×Dn−λ)/(x ∼ f(x)), (x ∈ ∂Dλ ×Dn−λ).
Let us denote by H̃t0 the t ≤ t0 part of H̃. Then, H̃t(p)+ǫ (ǫ > 0) just above the
critical point p of index λ is homeomorphic (in fact diffeomorphic, taking account
of the smoothing procedure) to that just below p, H̃t(p)−ǫ attached with a λ-handle,
H̃t(p)+ǫ ≈ H̃t(p)−ǫ ∪ h
if there are no other critical points satisfying t(p)−ǫ ≤ t ≤ t(p)+ǫ. The handlebody
itself is denoted by hλ as well.
Let us consider several examples. The 0-handle attachment does not need an
attaching map f . It simply corresponds to the emergence of the (n − 1)-sphere
Sn−1 ≈ ∂Dn as a black hole horizon ∂B(t). A typical example is the creation of a
black hole (Fig. 5): A black hole always emerges as 0-handle attachment. The other
Figure 4. The attachment of a 1-handle and a 2-handle to a
3-manifold N creates a new 3-manifold N ∪ h1 ∪ h2.
Figure 5. The emergence of a black hole through a 0-handle attachment.
Figure 6. The emergence of a bubble in the black hole region by
0-handle attachment, which does not occur in the real black hole
space-times.
Figure 7. The collision of a pair of black holes, creating a single
black hole, is realized through 1-handle attachment.
possiblity is the creation of a bubble that is subset of J−(I +) in a black hole region
(Fig. 6). One might think that this corresponds to wormhole creation between the
internal and external regions of the event horizon. Although in the framework of the
standard Morse theory on H̃ , these two examples are indistinguishable, we below
see that the latter process is in fact impossible.
Next, we consider 1-handle attachment. A typical example is the collision of two
black holes. A 1-handle serves as a bridge connecting black holes, or it corresponds
to taking the connected sum of each component of multiple black holes (Fig. 7).
Figure 8. The bifurcation of one black hole into two is represeted
by an (n − 1)-handle attachment. This, however, never occurs in
real black hole space-times.
Figure 9. The structure of λ-handle. The core Dλ × {0} corre-
sponds to the stable submanifold with respect to the flow gener-
ated by the gradient-like vector field, and the co-core {0} ×Dn−λ
corresponds to the unstable submanifold.
The time reversal of the collision of black holes consists of the bifurcation of one
black hole into two. This would be realized through an (n− 1)-handle attachment,
if such a process were possible (Fig. 8). It is, however, well known that such a
process is forbidden. [2] In general, the time reversal of the λ-handle attachment
corresponds to (n− λ)-handle attachment.
Before discussing general cases, let us consider the structure of a handlebody.
Recall that a λ-handle consists of the product space Dλ × Dn−λ. The subset
Dλ × {0} ⊂ Dλ × Dn−λ is called the core of the handlebody, and {0} ×Dn−λ ⊂
Dλ ×Dn−λ is called the co-core. The core and co-core intersect transversely at a
point. This point can be regarded as a critical point p.
Let us refer to the subset Ws(p) of H̃
(1) Ws(p) = {q ∈ M | lim
expq tX = p}
which consists of points that converge to p along the flow generated by the gradient-
like vector field X , as the stable manifold with respect to the critical point p. The
stable manifold Ws(p) is homeomorphic to R
λ if the index of p is given by λ. [17]
Similarly, let us refer to the subset Wu(p) ⊂ H̃ consisting of points which converge
to p along the flow generated by (−X) as the unstable manifold with respect to p.
For the unstable manifold, Wu(p) ≈ R
n−λ holds. The portions of the stable and
unstable manifolds in the handlebody can be regarded as corresponding to the core
and co-core, respectively.
