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arXiv:gr-qc/9508001v1 1 Aug 1995
Existence of maximal hypersurfaces in some spherically symmetric spacetimes
Gregory A. Burnett
Department of Physics, University of Florida, Gainesville, Florida 32611, USA
Alan D. Rendall
Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, 91440 Bures sur Yvette, France
(1 August 1995)
We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov
equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature S1×S2
Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime
can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing
that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a
bound for the volume of any Cauchy surface Σ in any spacetime satisfying the timelike convergence condition in terms of the
volume and mean curvature of a fixed Cauchy surface Σ0and the maximal distance between Σ and Σ0. In particular, this
shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot
attain an arbitrarily large spatial volume.
04.20.-q, 04.20.Dw
I. INTRODUCTION
Given an initial data set for the gravitational field and
any matter fields present, what can be said of the space-
time evolved from this initial data?
In the asymptotically flat case, one would like to know
such things as how much gravitational energy is radi-
ated to null infinity, the final asymptotic state of the sys-
tem, whether black holes are formed, the nature of any
singularities produced, and whether cosmic censorship is
violated. For example, it is known that the maximal de-
velopment of sufficiently weak vacuum initial data is an
asymptotically flat spacetime that is free of singularities
and black holes [1]. In this case the gravitational waves
are so weak that they cannot coalesce into a black hole;
instead they scatter to infinity. Further it is known that
an initial data set containing a future trapped surface or
a future trapped region must be singular, provided the
null-convergence condition holds [2,3]. In these cases, the
gravitational field is already sufficiently strong that col-
lapse is inevitable.
In the cosmological case (spacetimes with compact
Cauchy surfaces), the questions one asks are a bit differ-
ent as one expects these spacetimes to be quite singular.
In fact, it is known that spacetimes with compact Cauchy
surfaces are singular, provided a genericity condition and
the timelike-convergence condition hold [2,3]. So, here
one would like to know such things as the nature of the
singularities, if the spacetime has a finite lifetime (in the
sense that there is a global upper bound on the lengths of
all causal curves therein), whether it expands to a max-
imal hypersurface and then recollapses or is always ex-
panding (contracting), and whether cosmic censorship is
violated. For example, it is known that if the initial data
surface is contracting to the future (past), then any de-
velopment satisfying the timelike-convergence condition
must end within a finite time to the future (past) [2,3].
Can more be said about the behavior of the cosmological
spacetimes?
The closed-universe recollapse conjecture asserts that
the spacetime associated with the maximal development
of an initial data set with compact initial data surface ex-
pands from an initial singularity to a maximal hypersur-
face and then recollapses to a final singularity (all within
a finite time), provided that the spatial topology does
not obstruct the existence of a maximal Cauchy surface
(e.g., S3or S1×S2) and provided the matter satisfies cer-
tain energy and regularity conditions [4,5,6]. It has also
been conjectured that such spacetimes admit a unique
foliation by constant mean curvature (CMC) Cauchy sur-
faces with the mean curvatures taking on all real values.
(See, e.g., conjecture 2.3 of [7] and the weaker conjec-
ture C2 of [8].) Just what energy conditions the matter
must satisfy is an open problem. However, in the study
of the weak form of this conjecture (which merely asserts
that the spacetime has a finite lifetime), the dominant en-
ergy and non-negative pressures conditions together have
proven sufficient for the cases studied [9,10]. More subtle
is the problem of what regularity conditions the matter
needs to satisfy. The difficulty here is that the maximal
development of an Einstein-matter initial data set may
not contain a maximal hypersurface because of the de-
velopment of a singularity in the matter fields, such as
a shell-crossing singularity in a dust-filled spacetime, be-
fore the spacetime has a chance to develop a maximal
hypersurface. While not for certain, it is thought that
those matter fields that do not develop singularities when
evolved in fixed smooth background spacetimes will not
lead to the obstruction of a maximal hypersurface.
Here, we study the maximal development of spherically
1
symmetric constant mean curvature initial data sets with
S1×S2Cauchy surfaces and matter consisting of either
collisionless particles of unit mass (whose evolution is de-
scribed by the Vlasov equation) or a massless scalar field
(whose evolution is described by the massless wave equa-
tion). It has already been established that if the mean
curvature is zero on the initial data surface, i.e., it is
a maximal hypersurface, then its maximal evolution ad-
mits a foliation by CMC Cauchy surfaces with the mean
curvature taking on all real values [11]. Further, it is
known that if the mean curvature is negative (positive)
then the initial data can be evolved at least to the extent
that the spacetime can be foliated by CMC spatial hy-
persurfaces taking on all negative (positive) values [11].
Left unresolved was whether the maximal evolution in
the latter two cases actually contains a maximal spatial
hypersurface and, hence, can be foliated by CMC hyper-
surfaces taking on all real values. The nonexistence of a
maximal spatial hypersurface would be reasonable if such
spacetimes could expand (contract) indefinitely, however,
it is known that these spacetimes have finite lifetimes
[9,10]. Therefore, it would seem that their maximal de-
velopment should contain a maximal Cauchy surface. We
show that it does.
