INSTABILITY OF HORIZONS OF TYPE II

CHAPTER2

IV. INSTABILITY OF HORIZONS OF TYPE II

~ ~ ~ ~

this holds prior to Sand «!> and ~1 ^(~i) are smooth, ~1 ^(~;) will carry this region on which«!> is nonzero arbitrarily down into the past along the generators of S But when we move a sufficiently large affine distance into the past along these generators we clearly enter the region J-[{ v =v (p _{0 )}]} in which the generators along P (C) are past endless generators of j\zPo) and hence of j+(N₂). Therefore any neighborhood in M of P (C) in this region intersects a piece of M not contained in J±(N_{2 )} [Fig. 3]. But

~ ~

again by the smoothness of«!> and as ~1(«!>)=0 (~; («j>)=:O), ^«1> will be nonzero at all points of P ( C) in this region and thereby be nonzero in a neighborhood in M of any point of P (C) there, contradicting the condition (b) of the theorem. Thus the assumption that «!> is smooth near S=( u =!} is contradictory and must be false, and the field «!>

must develop singularities on S proving the theorem. D

The singularity of ^«1> on S will in most cases be of the form «!>~ on P ( C) for v (p ₀)<v <v (p ₁⁾ (bounded or unbounded) whereas

«1>=0

on S outside P (C), with pos- sibly an added smooth background field on Swhich satisfies «j>8 (p )=0 V p e P( C). Thus even though the field itself might be bounded near S, some of its derivatives will diverge on the two-surface P (curve C) in S However, if the field equations are

~

linear, exactly the same argument we will use in proving theorem 2 will imply (as ~1

~~

or ~1^,~2 vanish on the surface P or the curve C in S) that ^«1> actually diverges on the set P(or on C) inS

a) The equations are linear.

b) There is a consistent (noncharacteristic) initial-value formalism for the field

( cr }

and the evolution equations it satisfies, with local existence and uniqueness holding for both the general and the plane-symmetric initial-value problems.

If these conditions are satisfied, then there exists a spacelike partial Cauchy sur- face L in/-( S) such that the evolution of any generic, plane-symmetric initial data for

( (7l } on L results in singularities on the Killing-Cauchy horizon S.

Remarks:

(i) As will be clear from the proof, the assumptions of the theorem need only hold on r ( S)u Sin M .

(ii) The condition of genericity for the initial data on L will be formulated in the proof.

(iii) When studying the proof the reader may find it helpful to carry along and look at the prototype example of a type II horizon discussed in the Introduction [Eqs.

(1.10)-(1.16)].

Proof of Theorem 2. We can set up the canonical local tetrad (3 .1) on ( M,g) in which the metric will be of the form

g - -1

- -dudv+A (u ,v )du ²+B ^(u ,v )dv²+M^2(u ,v )dx²+ R (u ,v)

where R (u ,v) is positive, bounded and nonzero

(4.1)

on

s.

Put

<2

>g=-dudv+RAdu²+RBdv². Find local functions t(u,v),z(u ,v) such that t=O on S and

(4.2) where P (>0) is the conformal factor. (This can be done, for example, by solving the initial value problem {t=O on S, <²>Dt=O } which in general has nonunique solutions, and then finding a "conjugate" z (u ,v) such that (4.2) is satisfied.¹⁵) Then the metric (4.1) becomes

g

+L; (t ,z )dtdxi +f; (t ,z )dzdxi , (4.3)

where

R

(t ,z) is again positive and bounded on S. In both coordinate systems (4.1)

-t .

and (4.3) the Killing vectors are l;;=dldx' . Our definition oft guarantees that Sis given by {t=O}; and by choosing -t

1;

₁ to be the Killing vector that becomes null and tangent to Son S, we find that F (t=O,z )=0. Note that Vt=-R dldt is a timelike vee- tor field which blows up on S while at the same time becoming tangent to S. It will not be necessary in what follows to fix the coordinate (gauge) freedom further than that of Eq.(4.3).

Now we claim that since Sis a Killing-Cauchy horizon of type IT, some null generators of S must have future endpoints on S. Null generators of S by definition have no past endpoints; if they do not have any future endpoints either, then one can globally express Sin the form {/ (u ,v )=const} where

V f

is perfectly smooth and everywhere nonzero on S, contradicting our assumption that Sis of type II. To see this, assume null generators of S have no endpoints. Take a spacelike two-

- t - t

dimensional section Z of S, and take a smooth field of (spacelike) basis fields k _1, k₂ on Z in S. (This can be done since by plane symmetry the spacelike sections of S

- t - t

will not have spherical topology.) Propagate k; along null generators l of Sby paral- lel transport to all of S Since generators of S are both past and future complete in

- t

M by definition, k; will be smooth on Sand will have smooth extensions to a neigh- borhood of S in M . Construct a null vector field

it

^{on S} satisfying

- t - t - t - t - t

g (n ,k; )=0, g (n ,l )=-1. n will be a smooth vector field on Sand will have a smooth extension (as a null vector) to a neighborhood of Sin M . Then take

f

to be the affine parameter along geodesics in the

it

direction, so that S= {f =0} and

V ^f

^{on Sis equal}

- t - t

to -l and hence is smooth, null and everywhere nonzero on S By choosing I, hence

it

and the (now not necessarily affine) parameter

f

such that

f

is constant on a family of parallel null surfaces near S,

V ^f

will retain these properties over a neighborhood of S in M. Finally, by the same argument as we gave just before the statement of theorem 1,

f

can be chosen to be a function of only u and v .