The effect of smoothing the event horizonH to H̃ is to deform the null vector field
generating H into a gradient-like vector field X . The primary difference between
the null geodesic generators and the flow generated by X is that the former does
not have future endpoints, but the latter can. Thus, there are admissible and
inadmissible processes for the smoothed manifold H̃. An admissible process is
given by H̃ , which is obtained from an in priciple realizable event horizon, while an
inadmissible one is constructed from a spurious event horizon, i.e., one that consists
of the null hypersurface containing null geodesic generators with a future endpoint.
3. The structure of the critical points
The spatial topology of a black hole changes only when the time function takes a
critical value. The time evolution of the black hole topology can be understood by
considering its local structure around critical points. To determine whether a given
topological change is admissible or inadmissible, it is not sufficient to consider only
the intrinsic structure of the event horizon. Rather, it is required to take account
of its imbedding structure relative to the space-time.
In a time slice, any point separate from H̃ belongs to either of the black hole
or the exterior of the black hole region. It is useful to consider the local behavior
of the black hole region or the exterior region near the critical point p. Let us call
the exterior E of the black hole region simply the exterior region, for brevity. The
exterior region is slightly deformed by the smoothing procedure. The deformed
exterior region is denoted by Ẽ , and the deformed exterior region at the time t by
Ẽ (t) = Ẽ ∩ Σ(t) = Σ(t) \ B̃(t).(2)
The 0-handle is placed at some t ≥ t(p). Such an attachment describes the emer-
gence of the black hole region at the critical point p and its expansion with time.
The emergence of a bubble, which consists of a part of J−(I +), in the background
of the black hole region would also be described by a 0-handle attachment. This,
however, never occurs, as we explain below in detail. Hence, a 0-handle attachment
always describes the creation of a black hole homeomorphic to the n-disk.
An n-handle attachment corresponds to the time reversal of a 0-handle attache-
ment. This process, however, never occurs in real black hole space-time. An n-
handle is defined for t ≤ t(p), which means that it terminates at the critical point
p. The crease set is isolated into critical points during the course of the smoothing
procedure. The gradient-like vector field, which can be regarded as being tangent
to the generator of the deformed event horizon H̃ , may have several inward (con-
verging) directions at the critical point due to this smoothing procedure, while the
original null generator of the event horizon does not have an inward direction at
the crease set. In the case of the n-handle, all the directions become inward at the
critical point. This implies that the null generators of the event horizon H must
have future endpoints at the critical point, which is, of course, impossible. It is
thus seen that an n-handle attachment never occurs in real black hole space-times.
The remaining cases are λ-handle attachments for 1 ≤ λ ≤ n − 1. In these
cases, the λ-handle lies on either side of the critical point p both in the future
[t > t(p)] and in the past [t < t(p)]. Then, we consider the case in which the handle
exists during the sufficiently small time interval t ∈ [t(p) − δ, t(p) + δ] (δ > 0), to
understand the topological change of the black hole region at the critical point p.
Figure 10. The neighborhood U of p is separated by hλ into the
future region, U+, and the past region, U−.
First, we introduce a coordinate system {t, xi} (i = 1, · · · , n) in the neighborhood
U of p, where t is a given function of time, and {xi} is the extension over U of
the cannonical coordinate appearing in the Morse Lemma such that each curve
(x1, · · · , xn) = [const] is timelike in U . We assume that U is the solid cylinder
given by t ∈ [t(p)−δ, t(p)+δ],
(xi)2 ≤ δ. In this coordinate system, the λ-handle
hλ is given by the saddle surface
t = t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2
in U , which is an acausal set if the constant δ is taken sufficiently small, since hλ
is tangent to the space-like hypersurface t = t(p) at p. Therefore, hλ separates
U into two open subsets, the future and past regions U+ and U− of U , where
U+ and U− are the subsets lying chronological future and past, respectively, of
hλ: U± = I±(hλ) ∩ U . Explicitly, the future and past regions U± are the regions
satisfying
t ≷ t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2
in U , respectively (Fig. 10).
Because the λ-handle is a subset of the black hole boundary H̃ , one of U± is
contained in the black hole region, B̃, and the other in the exterior region, Ẽ .