Theorem 1. The maximal development of any spher-
ically symmetric spacetime with collisionless matter
(obeying the Vlasov equation) or a massless scalar field
(obeying the massless wave equation) that possesses a
CMC S1×S2Cauchy surface Σ admits a unique folia-
tion by CMC Cauchy surfaces with the mean curvature
taking on all real values. In particular, it contains a
maximal Cauchy surface and its singularities are crush-
ing singularities.
By the maximal development of a globally hyperbolic
spacetime, we mean the maximal development of an ini-
tial data set induced on a Cauchy surface in the space-
time. This is well-defined as the maximal developments
associated with any two Cauchy surfaces are necessar-
ily isometric [12]. Further, recall that a spacetime with
compact Cauchy surfaces is said to have a future (past)
crushing singularity if the spacetime can be foliated by
Cauchy surfaces such that the mean curvature of these
surfaces tends to infinity (negative infinity) uniformly to
the future (past). That the future and past singularities
associated with the spacetimes of theorem 1 are crushing
is then a simple consequence of the existence of a CMC
foliation taking on all real values.
As a consequence of theorem 1, the maximal develop-
ment of the spacetimes studied is rather simple. They
expand from an initial crushing singularity to a maxi-
mal hypersurface and then recollapse to a final crushing
singularity—all in a finite time. That is, they satisfy
the closed-universe recollapse conjecture in its strongest
sense as well as the closed-universe foliation conjecture.
While the maximal development of the spacetimes in
theorem 1 is about as complete as one could expect given
the existence of a complete CMC foliation, these space-
times may still be extendible (though there is no glob-
ally hyperbolic extension). In other words, theorem 1
does not eliminate the possibility that these spacetimes
violate cosmic censorship. In fact, cosmic censorship is
violated in the vacuum case. This is easily seen by re-
alizing that the maximal development in this case is ei-
ther of the of the two regions where r < 2Mof an ex-
tended Schwarzschild spacetime of mass M(ris the areal
radius), modified by identifications so that the Cauchy
surface topology is S1×S2. Although the “singular-
ity” corresponding to r→2Mis a crushing singularity,
this is actually a Cauchy horizon. Is this vacuum case
exceptional? It is worth noting that if a crushing singu-
larity corresponds to r→0, then the singularity must in
fact be a curvature singularity. This follows easily from
the fact that RabcdRabcd ≥(4m/r3)2, for any spherically
symmetric spacetime satisfying the null-convergence con-
dition, and the fact that the mass function mis bounded
away from zero by a positive constant in our case [10].
If we could show that rmust go to zero (uniformly) at
the extremes of our foliation, then the spacetime would
indeed be inextendible, thereby satisfying the cosmic cen-
sorship hypothesis. Establishing such a result appears to
be difficult and the vacuum case shows that such a result
will not always hold (though this case may be excep-
tional). Using a different approach, Rein has shown that
for an open set of initial data, there is a crushing singu-
larity in which r→0 uniformly, and which, therefore, is
a curvature singularity [13]. While this is encouraging,
the extent to which the spacetimes of theorem 1 satisfy
cosmic censorship remains to be seen.
The proof of theorem 1 involves a combination of
three ideas. First, it is known that spherically symmet-
ric spacetimes with S1×S2or S3Cauchy surfaces and
satisfying the dominant energy and non-negative pres-
sures (or merely “radial” non-negative pressure) condi-
tions have finite lifetimes [9,10]. Second, using a gen-
eral theorem (which is independent of symmetry assump-
tions) established in Sec. III, it follows that the spatial
volumes of Cauchy surfaces in the spacetime are bounded
above, which allows us to bound various fields describing
the spacetime geometry. Third, introducing a new time
function to avoid the problems associated with “degen-
erate” maximal hypersurfaces (i.e., surfaces where the
mean curvature cannot be used as a good coordinate),
the theorem then follows using the methods developed
in [11]. Furthermore, it is worth noting that our method
uses only a few properties of the matter fields themselves.
Namely, we use the fact that they satisfy the dominant
energy and “radial” non-negative pressures conditions
and, roughly speaking, the fact that the matter fields are
nonsingular as long as the spacetime metric is nonsingu-
lar. This latter property has not been given a precise for-
mulation, as it seems difficult to do so, and serves merely
as a heuristic principle—the arguments for collisionless
matter and the massless scalar field in [11] providing an
example of what it means in practice.
