1 Ko Pitt 0pt20 and p1p2A Js

We shall say that the energy-momentum tensor satisfies the strong energy condition if it obeys the above inequality for A = 0. This is a stricter requirement than the weak energy condition but it is still physically reasonable for the total energy-momentum tensor. For the general case, type I, it would be violated only by a negative energy density or a large negative pressure (e.g. for a perfect fluid with density 1 gm cm-3 it can only be violated if p < —1016 atmospheres). It holds for the electromagnetic field and for the scalar field with m zero (in particular, it holds for the scalar field of Brans and Dicke). For m non-zero, the energy-momentum tensor of a scalar field has the form (§3.3): ^ =

Thus if Wa is a unit timelike vector

1 m2

which may be negative. However by the equation of the scalar field

Inserting this in (4.37) and integrating over a region one obtains

The first term will be non-negative since gab + 2WaWb is a positive definite metric and the second term will be small compared to the first if the region is large compared to the wavelength h/m. For it mesons, which may be described classically by a scalar field with m — 6 x 10_25gm, this wavelength is 3 x 10_13cm. Thus although the energy-momentum tensor of tt mesons may not satisfy the strong energy condition at every point, this should not affect the convergence of timelike geodesies over distances greater than 10-12 cm. This might possibly lead to a breakdown of the singularity theorems in chapter 8 when the radius of curvature of space-time becomes less than 10-12 cm but such a curvature would be so extreme that it might well count as a singularity (§10.2).

4.4 Conjugate points

In §4.1 we saw that the components of the vector which represented the separation between a curve y(s) and a neighbouring curve in a congruence of timelike geodesies, satisfied the Jacobi equation:

A solution of this equation will be called a Jacobi field along y(s). Since a solution may be specified by giving the values of Za and dZ°/ds at some point on y(s) there will be six independent Jacobi fields along y(s). There will be three independent Jacobi fields which vanish at some point q of y(s). They may be expressed as:

and Aap(s) is a 3 x 3 matrix which vanishes at q. These Jacobi fields may be thought of as representing the separation of neighbouring geodesies through q. As before one may define the vorticity, shear and expansion of the Jacobi fields along y(s) which vanish at g:

These will obey the equations derived in § 4.1, with ^ = 0. In particular will be constant along y(s). But it vanishes at g where Aafl is zero. Thus o)ap will be zero wherever is non-singular.

We shall say that a pointy on y(s) is conjugate to q along y(s) if there is a Jacobi field along y{s), not identically zero, which vanishes at q and p. One may think of p as a point where infinitesimally neighbouring geodesies through q intersect. (Note, however, that it may be only infinitesimally neighbouring geodesies which intersect aty; there need not be two distinct geodesies from q passing through y.) The Jacobi fields along y(s) which vanish at g are described by the matrix Aap. Thus a pointy is conjugate to g along y (s) if and only if-<4^ is singular aty. The expansion 6 is defined as (det A)-1 d (det A)/ds. Since Aaf? obeys (4.39) where Raiyiis finite, d (det A)/ds will be finite. Thus a point y will be conjugate to g along y(s) if 6 becomes infinite there. The converse will also be true since 6 = d log (det A)/ds and Aap can be singular only at isolated points or else it would be singular everywhere.

Proposition 4.4.1

If at some point y^) («j > 0), the expansion 6 has a negative value 6X < 0 and if Rab Va Vb ^ 0 everywhere then there will be a point conjugate to g along y{s) between y(Sj) and y^ + (3/— dj)), provided that y(s) can be extended to this parameter value. (This may not be possible if space-time is geodesically incomplete. In chapter 8 we shall interpret such incompleteness as evidence of the existence of a singularity.)

The expansion 6 of the matrix Aap obeys the Raychaudhuri equation

where we have used the fact that the vorticity is zero. All the terms on the right-hand side are negative. Thus for s > sy

So 6 will become infinite and there will be a point conjugate to q for some value of s between and st + (3l~61). □

In other words, if the timelike convergence condition holds and if the neighbouring geodesies from q start converging on y(s), then some infmitesimally neighbouring geodesic will intersect y(s) providing that y(s) can be extended to large enough values of the parameter s.

Proposition 4.4.2

IiRabVaVb Si 0 and if at some pointy = y(s1) the tidal force Rabcd Vb Vd is non zero, there will be values s0 and s2 such that q = y(s0) and r = y(s2) will be conjugate along y(s), providing that y(s) can be extended to these values.

A solution of (4.39) along y(s) is uniquely determined by the values of Aap and dA^/ds at p. Consider the set P consisting of all such solutions for which Aafi\p = 6afi, (dA^jds)\p is symmetric with trace 6\p < 0. For each solution in P there will be some s3 > Sj for which Aap(sa) is singular, since either 6\p < 0, in which case this follows from the previous result, or 6\p — 0, in which case (d<7a^/ds)|p is non-zero which will then cause cr2 to be positive and so cause 6 to become negative for s > sv The members of the set P are in one-one correspondence with the space S of all symmetric 3x3 matrices with non-positive trace (i.e. with the values of dAa^jds)\p). There is thus a map 7j from S to y(s) which assigns to each initial value (dAafi[ds)\p the point on y(s) where A^ first becomes singular. The map 7) is continuous. Further if any component of (dAalj/ds)|p is very large, the corresponding point on y(s) will lie near p, since in the limit the term Raiyi in (4.39) becomes irrelevant and the solution resembles the flat space case. Thus there is some C > 0 and somes 4 > Sj such that if any component of (dAa^ds)\p is greater than C, the corresponding point on y(s) will be before y(«4). However the subspace of S consisting of all matrices all of whose components are less than or equal to C, is compact. This shows that there is some s6 > st such thati/(/S) is contained in the segment from y(«i) to y(s8). Consider now a point r = y(s2) where s2 > s5. If there is no point conjugate to r between r and p, the Jacobi fields which are zero at r must have an expansion 6 which is positive atp (otherwise they would be in the set P which represents all families of Jacobi fields with zero vorticity which have non-positive expansion at^>). It follows from the previous result that there is then a point q = y(s0) (s„ < sj which is conjugate to r along y(s). □