However, the future region U+ of U is always included in the black hole region, i.e.
U+ ⊂ B̃, and hence we have U− ⊂ Ẽ , since the horizon is the boundary of the past
set, J−(I +). Therefore, the black hole region B̃(t(p) − ǫ) ∩ U in U at the time
t = t(p)− ǫ just before the critical time is given by
(x1)2 + · · ·+ (xλ)2 > (xλ+1)2 + · · ·+ (xn)2 + ǫ,
which is homotopic to the (λ − 1)-sphere Sλ−1. (For λ = 1, S0 simply consists of
two points.) Similarly, B̃(t(p) + ǫ) ∩ U just after the critical time is given by
(x1)2 + · · ·+ (xλ)2 + ǫ > (xλ+1)2 + · · ·+ (xn)2,
which is homotopic to the n-disk. In this way, the black hole region restricted to
the small neighborhood of the critical point p is initially homotopic to a sphere.
Then, the internal region of the sphere is filled up at the critical time t = t(p) and
eventually becomes homotopically trivial. The exterior region, Ẽ (t) ∩ U , in U is
initially homotopic to an n-disk for t = t(p) − ǫ. Then, its (n − λ)-dimensional
direction is penetrated by the black hole region at t = t(p), and thus it becomes
homotopic to an (n− λ− 1)-sphere Sn−λ−1 for t = t(p) + ǫ. If the spurious event
horizon is also taken into account, the future region U+ might be a subset of Ẽ , and
therefore the past region U− might be a subset of B̃. Then, the black hole region in
the λ-handle might be homotopic to an n-disk initially and become homotopic to an
(n−λ−1)-sphere finally, and vice versa for the exterior region. Let us refer to such
a topological change of the black hole region B̃(t)∩U from a region homotopic to a
sphere to a region homotopic to a disk as a black handle attachment, and that from
a region homotopic to a disk to a region homotopic to the sphere as a white handle
attachment. The above observation shows that only a black handle attachment
occurs if a sufficiently small neighborhood of the critical point is considered. For
example, a collision of black holes corresponds to a black 1-handle attachment, while
the bifurcation of a black hole corresponds to a white (n − 1)-handle attachment
in the sense that the homotopy type of the exterior region Ẽ (t) ∩ U changes from
that of Sn−2 to that of Dn. This local argument also elucidates te reason that a
black hole collision is admissible while a black hole bifurcation, which is its time
reversal, is inadmissible. We also note that the effect of time reversal is to convert
a black λ-handle attachment into a white (n− λ)-handle attachment.
It is appropriate to refer to the 0-handle attachment corresponding to the cre-
ation of a black hole as a black 0-handle attachment. Then, the proposition above
also applies to a 0-handle attachment.
4. Connectedness of the exterior region
There also exist processes that are unrealizable due to global conditions. Let
us, for a moment, consider the event horizon in maximally extended Schwarzschild
space-time. Though we are interested in the event horizon defined with respect to
a specific asymptotic end, for the purpose of explanation, we examine the event
horizon defined with respect to a pair of asymptotic ends in Schwarzschild space-
time (Fig. 11).
Let I +1 and I
2 be the pair of future null infinities of the maximally extended
Schwarzschild space-time. The event horizon here is defined by H = ∂J−(I +1 ∪
2 ), which is nondifferentiable at the bifurcate horizon F = ∂J
−(I +1 )∩∂J
−(I +2 ).