In theorem 1 we have restricted ourselves to space-
times with S1×S2Cauchy surfaces and have not con-
2
sidered similar spacetimes with S3Cauchy surfaces. The
problem with the S3case is that there exist two time-
like curves on which the symmetry orbits degenerate
to points. When we then pass to the quotient of our
spacetime by the symmetry group, the field equations on
the quotient spacetime are singular on boundary points
corresponding to the degenerate orbits. Experience has
shown that this degeneracy can have nontrivial conse-
quences on the evolution of the spacetime. For example,
in the study of the spherically symmetric asymptotically
flat solutions of the Einstein-Vlasov equations, it has
been shown that if a solution of these equations develops
a singularity, then the first singularity (as measured in
a particular time coordinate) is at the center [14]. How-
ever, currently it is not known how to decide when a cen-
tral singularity must occur. In the case of asymptotically
flat spherically symmetric solutions of the Einstein equa-
tions coupled to a massless scalar field, Christodoulou
has shown that naked singularities do form in the center
of symmetry for certain initial data (and that they can
form nowhere else) [15]. Note that the degeneracy of the
orbits in these spacetimes is of the same type that occurs
in the spherically symmetric spacetimes with S3Cauchy
surfaces. Similar problems occur in the study of the vac-
uum spacetimes with U(1) ×U(1) symmetry and having
S3or S1×S2Cauchy surfaces. Here the dimension of
the orbits is non-constant and, consequently, this case
is much harder to analyze than the T3case, which has
orbits of constant dimension [16]. The spherically sym-
metric spacetimes with S1×S2Cauchy surfaces, having
no degenerate orbits, avoid these complications.
It would, of course, be preferable to strengthen the-
orem 1 by removing the requirement that there exist a
CMC Cauchy surface in the spacetime. While such a re-
sult seems plausible, the methods currently used are not
adequate to cover this more general case. Strengthen-
ing our results in this direction is a sub ject for future
research.
Our conventions are those of [3], with the notable ex-
ception that trace Hof the extrinsic curvature Kab of
a spatial hypersurface measures the convergence of the
hypersurface to the future. Thus, surfaces with negative
Hare expanding to the future, while those with positive
Hare contracting to the future.
II. PROOF OF THEOREM 1
Fix a spacetime (M, g ) satisfying the conditions of the-
orem 1. Both classes of spacetimes considered here (the
Einstein-Vlasov and massless scalar field spacetimes) sat-
isfy the dominant energy condition (the Einstein tensor
Gab satisfies Gabvawb≥0 for all future-directed timelike
vectors vaand wb) as well as the timelike-convergence
condition (the Ricci tensor satisfies Rab tatb≥0 for all
timelike ta). While the Einstein-Vlasov spacetimes also
satisfy the non-negative pressures condition (Gab xaxb≥
0 for all spacelike xa), in general the massless scalar field
spacetimes do not. However, they do satisfy the weaker
“radial” non-negative pressures condition (Gab xaxb≥0
for all spatial vectors xaperpendicular to the spheres
of symmetry). It was shown in [9,10] that the spheri-
cally symmetric spacetimes with S3or S1×S2Cauchy
surfaces satisfying the dominant energy and the non-
negative pressures conditions (or merely the “radial”
non-negative pressures condition) have a finite lifetime,
i.e., the supremum of the lengths of all causal curves is
finite. Therefore, our spacetime (M, g) has a finite life-
time. It then follows immediately from lemma 2 (estab-
lished in Sec. III) that the volumes of all spatial Cauchy
surfaces in (M, g) are bounded above.
Denote the mean curvature of the Cauchy surface Σ
by t0. This initial data surface must be spherically sym-
metric. In the case t06= 0, this follows from the unique-
ness theorem for such hypersurfaces (see, e.g., theorem 1
of [4]) since if a rotation did not leave Σ invariant, we
would have a distinct CMC Cauchy surface with identi-
cal (nonzero) constant mean curvature. The case where
t0= 0 then follows from the fact that there is a neigh-
borhood Nof Σ in Msuch that Ncan be foliated by
CMC hypersurfaces, each having a different CMC, and
the fact that those with non-zero CMC must be spheri-
cally symmetric. As the theorem has already been proven
in the case where t0= 0 (Σ is a maximal hypersurface)
[11], we shall take t0to be negative (Σ is expanding to
the future). The case where the mean curvature is ini-
tially positive follows by a time-reversed argument. As
was shown in [11], in a neighborhood of the hypersur-
face Σ, the spacetime can be foliated by CMC Cauchy
surfaces. Define the scalar field tat any point to be the
value of the mean curvature of the CMC hypersurface
passing through that point, i.e., so level surfaces of tare
CMC hypersurfaces and, in particular, the surface t=t0
is Σ. A further scalar field xcan then be introduced so
that the spacetime metric gis given by
g=−α2dt2+A2[(dx+βdt)2+a2Ω],(2.1)
where Ω is the natural unit-metric associated with the
spheres of symmetry. The functions α,β, and Adepend
only on tand x(being spherically symmetric) and are
periodic in x. The function adepends only on t. The
fields can be chosen so that Rβ(t, x) dx= 0 for each t,
where the integral is taken over one period of a surface
of constant t.
It was shown in [11] that the initial data induced on Σ
can be evolved so that tcovers the interval (−∞,0) and
that, if it can be evolved to the closed interval (−∞,0],
i.e., a maximal hypersurface is attained, the spacetime
can be extended and foliated by CMC spatial hypersur-
faces taking on all real values. Therefore, our task is to
establish the existence of a maximal hypersurface. To
this accomplish this, we establish the existence of upper
bounds on a,A, and their inverses on the interval [t0,0).