Therefore, there is a nonempty subset C of S which consists of the endpoints of null generators of S (As Sis achronal and edgeless it is a closed set and must contain

- t

these endpoints.) Now our Killing field

S,

₁ becomes null and tangent to Son S, point- ing along its null generators. But since Sis a Killing horizon, the convergence and shear of its null geodesic generators must identically vanish on S, and since S has no edge⁸·14 the only way these generators can have endpoints on Sis by intersecting other non-neighboring geodesic generators. Therefore at any point in C, there are at least two distinct null directions pointing to the past along two distinct generators of

- t

S. Then, as

s

1 is smooth and parallel to these generators on S, it has to vanish at all points in C c S (This is also expected because the set C represents an isolated set of points with a special geometric property that would be left invariant under the action

- t

of

S,

₁ if it were nonzero on C.) Thus, we have a nonempty subset C of Son which

-,-t

the Killing field

S,

₁ vanishes (that is, Cis the bifurcation set for the Killing horizon

S).

We now note that, as before we only need to prove the theorem in the case ( 00}

is a scalar field

.p.

As each component of a multi-index field ( 00} in the basis frame field (d₁ ,dz ,dx ,dy ) or in the spin basis corresponding to the tetrad (3.1) (and similarly

~

in any local basis field Lie parallel along the ~; or in the spin basis corresponding to

~

any null tetrad Lie parallel along the ~; so that the argument we gave in the proof of theorem 1 applies without modification) behaves like a scalar field under Lie transport

~

by ~;, exactly the same arguments that prove the singularity of

.p

on Swill prove the singularity of an arbitrary field ( 00 } (by constructing a suitable basis field Lie paral-

~

lel along the ~; for each such field ( 00 } ) when the initial data satisfy the conditions of the theorem.

Now consider the spacelik:e partial Cauchy surface L=(t=-c } in /-(S) where c >0 is sufficiently small so that :E lies within the region of strict plane symmetry W [Fig. 4]. Since Shas past endless null generators and S=H+(L), :E has no edge, i.e., it is infinite in the Killing ~; ~ directions.⁸ Consider generic, plane symmetric initial data for our scalar field

.p

on :E. We will adopt the following notion of genericity:

Plane symmetric initial data for

.p

on :E are generic, if we can find an arbitrarily large number L and coordinate values x =a , x =b with b -a =L such that if we cutoff the data for

.p

^{on :E} except on the portion of :E between x=a and x=b (thereby break- ing the plane symmetry), then the solution .p<L > to the initial value problem with data

( .p<L>=O, ~(L)=O on L, except on the strip between ^x=a and ^x=b where they are equal to the data of

.p}

will be nonzero at least on some points of the subset Con S. (Note that, even though the data for .p<L) on :E are cutoff in the x -direction, they still extend infinitely far in the other Killing (y-) direction.) In the case of a multicomponent field ( 00 } , plane-symmetric initial data on :E are called generic if there is an arbitrarily

large L =b -a so that the solution developing from the truncation of these initial data in the manner described above takes nonzero tensor (or spinor) values at some points on the subset C in S. As before, the values of the solution <I>(L > at the points on the subset C in S are defined as the limiting values of the solution on /-( S) as the field points approach the set C in S. Again to be more precise, we will call the initial data

for <1> on L generic if either this limit does not exist for <I><L

>,

or it does exist and is

nonzero somewhere on the subset C in S. In case this limit does not exist, the solu- tion <I> is clearly singular (and divergent) on the horizon Sand the theorem is proved.

Therefore, in the following we will assume that this limit does exist for <I>(L) and takes nonzero values somewhere in the subset C of S.

- t

But now consider the action of the symmetry group generated by

l;l>

given by GL: (x ,y ,z ,t) ~ (x+L ,y ,z ,t). By assumption (b) of the theorem, if we Lie transport the initial data truncated in the manner of the preceding paragraph with the Killing

- t - t - t

vector field

S1>

then the solution will be Lie transported by

1;

_1. But

1;

₁ vanishes on C

- t - t

and L ~~ <1>=1;₁(<1>); therefore the action of

1;

₁ leaves the value of <I>(L) on C invariant.

However, by the linearity of the field equations, the solution for the original plane symmetric initial data will be

n=-oo

hence on C, since GL (<I>(L)) (C)=<I><L> (C),

and thus <1> diverges on Cas <I>CL > ( C):;eO by genericity; and the theorem is proved. 0