Let t be a global time function and χ be a global radial coordinate function such
that each two-surface t, χ = [const] is invariant under the SO(3) isometry. These
coordinates are chosen such that the bifurcation surface F is located at t = χ = 0
and the event horizon H is determined by t = |χ| around F . The smoothed event
horizon H̃ is also taken to be invariant under the SO(3) isometry. Due to the
symmetry of the configuration, the time function t has critical points of degenerate
type. In fact, any point on bifurcate horizon F is critical. Here, we are not interested
in such a nongeneric situation. Instead, we consider a slightly different time slicing
determined by the new time function
t′ = t+ ǫ sin2
where ǫ > 0 is a sufficiently small positive constant and ϑ, which satisfies 0 ≤ ϑ ≤ π,
is the usual polar coordinate of the 2-sphere. Then, there appears only a pair of
isolated critical points at the north pole (ϑ = 0) and the south pole (ϑ = π) on
the bifurcate horizon F , and the time function t′ becomes the Morse function on
H̃. At the time t′ = 0, the black hole appears at the north pole. This is the
0-handle attachment. The black hole formed there grows into a geometrically thick
spherical shell with a hole at the south pole, which is nevertheless a topological
3-disk. At the time t′ = ǫ, the puncture at the south pole is filled, and the black
hole region becomes topologically S2 × [0, 1]. The deformed event horizon H̃ splits
into a disjoint union of a pair of 2-spheres. This is the 2-handle attachment.
This kind of 2-handle attachment occurs because the event horizon is defined
with respect to the two asymptotic ends, which is in general inadmissible if the
future null infinity is connected, as we assume from this point. To understand the
above statement, it should be noted that there is no process through which the
several connected components of the exterior region Ẽ (t) = Ẽ ∩ Σ(t) at time t
merge together at a later time because such a handle attachment is not admissible.
It is also seen that no connected component of Ẽ (t) disappears, because possi-
ble n-handle attachments are inadmissible. These facts imply that the number of
connected components of the exterior region Ẽ (t) cannot decrease with the time
function t. On the other hand, there is only one connected component of the exte-
rior region Ẽ (t) for sufficiently large t, because of the connectedness of I +. This
observation shows that the exterior region Ẽ (t) remains connected in any process.
The only possible process through which the number of connected components
of the exterior region Ẽ (t) changes is an (n− 1)-handle attachment, as constructed
above in the Schwarzschild space-time. This is because the subset Dλ × ∂Dn−λ of
H2 H1
t' = t' (p)
B (t'(p))
II 12
Figure 11. The figure on the left is a conformal diagram of the
maximally extended Schwarzschild space-time. The structure of
the event horizon defined with respect to the two asymptotic ends
is depicted on the right, with one dimension omitted. The shaded
region represents the black hole region at the critical time t = t(p).
This corresponds to the 2-handle attachment, where the exterior
region is separated into a pair of connected components.
the boundary of the λ-handle
∂hλ ≈ (∂Dλ ×Dn−λ) ∪ (Dλ × ∂Dn−λ),
namely the part of ∂hλ which is the complement of the preimage of the attaching
f : ∂hλ ⊃ ∂Dλ ×Dn−λ → H̃t,
is disconnected only when λ = n − 1. In this case, the homotopy type of the
exterior region Ẽ (t) changes from that of an n-disk to that of S0, namely two points.
Note, however, that this does not imply that the exterior region Ẽ (t) is always
separated into two disconnected parts through the (n− 1)-handle attachment. For
example, a transition from the black ring horizon ≈ Sn−2 × S1 to the spherical
black hole horizon ≈ Sn−1 is realized through a black (n− 1)-handle attachment,
which pinches the longitude {a point}×S1 ⊂ Sn−2 ×S1 into a point. The exterior
region Ẽ (t) remains connected all the while. Thus, there are both admissible and
inadmissible processes for (n−1)-handle attachments. An (n−1)-handle attachment
is inadmissible if it separates the exterior region Ẽ (t).
5. Concluding remarks
The arguments given in this paper are summerized by the following rules. As-
sume that (i) an (n+ 1)-dimensional space-time M is asymptotically flat and the
future null infinity I + is connected, or the event horizon H = ∂J−(I +) is defined
with respect to a single asymptotic end, (ii) the space-time M admits a smooth
global time function t, (iii) the event horizon H can be deformed so that the black
hole B̃(t) deformed accordingly at each time t is smooth and homeomorphic to orig-
inal one B(t) at each time t and the time function t becomes the Morse function
on H̃. Then, the topological evolution of the event horizon can be regarded as a
λ-handle attachment (0 ≤ λ ≤ n) subject to the following rules:
(1) The n-handle attachment is inadmissible.