We then introduce a new time function τ=f◦tby intro-
ducing a function fthat allows us to avoid the problem
3
associated with tbeing a bad coordinate on maximal hy-
persurfaces. Once this has been accomplished, theorem 1
will follow from an argument similar to that used in [11].
First, we establish upper bounds on the area radius
r=aA, the mass function m=1
2r(1− ∇ar∇ar), the vol-
ume V(t) of level surfaces of t, and their inverses. That
rand m−1are bounded above follows from the results
of [10]. (Note, mis positive.) Further, the technique
introduced in [17] was used in [11] to show that m/r is
bounded above on [t0,0). Therefore, mand r−1are also
bounded above on [t0,0). (That is, the mass mcannot
become arbitrarily large and rcannot become arbitrarily
small in this portion of the spacetime. This is nontriv-
ial as both mand r−1can become arbitrarily large on
unbounded intervals, e.g., near an initial or final singu-
larity.) As we have already established that the volume
of all spatial Cauchy surfaces are bounded above, V(t)
is bounded above. Using the fact that ∂tV(t) is positive
on [t0,0), as these hypersurfaces are everywhere expand-
ing, shows that Vis bounded from below by a positive
constant, and hence V−1is bounded above on [t0,0).
Next, that a,A, and their inverses are bounded above
on [t0,0) now follows easily from the facts that r=aA,
V(t) = 4πZa2A3dx= 4πa−1Zr3dx, (2.2)
and our upper bounds for V,r, and their inverses.
Next, we bound α′using the lapse equation
−A−3(Aα′)′+ (KabKab +Rabnanb)α= 1,(2.3)
where Kab is the extrinsic curvature of the CMC hyper-
surface, nais a unit timelike normal to the CMC hyper-
surface, and a prime denotes a derivative by ∂x. (This
is equation (2.4) in [11].) Using the fact that Kab Kab is
manifestly non-negative and Rab nanb≥0 by the timelike
convergence condition, it follows that (Aα′)′≥ −A3. Us-
ing the fact that Ais bounded above and integrating in a
CMC hypersurface, we find that (Aα′)|p−(Aα′)|q≥ −C1
for some positive constant C1and any two points pand
qin the hypersurface. Choosing qwhere αis extremal
on the surface (so α′(q) = 0) and using the fact A−1
is bounded above shows that α′is bounded from be-
low. Choosing pwhere αis extremal on the surface (so
α′(p) = 0) and using the fact A−1is bounded above
shows that α′is bounded from above. Therefore, there
exists a constant C2such that |α′| ≤ C2. Thus, even if
αis unbounded, it must diverge in a way that is uniform
in space: For any two points pand qin a CMC hyper-
surface, |α(p)−α(q)|=|Rq
pα′dx| ≤ Rq
p|α′|dx ≤πC2.
If we knew that αwere bounded above on [t0,0), we
could then proceed to argue as in [11]. While such a
bound can be established rather easily for fields satisfying
the dominant energy and non-negative pressures condi-
tions, such an argument fails for the massless scalar field.
The difficulty in establishing an upper bound on αis
linked to the possibility that dtmay be zero on a maximal
hypersurface, and thus tbeing a bad coordinate. Note
that this can only occur if Kab = 0 everywhere on Σ (i.e.,
Σ is momentarily static) and Rabnanb= 0 everywhere on
Σ. If the non-negative energy condition (Gab tatb≥0 for
all timelike ta) and non-negative sum-pressures condition
[Gab(tatb+gab)≥0 for all unit-timelike ta] are satisfied,
then Rabnanb= 0 implies that Gab nanb= 0 and, hence,
by the Hamiltonian constraint equation, the Ricci scalar
curvature of the metric induced on Σ must be zero. How-
ever, it is easy to show that there are no such spherically
symmetric geometries on S1×S2. Thus, the Einstein-
Vlasov spacetimes do not admit such surfaces. However,
it can be shown that there are massless scalar field space-
times with such “degenerate” maximal hypersurfaces. To
avoid this difficulty, we change our time function to one
that is guaranteed to be well-behaved even on a maximal
hypersurface with dt= 0.
Fix any inextendible timelike curve γthat is every-
where orthogonal to the CMC hypersurfaces. The length
of the segment of γbetween any two CMC hypersurfaces
t=t1and t=t2is then simply Rt2
t1α(γ(u)) du. Using
the fact that there is a finite upper bound on the lengths
of all timelike curves in our spacetime, the integral
Z0
t1
α(γ(u)) du= lim
t2→0Zt2
t1
α(γ(u)) du(2.4)
must exist, i.e., α(γ(t)) is integrable on any interval of the
form [t1,0). Fix some value x0of xand consider the func-
tion α(t, x0). Since α′is bounded there is a constant C
such that α(t, x0)≤α(γ(t)) + C. It follows that α(t, x0)
is also integrable on any interval of the form [t1,0). Using
this fact, define the function fon (−∞,0) by setting
f(λ) = λ−Z0
λ
α(u, x0) du. (2.5)
Noting that f′(λ) = 1 + α(λ, x0) and limλ→0f(λ) = 0,
we see that fis an orientation-preserving diffeomorphism
from (−∞,0) to (−∞,0). Hence,
τ=f◦t(2.6)
is a new time function on our spacetime. Note that
∂τ /∂t = 1 + α(t, x0).
