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physical significance of curvature

Figure 9. A compact region It of space-time with past and future non-timelike boundaries (8%)t and timelike boundary (8%)s. The part of lying to the past of the surface ^f(i') (defined by t = t') is <%(t').

Figure 9. A compact region It of space-time with past and future non-timelike boundaries (8%)t and timelike boundary (8%)s. The part of lying to the past of the surface ^f(i') (defined by t = t') is <%(t').

function t whose gradient is everywhere timelike. (It will be shown in § 6.4 that such a function will exist provided space-time is not on the verge of violating causality.) The boundary dlV of the compact region consists of a part (8&)v whose normal form n is non-spacelike and such that nat.bgab is positive, a part (S<^)2 whose normal form n is non-spacelike and such that nat.bg°b is negative, and a remaining part {d$/)3 (which may be empty). The sign of the normal form n is given by the requirement that (n, X) be positive for all vectors X which point out of aU (cf. § 2.8), Jf(i') denotes the surface t = t' and aM{t') denotes the region of for which t < t'. For later use in § 7.4 we shall establish an inequality which holds not only for the energy-momentum tensor T"* but also for any symmetric tensor iS"6 which satisfies the dominant energy condition. Applied to the energy-momentum tensor this inequality will show that T"*' vanishes everywhere on ^ if it vanishes on and on the initial surface (8^i)v

Lemma 4.3.1

There is some positive constant P such that for any tensor S^ which' satisfies the dominant energy condition and vanishes on (8^/)a,

Sabt.adab

Consider the volume integral

I(t) = f (S°»t.a).b dv = f S°Hiab&>+ f S°».bt.adv. J *«) J *«) J *«)

By Gauss' theorem this can be transformed into an integral over the boundary of W(t): ,

The boundary of <%(t) will consist of <2r(f) n 8<% and n ^(t). Since S^ is zero on m = f + f + f

By the dominant energy condition, S^t.^ is a non-spacelike vector such that S^t.^t-h > 0. As the normal form to is non-spacelike and such that nat. bg°-b < 0, the second term on the right will be nonnegative. Thus f 8abt.adub - f S°»t.adcrb

Since fy is compact there will be some upper bound to the components of t;ab in any orthonormal basis whose timelike vector is in the direction of t. Thus there will be some P > 0 such that on

&»ttab PS°H.at.b for any S^ which obeys the dominant energy condition. The volume integral over can be decomposed into a surface integral over Jif(t') n % followed by an integral with respect to t' :

f (P^i;oi;6 + ^.6i:o)dt;= f(f (PS^t.h + S"6.6) dcr0| dt', J*(i) ' J Ujr((')"* )

where dcr0 is the surface element of Jf(t'). Thus f S°H.adcrb < - f S°H.atorb J Jf (i) n * J *(() n (»),

+ Pf(f S*t:odcr6W+f(f S°»:ad*b)dt'. □ J wjr«')"® / J \JjfU')"* /

As an immediate consequence of this result one has:

The conservation theorem

If the energy-momentum tensor obeys the dominant energy condition and is zero on (&iV)3 and on the initial surface (8tf)lt then it is zero everywhere on

Then the above lemma gives dz/di < Px. But for sufficiently early values of t, 3V(t) will not intersect and so x will vanish. Thus x will vanish for all t which implies that Tab is zero on □

From the conservation theorem it follows that if the energy-momentum tensor vanishes on a set ¿f, then it also vanishes on the

future Cauchy development which is defined as the set of all points through which every past-directed non-spacelike curve intersects (figure 10) (cf. § 6.5). For if q is any point of D+(£P), the region of D+(£f) to the past of q is compact (proposition 6.6.6) and may be taken as CU. This result may be interpreted as saying that the dominant energy condition implies that matter cannot travel faster than light.

For our consideration of singularities, the importance of the weak energy condition is that it implies that matter always has a converging (or more strictly nondiverging) effect on congruences of null geodesies. If the vorticity vanishes, the expansion 6 obeys the equation:

Thus in this case 6 will monotonically decrease along the null geodesic if R^ WaWb> 0 for any null vector W. We shall call this the null convergence condition. From the Einstein equations,

it follows that this condition is implied by the weak energy condition, independent of the value of A.

From (4.26) it can be seen that the expansion 6 of a timelike geodesic congruence with zero vorticity will monotonically decrease along a geodesic if Rab WaWb ^ 0 for any timelike vector W. We shall call this the timelike convergence condition. By the Einstein equation, this condition will be satisfied if the energy-momentum tensor obeys the inequality, , .

This will hold for type I if fi+Pa>°> fi + 2,pa-~A2> 0, and for type II if

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