ABSTRACT. We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume.

Date: April 18, 2002.

2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.

Key words and phrases. Lorentzian manifold, volume estimates, cosmological spacetime, general relativity, constant mean curvature, CMC hypersurface.

0. Introduction

Let N be a (n + 1)-dimensional Lorentzian manifold and suppose that N can be decomposed in the form

(0.1)                  N  = N0   U  N-  U  N+,

where N0 has finite volume and N- resp. N+ represent the critical past resp. future Cauchy developments with not necessarily a priori bounded volume. We assume that N+ is the future Cauchy development of a Cauchy hypersurface M1, and N- the past Cauchy development of a hypersurface M2, or, more precisely, we assume the existence of a time function x0, such that

           N+  =  x0  ([t1,T+)),       M1 =  {x0 = t1},
(0.2)               0-1                        0
           N - =  x   ((T-,t2]),      M2 =  {x  = t2},

and that the Lorentz metric can be expressed as

                2    2y      02       0      i  j
(0.3)          ds  = e  { -dx   + sij(x ,x)dx dx },

where x = (xi) are local coordinates for the space-like hypersurface M 1 if N+ is considered resp. M2 in case of N-.

The coordinate system (xa) 0<a<n is supposed to be future directed, i.e. the past directed unit normal (na) of the level sets

(0.4)                    M  (t) = {x0 = t}

is of the form

(0.5)                (na) = - e-y(1, 0,...,0).

If we assume the mean curvature of the slices M(t) with respect to the past directed normal--cf. [4] for a more detailed explanation of our conventions--is strictly bounded away from zero, then, the following volume estimates can be proved

Theorem 0.1. Suppose there exists a positive constant e0 such that

(0.6)        H(t) >  e                 A t < t < T  ,
                      0                 1        +

(0.7)        H(t) <  - e0              A T - < t < t2,


(0.8)                   |N+ | < e |M (t1)| ,
(0.9)                   |N -| < --|M (t2)| .

These estimates also hold locally, i.e. if Ei < M(ti), i = 1, 2, are measurable subsets and E1+,E 2- the corresponding future resp. past directed cylinders, then,

                            +    1-
(0.10)                   |E1 |<  e |E1|,


                            -    1
(0.11)                   |E2 |<  --|E2|.

1. Proof of Theorem  0.1

In the following we shall only prove the estimate for N+, since the other case N- can easily be considered as a future development by reversing the time direction.

Let x = x(q) be an embedding of a space-like hypersurface and (na) be the past directed normal. Then, we have the Gauß formula

(1.1)                       xa = hijna.

where (hij) is the second fundamental form, and the Weingarten equation

(1.2)                       nai = hkixak.

We emphasize that covariant derivatives, indicated simply by indices, are always full tensors.

The slices M(t) can be viewed as special embeddings of the form

(1.3)                      x(t) = (t,xi),

where (xi) are coordinates of the initial slice M(t 1). Hence, the slices M(t) can be considered as the solution of the evolution problem

(1.4)               x = - eyn,     t1 < t < T+,

with initial hypersurface M(t1), in view of ( 0.5).

From the equation ( 1.4) we can immediately derive evolution equations for the geometric quantities gij,hij,n, and H = gijh ij of M(t), cf. e.g. [3, Section 4], where the corresponding evolution equations are derived in Riemannian space.

For our purpose, we are only interested in the evolution equation for the metric, and we deduce

(1.5)            gij = <xi,xj> + <xi,xj > = - 2e hij,

in view of the Weingarten equation.

Let g = det(gij), then,

                            ij         y
(1.6)                 g = gg  gij = - 2e Hg,

and thus, the volume of M(t),|M(t)|, evolves according to

                          integral                integral 
             -d                d- V~ --           y
(1.7)         dt|M (t)|=        dt  g = -       e H,
                          M(t1)            M(t)

where we shall assume without loss of generality that |M(t1| is finite, otherwise, we replace M(t1) by an arbitrary measurable subset of M(t1) with finite volume.

Now, let T  (- [t1,T+) be arbitrary and denote by Q(t1,T) the cylinder

(1.8)           Q(t1, T) = {(x0, x): t1 < x0 < T },


                                  integral  T  integral 
(1.9)                 |Q(t1,T )|=         ey,
                                  t1  M

where we omit the volume elements, and where, M = M(x0).

By assumption, the mean curvature H of the slices is bounded from below by e0, and we conclude further, with the help of ( 1.7),

                            1   integral  T integral 
                |Q(t1, T)|<  --        eyH
                            e0  t1   M
(1.10)                    =  e {|M (t1)|- |M (T )| }
                         <  1-|M (t1)| .

Letting T tend to T+ gives the estimate for |N+|.

To prove the estimate ( 0.10), we simply replace M(t1) by E1.

If we relax the conditions ( 0.6) and ( 0.7) to include the case e0 = 0, a volume estimate is still possible.

Theorem 1.1. If the assumptions of Theorem  0.1 are valid with e0 = 0, and if in addition the length of any future directed curve starting from M(t1) is bounded by a constant g1 and the length of any past directed curve starting from M(t2) is bounded by a constant g2, then,

(1.11)                  |N+ | < g1|M (t1)|


(1.12)                  |N -| < g2|M (t2)| .