The level surfaces of τclearly coincide with those of
tand so are CMC hypersurfaces. As a consequence the
field equations for the geometry and the matter written
in terms of τlook very similar to those written in terms
of t. Using τin place of t, the metric has the same form
as before
g=−˜α2dτ2+A2[(dx+˜
βdτ)2+a2Ω],(2.7)
where the new lapse function ˜αis given by
˜α=α∂t
∂τ =α
1 + α(t, x0),(2.8)
and similarly for the new shift ˜
β. In terms of our new
coordinates (τreplacing t) and new variables (˜αand ˜
β
4
replacing αand β, respectively), the field equations are
the same as in [11] with ∂τreplacing ∂t, ˜αreplacing α,
˜
βreplacing β, and ∂t/∂τ replacing the right-hand side
of equation (2.3). Explicit occurrences of tin the equa-
tions are left unchanged, tbeing simply considered as a
function of τ, determined implicitly by equation (2.6).
Using equation (2.8), it is straightforward to show that
∂t/∂τ = 1 −˜α(τ, x0). With this, the lapse equation can
be written as
−A−3(A˜α′)′+ (KabKab +Rabnanb)˜α= 1 −˜α(τ, x0).
(2.9)
Using the fact that α′is bounded, as argued above, it
follows that α(t, x)≤α(t, x0)+ C, where Cis a constant.
Therefore, by equation (2.8), ˜αis bounded above.
It is now possible to apply the same type of arguments
to the system corresponding to the time coordinate τas
were applied in [11] to the system corresponding to the
time coordinate tto show that all the basic geometric and
matter quantities in the equations written with respect to
τare bounded and that the same is true for their spatial
derivatives of any order. Bounding time derivatives of all
these quantities requires some more effort. All but one of
the steps in the inductive argument used to bound time
derivatives in [11] apply without change. (Note that in
[11], derivatives with respect to twere bounded, whereas
here, derivatives with respect to τare bounded.) The ar-
gument that does not carry over is that which was used to
bound time derivatives of αand α′. To see why, consider
the equation obtained by differentiating equation (2.9) k
times with respect to τ
−A−3(A(Dk
τ˜α)′)′+ (KabKab +Rab nanb)Dk
τ˜α
+Dk
τ˜α(τ, x0) = Bk,(2.10)
where Dk
τ=∂k
τdenotes the k-th partial derivative with
respect to τ. Here Bkis an expression which is already
known to be bounded when we are at the step in the in-
ductive argument to bound Dk
τ˜αand Dk
τ˜α′. In lemma 3.4
of [11], Dk
tαwas bounded by using the fact that twas
bounded away from zero. The analogous procedure is
clearly not possible in the present situation, where tis
tending to zero. This kind of argument was also used
in [11] to bound time derivatives of higher order spa-
tial derivatives of α, but that is unnecessary, since such
bounds can be obtained directly by differentiating the
lapse equation once the time derivatives of αand α′have
been bounded. The same argument applies here, so all
we need to do is to prove the boundedness of Dk
τ˜αand
Dk
τ˜α′using equation (2.10) under the hypothesis that
Bkis bounded. This follows by simply noting that equa-
tion (2.10) has the same form for each value of kand the
following lemma.
Lemma 1. Consider the differential equation
(au′)′=bu +c+du(x0) (2.11)
where a,b,c,d, and uare 2π-periodic functions on the
real line and x0is a point therein. Suppose that a > 0,
b≥0, d≥0, and that dis not identically zero. Then |u|
and |u′|are bounded by constants depending only on the
quantities K1= max{a−1(x)}>0, K2=R2π
0|c(x)|dx≥
0, K3=R2π
0d(x) dx > 0, and K4=R2π
0b(x) dx≥0.
Proof. First, if u(x0)>2πK1K2, then u > 0 every-
where. To see this, suppose otherwise and let x1be a
point where uachieves its maximum, so u(x1)≥u(x0)>
2πK1K2and let x2be that number such that u > 0 on
[x1, x2) and u(x2) = 0 (so x1< x2< x1+ 2π). Then,
on the interval [x1, x2], we have (au′)′≥c, from which
it follows that u′≥ −K1K2on [x1, x2]. Integrating this
and using the fact that u(x2) = 0, we find that u(x1)≤
2πK1K2, contradicting the fact that u(x1)>2πK1K2.
Therefore, as uis everywhere positive, it follows that
(au′)′≥c. Integrating this inequality starting (or end-
ing) at a point where u′= 0 shows that |u′| ≤ K1K2. In-
tegrating equation (2.11) from 0 to 2πand using the fact
that uis positive gives u(x0)R2π
0d(x) dx≤R2π
0|c(x)|dx,
and hence, |u(x0)| ≤ K2K−1
3. Using this and the fact
that |u′| ≤ K1K2shows that |u| ≤ K2K−1
3+ 2πK1K2.