(2) Only the black λ-handle attachment (0 ≤ λ ≤ n − 1), where the black
hole region in the neighborhood of the critical point varies from the region
homotopic to the sphere Sλ−1 (regarded as the empty set for λ = 0) to the
n-disk Dn, is admissible.
(3) The (n − 1)-handle attachment which separates the spatial section of the
exterior region of the black hole is inadmissible.
The first rule simply states that no connected component of a black hole disap-
pears. It also implies that if a bubble of the exterior region forms within the black
hole region, it does not vanish.
The second rule is concerned with the imbedding structure of the event horizon
relative to the space-time manifold. The neighborhood of the critical point is sep-
arated into two regions by the event horizon. One changes homotopically from a
sphere to a disk and the other from a disk to a sphere. We call it a black handle at-
tachment when the former corresponds to the black hole region and a white handle
attachment otherwise. Then, the second rule states that a white handle attach-
ment never occurs. The reverse process, in which a black hole region homotopically
changes from a disk to a sphere, is ruled out. A white 0-handle attachment, which
Figure 12. Black ring formation from a spherical black hole must
be non-axisymmetric in real black hole space-times.
describes the emergence of the exterior region, is also forbidden. This gives an-
other reason for the well-known result that a black hole cannot bifurcate, because
it corresponds to a white (n− 1)-handle attachment.
The second rule applies to more general situations. For example, let us consider
the topological evolution of the event horizon from Sn−1 to Sn−2 × S1 in (n+ 1)-
dimensional space-time (n ≥ 3). When it is realized with a single critical point, it
corresponds to a 1-handle attachment. Here, one might expect two possibilities if
the second rule is not considered. One possibility is that the 1-handle is attached in
the exterior region of the black hole. This is locally equivalent to the merging of a
pair of black holes, where these two black holes are connected elsewhere irrelevant.
The other possibility is that it is attached from the inside such that the 1-handle
pierces the black hole region. In asymptotically flat space-times, only the latter
includes axisymmetric configurations such that a spherical black hole is pinched
out along the symmetric axis; here the axisymmetric configuration is such that
the space-time possesses the SO(n − 1) isometry and the time slicing respects
this symmetry. However, this latter possibility corresponds to a white 1-handle
attachment, which is impossible, and only the former, which corresponds to a black
1-handle attachment, is possible. In particular, a transition from a spherical event
horizon (≈ Sn−1) to a black ring horizon (≈ Sn−2×S1) in asymptotically flat space-
times is always non-axisymmetric in the sense that such a configuration cannot
possess SO(n− 1) symmetry (Fig. 12).
While the apparent horizon must be diffeomorphic to a two-sphere in four-
dimensional space-times under the dominant energy condition, a torus event hori-
zon may appear, even under the dominant energy condition, via a black 1-handle
attachment to the spherical horizon. More generally, an event horizon with an
arbitrary number of genura may be formed by several black 1-handle attachments.
The third rule is not directly determined by the local structure of the critical
point. It states that the exterior region E (t) = E ∩ Σ(t) at each time is always
connected under the assumption that I + is connected. Thus, the possibility that
there forms a bubble of the exterior region inside the black hole horizon is ruled
out. It should, however, be noted that such a process is possible if I + consists
of several connected components. This may also be related to the topological
censorship theorem. [19] The topological censorship theorem states that all causal
curves from I − to I + are homotopic under the null energy condition. This also
forbids the formation of a bubble of the exterior region inside the black hole, because
otherwise there would be two nonhomotopic causal curves from I − to I +, one
passing inside the horizon and the other outside. Our argument, however, does not
depend on energy conditions.
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	1. Introduction
	2. The Morse theory for event horizons
	3. The structure of the critical points
	4. Connectedness of the exterior region
	5. Concluding remarks
	References