Proof. As before, we only consider the estimate for N+.

From ( 1.6) we infer that the volume element of the slices M(t) is decreasing in t, and hence,

                    ----     -----
(1.13)              V~  g(t) <  V~  g(t )    A t < t.
                                1        1

Furthermore, for fixed x  (- M(t1) and t > t1

                             t y
(1.14)                         e  <  g1

because the left-hand side is the length of the future directed curve

(1.15)              g(t) = (t,x)    t1 < t < t.

Let us now look at the cylinder Q(t1,T) as in ( 1.8) and ( 1.9). We have

(1.16)            integral    integral                    integral   integral 
                  T        y V~  ------     T        y V~ -------
    |Q(t1, T)| =           e    g(t,x) <            e   g(t1,x)
                 t1  M(t1)               t1   M(t1)
                   integral       V~ -------
              < g1         g(t1,x) = g1| M  (t1)|

by applying Fubini’s theorem and the estimates ( 1.13) and ( 1.14). []

2. Cosmological spacetimes

A cosmological spacetime is a globally hyperbolic Lorentzian manifold N with compact Cauchy hypersurface S0, that satisfies the timelike convergence condition, i.e.

(2.1)             Rabnanb  >  0     A  <n,n> = - 1.

If there exist crushing singularities, see [1] or [2] for a definition, then, we proved in [2] that N can be foliated by spacelike hypersurfaces M(t) of constant mean curvature t, - oo < t <  oo ,

(2.2)                 N  =       M  (t )  U  C0,
                           0/=t (- R

where C0 consists either of a single maximal slice or of a whole continuum of maximal slices in which case the metric is stationary in C0. But in any case C0 is a compact subset of N.

In the complement of C0 the mean curvature function t is a regular function with non-vanishing gradient that can be used as a new time function, cf. [4] for a simple proof.

Thus, the Lorentz metric can be expressed in Gaussian coordinates (xa) with x0 = t as in ( 0.3). We choose arbitrary t 2 < 0 < t1 and define

                 N0  = {(t, x): t2 < t < t1},

(2.3)             N - = {(t, x): -  oo  <  t < t2},
                 N   = {(t, x): t < t <   oo  }.
                   +            1

Then, N0 is compact, and the volumes of N-,N+ can be estimated by

(2.4)                   |N+  |< t  |M  (t1)|,
(2.5)                   |N - |< ----|M  (t2)| .

Hence, we have proved

Theorem 2.1. A cosmological spacetime N with crushing singularities has finite volume.

Remark 2.2. Let N be a spacetime with compact Cauchy hypersurface and suppose that a subset N- < N is foliated by constant mean curvature slices M(t) such that

(2.6)                   N - =        M (t)
                              0<t <t2

and suppose furthermore, that x0 = t is a time function--which will be the case if the timelike convergence condition is satisfied--so that the metric can be represented in Gaussian coordinates (xa) with x0 = t.

Consider the cylinder Q(t,t2) = {t < x0 < t 2} for some fixed t. Then,

                        integral  t2 integral        integral  t2     integral 
(2.7)       |Q(t, t2)| =         ey =     H - 1   Hey,
                        t    M       t        M

and we obtain in view of ( 1.7)

(2.8)           t-21{|M (t)|-  |M  (t2)| } < |Q(t, t2)| ,

and conclude further

(2.9)             ltim-->0 |M  (t)|< t2|N -|+ |M (t2)|,


(2.10)           lim |M  (t)|=  oo   ===>   |N  |=  oo .
                t-->0                     -

3. The Riemannian case

Suppose that N is a Riemannian manifold that is decomposed as in ( 0.1) with metric

(3.1)           ds2 = e2y{dx02 + sij(x0,x)dxidxj}.

The Gauß formula and the Weingarten equation for a hypersurface now have the form

(3.2)                      xaij = - hijna,

                           a     k a
(3.3)                      ni = h ixk.

As default normal vector--if such a choice is possible--we choose the outward normal, which, in case of the coordinate slices M(t) = {x0 = t} is given by

                       a     -y
(3.4)                 (n ) = e   (1,0,...,0).

Thus, the coordinate slices are solutions of the evolution problem

(3.5)                        x = eyn,

and, therefore,

(3.6)                      gij = 2e hij,

i.e. we have the opposite sign compared to the Lorentzian case leading to

                       d           integral 
(3.7)                   --| M  (t)| =    eyH.
                       dt          M

The arguments in Section  1 now yield

Theorem 3.1. (i) Suppose there exists a positive constant e0 such that the mean curvature H(t) of the slices M(t) is estimated by

(3.8)        H(t) >  e0                A t1 < t < T+,


(3.9)        H(t) <  - e0              A T - < t < t2,


(3.10)                |N+ | < e0 lt-->iTm+ |M (t)| ,

(3.11)                |N -| < -- lim |M (t| .
                             e0 t-->T -

(ii) On the other hand, if the mean curvature H is negative in N+ and positive in N-, then, we obtain the same estimates as Theorem  0.1, namely,

(3.12)                  |N+ | < -1|M (t1)| ,

(3.13)                  |N -| < e |M (t2)| .


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