Second, if u(x0)<−2πK1K2, a similar argument shows
that uis everywhere negative and we again obtain the
same bounds on |u′|and |u|. Third, suppose that
|u(x0)| ≤ 2πK1K2. If max(u)>2πK1K2(1 + 2πK1K3),
using the inequality (au′)′≥c+du(x0), we can argue
much as before to see that uis everywhere positive and
again obtain the same bounds on |u′|and |u|. Similarly, if
min(u)<−2πK1K2(1+2πK1K3), it follows that uis ev-
erywhere negative and again we recover the same bounds
on |u′|and |u|. Next, if |u| ≤ 2πK1K2(1 + 2πK1K3) ev-
erywhere, |u|is already bounded, and to bound |u′|, we
note that we have bounds for all terms on the right hand
side of equation (2.11), so it suffices to integrate it start-
ing from a point where u′is zero to bound |u′|.
At this stage, we have indicated how all geometric and
matter quantities, expressed in terms of the new time
coordinate τ, can be bounded, together with all their
derivatives. In particular, this means that all these quan-
tities are uniformly continuous on any interval of the form
[τ1,0), where τ1is finite. It follows that all these quan-
tities have smooth extensions to the interval [τ1,0]. Re-
stricting them to the hypersurface τ= 0 gives a initial
data set for the Einstein-matter equations with zero mean
curvature. By the standard uniqueness theorems for the
Cauchy problem, the spacetime which, in the old coordi-
nates, was defined on the interval (−∞,0) is isometric to
a subset of the maximal development of this new initial
data set. It follows that the original spacetime has an
extension which contains a maximal hypersurface.
Lastly, that the foliation is unique now follows from the
fact that compact CMC Cauchy surfaces with non-zero
mean curvature are unique [4] and that the spacetime is
indeed maximal follows from the fact that any spacetime
admitting a complete foliation by compact CMC Cauchy
surfaces is maximal [7].
5
III. A BOUND FOR THE VOLUME OF SPACE
It is well known that as we transport an “infinitesi-
mal” spacelike surface Salong the geodesics normal to
itself, the ratio νof its volume of to its original volume
is governed by the Raychaudhuri equation
d2
dt2ν1/3+1
3Rabtatb+σab σabν1/3= 0,(3.1)
where tis the proper time measured along the geodesics
normal to S,Rab is the Ricci tensor, and σab is the shear
tensor associated with the geodesic flow [2,3,18]. (This
equation is usually written in terms of the divergence of
the geodesic flow θ=ν−1dν/dt.) On the surface S,ν
satisfies the initial condition ν= 1 and dν /dt =−H(p),
where H(p) is the trace of the extrinsic curvature of Sat
the point pwhere the geodesic intersects S. Therefore, if
the spacetime satisfies the timelike-convergence condition
(Rabtatb≥0 for all timelike ta), it follows that as long
as νremains non-negative,
d2
dt2ν1/3≤0,(3.2)
from which we find that
ν(t)≤1−1
3H(p)(t−t0)3
.(3.3)
This equation bounds the growth of the volume of a local
spatial region in the spacetime.
Using this result, it is not difficult to show that, in a
spacetime satisfying the timelike-convergence condition,
if we fix a Cauchy surface Σ0and construct from it a
second Cauchy surface Σ by transporting Σ0to the future
along the flow determined by the geodesics normal to Σ0,
as long as these flow lines do not self-intersect (which will
be true if Σ is sufficiently close to Σ0), then
vol(Σ) ≤vol(Σ0)1 + 1
3sup
Σ0
(−H)T3
,(3.4)
where vol(S) denotes the three-volume of a Cauchy sur-
face Sand Tis the “distance” between the two surfaces
measured by the lengths of the geodesics normal to Σ0
(which will be independent of which geodesic is chosen
by the construction of Σ). Therefore, we have a bound
on the volume of Σ in terms of the volume of Σ0, the ex-
trinsic curvature of Σ0, and the distance between Σ0and
Σ. Does a similar result hold for more general Cauchy
surfaces Σ? For instance, a more general hypersurface Σ
may not be everywhere normal to the geodesics from Σ0,
some geodesics normal to Σ0may intersect one another
between Σ0and Σ, and parts of Σ may lie to the future of
Σ0while other parts may lie to the past. Can the simple
bound given by equation (3.4) be modified to cover these
cases? That it can is the subject of the following lemma.
Lemma 2. Fix an orientable globally hyperbolic space-
time (M, gab ) satisfying the timelike-convergence condi-
tion (Rabtatb≥0 for all timelike ta) and a smooth space-
like Cauchy surface Σ0therein. Then, for any smooth
spacelike Cauchy surface Σ,
vol(Σ) ≤vol(Σ0)1 + 1
3sup
Σ0
(|H|)∆(Σ0,Σ)3
,(3.5)
where vol(S) denotes the three-volume of a Cauchy sur-
face S,His the trace of the extrinsic curvature of Σ0
(using the convention that Hmeasures the convergence
of the future-directed timelike normals to a spacelike sur-
face), and ∆(Σ0,Σ) is the least upper bound to the
lengths of causal curves connecting Σ0to Σ (either fu-
ture or past directed). Further, for any Cauchy surface
Σ⊂D+(Σ0),
vol(Σ) ≤vol(Σ0)1 + 1
3sup
Σ0
(−H)∆(Σ0,Σ)3
.(3.6)
Note that for p, q ∈M, ∆(p, q) is not quite the distance
function d(p, q) as used in [2] as d(p, q) = 0 if q∈J−(p).
Instead, ∆(p, q) does not distinguish between future and
past: ∆(p, q) = ∆(q, p) = d(p, q) + d(q, p).
From lemma 2, we see that for a spacetime satisfy-
ing the timelike-convergence condition, possessing com-
pact Cauchy surfaces, and having a finite lifetime (in the
sense that d(p, q) [equivalently ∆(p, q)] is bounded above
by a constant independent of pand q), then the volume
of a Cauchy surface therein cannot be arbitrarily large.
Further, we see that if the spacetime admits a maximal
Cauchy surface Σ0(H= 0 thereon), we reproduce the re-
sult that there is no other Cauchy surface having volume
larger than Σ0(though there may be surfaces of equal
volume) [4].
In the following, dfdenotes the derivative map asso-
ciated with a differentiable map fbetween manifolds.
When viewed as a pull-back, we denote dfby f∗and,
when viewed as a push-forward, we denote dfby f∗. For
a map f:A→B,f[A] denotes the image of Ain B.
Lastly, A\Bdenotes the set of elements in Athat are
not in B.
A. Proof of lemma 2
To begin the proof of lemma 2, for each point p∈Σ0,
let γpdenote the unique inextendible geodesic containing
pand intersecting Σ0orthogonally. Parameterize γpby
tso that the tangent vector to γpis future-directed unit-
timelike and γp(0) = p. Then, define the map f: Σ0→
Σ, by
f(p) = γp∩Σ.(3.7)
Note that for each p∈Σ0,fis well defined since γp
intersects Σ at precisely one point as Σ is a spacelike
Cauchy surface for the spacetime.
6
Next, let Kbe the subset of Σ0defined by the property
that p∈ K if and only if the geodesic γpdoes not possess
a point conjugate to Σ0between Σ0and Σ (although
it may have such a conjugate point on Σ). Note that
this is precisely the condition that for each p∈ K the
solution νto equation (3.1) along γp, satisfying the initial
conditions ν= 1 and dν/dt =H(p) at p, be strictly
positive on the portion of γpbetween pand f(p). It
follows that Kis closed. Furthermore, fmaps Konto Σ.
To see this, recall that for any point q∈Σ there exists
a timelike curve µconnecting qto Σ0having a length
no less than any other such curve. Furthermore, such
a curve µmust intersect Σ0normally, is geodetic, and
has no point conjugate to Σ0between Σ0and q. (See
Theorem 9.3.5 of [3].) Therefore, the point p=µ∩Σ0
is in Kand µ⊂γp, so f(p) = γp∩Σ = µ∩Σ = q.
Therefore, fmaps Konto Σ. However, in general, fwill
not be one-to-one between Kand Σ.
Let Cdenote the set of critical points of the map f
on Σ0. That is, p∈Cif and only if its derivative map
f∗: (TΣ0)p→(TΣ)f(p)is not onto. Then, by Sard’s
theorem [19], f[C] (the critical values of f), and hence
f[K ∩ C], are sets of measure zero on Σ. Now, note that
Σ can be expressed as the union of f[K \ C] and a set
having measure zero. To see this, we write
Σ = f[K] = f[(K \ C)∪(K ∩ C)]
=f[K \ C]∪(f[K ∩ C]\f[K \ C]) .(3.8)
The last two sets are manifestly disjoint and the latter
is a set of measure zero (as it is a subset of a set of
measure zero). Therefore, we need only concern ourselves
the behavior of fon the set of regular points of fwithin
K. This is useful since, by the inverse function theorem
[19], fis a local diffeomorphism between K \ Cand f[K \
C]. As we shall see, for all p∈ K \ C, the point f(p)
is not conjugate to Σ0on γp, from which it follows that
K \ Cis an open subset of Σ0.
Denote volume elements associated with the induced
metrics on Σ0and Σ by eabc and ǫabc, respectively, cho-
sen so that eabc and ǫabc correspond to the same spatial
orientation class (which can be done as the spacetime is
both time-orientable and orientable). Then the Jacobian
of the map fis that unique scalar field Jon Σ0such that
(f∗ǫ)abc =Jeabc.(3.9)
Note that Jis zero on Cand positive on K \ C.
With these definitions, we have
vol(Σ) = Zf[K\C]
ǫ
≤ZK\C
(f∗ǫ)
≤"sup
K\C
(J)#ZK\C
e
≤"sup
K\C
(J)#vol(Σ0).(3.10)
The first step follows from the facts that Σ = f[K] and
f[K ∩ C] is a set of measure zero. That we have an
inequality in the second step follows from the fact that
although fis a local diffeomorphism, it may not be one-
to-one between K\Cand f[K\C]. The third step follows
from the definition of Jgiven by equation (3.9) and the
fact that Jis bounded above by its supremum. Lastly,
the fourth step follows from the fact that K\Cis a subset
of Σ0. So, to prove lemma 2, we need to show that, on the
set K\ C,Jis bounded above by the relevant expressions
in lemma 2.
To that end, define φ: Σ0×R→Mby setting φ(p, t) =
γp(t). Of course, if γpis not future and past complete,
this will not be defined for all t. Next, define T: Σ0→R
by setting T(p) to that number such that γp(T(p)) =
f(p), i.e., T(p) is the “time” along the geodesic γpat
which γpintersects Σ. Note that if f(p) lies to the future
of Σ0, then T(p) is positive, while if f(p) lies to the past
of Σ0, then T(p) is negative.
Fix a point p∈ K \ Cand define the map g: Σ0→M
by setting g(q) = φ(q, T (p)). Should γq(T(p)) not be de-
fined, then gis not defined for that point of Σ0. However,
it will always be defined for some neighborhood of pas
g(p) = f(p). Notice that gsimply “translates” points on
Σ0along the geodesics normal to Σ0a fixed distance T(p)
(independent of point), i.e., it is a translation along the
normal geodesic “flow”. Therefore, the derivative map
of gat a point is precisely the geodesic deviation map.
In particular, dgis injective (one-to-one) from (TΣ0)pto
(T M )f(p)if and only if f(p) is not conjugate to Σ0on γp
(by the definition of such a conjugate point).
Noting that fcan be written as f(q) = φ(q , T (q)), we
see that the derivative maps of fand gat p[both of
which are maps from (TΣ0)pto (T M )f(p)] are related by
(df)ab= (dg)ab+ta(dT)b,(3.11)
where tais the unit future-directed tangent vector to
γpat f(p). From this we see that dfis injective [from
(TΣ0)pto (T M )f(p)] if and only if dgis injective. There-
fore, on K \ C, not only is dfinjective, but dgis also
injective, and hence f(p) is not conjugate to Σ0on γp.
Define ˆeabc at f(p) by parallel transporting eabc at p
along γp. Then,
(f∗ˆe)abc = (g∗ˆe)abc =ν(T(p))eabc.(3.12)
The first equality follows from (3.11) and the fact that
taˆeabc = 0. The second equality follows by recogniz-
ing that the coefficient of the right-hand most term is
precisely the ratio of the volume of an “infinitesimal”
region in Σ0to its original volume as it is transported
along the geodesic flow normal to Σ0. As the transport
is done from pto f(p), the coefficient is ν(T(p)), where
νis the solution of equation (3.1) satisfying the stated
initial conditions. (In other words, ν(t) is the Jacobian
of the geodesic deviation map.)
Denote the future-directed normal to Σ at f(p) by na.
Then, there exists a unit-spacelike vector xa∈(TΣ)f(p)
7
such that ta=γ(na+βxa), where γ= (−tana) and
β=p1−γ−2. Then, for one of the two volume elements
ǫabcd on Massociated with the spacetime metric, we have
ǫabc =nmǫmabc and ˆeabc =tmǫmabc , which gives the
following relation between these two tensors at f(p),
ˆeabc =γǫabc +γβxmǫmabc .(3.13)
Therefore,
(f∗ˆe)abc =γ(f∗ǫ)abc ,(3.14)
where we have used (3.13) and the fact that the pull-back
of xmǫmabc by fmust be zero as xmis in the surface
Σ and the contraction of ǫabcd with four vectors all in
a three-dimensional subspace must be zero. Therefore,
using (3.14) and (3.12), we see that
(f∗ǫ)abc = (−tana)−1ν(T(p))eabc,(3.15)
which when compared to (3.9), gives
J(p) = (−tana)−1ν(T(p)).(3.16)
Since (−tana)−1≤1 and ν(T(p)) is bounded above by
(3.3), we have
J(p)≤1−1
3H(p)T(p)3
.(3.17)
So, if Σ ⊂D+(Σ0), we have 0 ≤T(p)≤∆(Σ0,Σ) and
−H(p)≤supΣ0(−H), and therefore,
sup
K\C
(J)≤1 + 1
3sup
Σ0
(−H)∆(Σ0,Σ)3
,(3.18)
which with (3.10) establishes equation (3.6). More gen-
erally, as
H(p)T(p)≤ |H(p)||T(p)| ≤ sup
Σ0
(|H|)∆(Σ0,Σ),(3.19)
we have
sup
K\C
(J)≤1 + 1
3sup
Σ0
(|H|)∆(Σ0,Σ)3
,(3.20)
which with (3.10) establishes equation (3.5). This com-
pletes the proof of lemma 2.
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8