In a physically realistic solution (though not necessarily in an exact one with a high degree of symmetry), one would expect every timelike geodesic to encounter some matter or some gravitational radiation and so to contain some point where Rabcd Vb Vd was non-zero. Thus it would be reasonable to assume that in such a solution every timelike geodesic would contain pairs of conjugate points, provided that it could be extended sufficiently far in both directions.

We shall also consider the congruence of timelike geodesies normal to a spacelike three-surface, 3ff. By a spacelike three-surface, 3/?, we mean an imbedded three-dimensional submanifold defined locally by / = 0 where/is a C2 function and tf^f-af-b < 0 when/ = 0. We define N, the unit normal vector to 3tf, by Na = (- g^f. bf. P)~lgod/; d and the second fundamental tensor x of Jf by Xab =" h,fhbdNc.<d, where hab = 9ab + NaNb is called the first fundamental tensor (or induced metric tensor) of Jf (cf. §2.7). It follows from the definition that x is symmetric. The congruence of timelike geodesies orthogonal to will consist of the timelike geodesies whose unit tangent vector V equals the unit normal N at Then one has:

The vector Z which represents the separation of a neighbouring geodesic normal to Jf from a geodesic y(s) normal to will obey the Jacobi equation (4.38). At a point q on y(s) at Jf it will satisfy the initial condition: ,

We shall express the Jacobi fields along y(s) which satisfy the above

condition as where

and at q, Aag is the unit matrix and

We shall say that a pointy on y(s) is conjugate to along y(s) if there is a Jacobi field along y(s) not identically zero, which satisfies the initial conditions (4.44) at q and vanishes at p. In other words, p is conjugate to Jf along y(s) if and only if Aafi is singular at p. One may think of p as being a point where neighbouring geodesies normal to Jif intersect. As before Aap will be singular where and only where the expansion 6 becomes infinite. At q, the initial value of Ayat)ytASp will be zero, therefore <i)ap will be zero on y(s). The initial value of 6 will be Xabtf*-

Proposition 4.4.3

If Rab VaVb > 0 and Xab9ab < °> there will be a point conjugate to Jf? along y(s) within a distance 3/(— Xab9ab) from Jif, provided that y(s) can be extended that far.

This may be proved using the Raychaudhuri equation (4.26) as in proposition 4.4.1. □

We shall call a solution of the equation:

along a null geodesic y(v), a Jacobi field along y{v). The components Zm could be thought of as the components, with respect to the basis Ex and E2, of a vector in the space Sq at each point q. We shall say that p is conjugate to q along the null geodesic y(v) if there is a Jacobi field along y{v), not identically zero, which vanishes at q and p. If Z is a vector connecting neighbouring null geodesies which pass through q, the component Z3 will be zero everywhere. Thus p can be thought of as a point where infinitesimally neighbouring geodesies through q intersect. Representing the Jacobi fields along y(v) which vanish at q by the 2x2 matrix Amn,

One has as before: AlmdlkAkn = 0, so the vorticity of the Jacobi fields which are zero at p vanishes. Also p will be conjugate to q along y(v)' if and only if ,

becomes infinite at p. Analogous to proposition 4.4.1, we have:

Proposition 4.4.4

If R^^K1 > 0 everywhere and if at some point y(vt) the expansion 6 has the negative value 8X < 0, then there will be a point conjugate to q along y(v) between y(v^ and y(vy + (2/ — 6y)) provided that y(v) can be extended that far.

The expansion 8 of the matrix Amn obeys (4.35):

^ = ~ R^ KaKb — 2a2 — and so the proof proceeds as before. □ Proposition 4.4.5

If RabKaKb > 0 everywhere and if at p = y{vx), KcKdK[aRb]cd[eKf] is non-zero, there will be t;0 and vt such that q = y(v0) and r = y[v2) will be conjugate along y(v) provided y(v) can be extended to these values.

If KcKdK[a Rb] cd[eKf] is non zero then so is Rmini- The proof is then similar to that of proposition 4.4.2. □

As in the timelike case, this condition will be satisfied for a null geodesic which passes through some matter provided that the matter is not pure radiation (energy-momentum tensor type II of §4.3) and moving in the direction of the geodesic tangent vector K. It will be satisfied in empty space if the null geodesic contains some point where the Weyl tensor is non-zero and where K does not lie in one of the directions (there are at most four such directions) at that point for which KcKdK{aCb] cd[e Kf] = 0. It therefore seems reasonable to assume that in a physically realistic solution every timelike or null geodesic will contain a point at which KaKbK[rRd]abuKf] is not zero. We shall say that a space-time satisfying this condition satisfies the generic condition.

Similarly we may also consider the null geodesies orthogonal to a spacelike two-surface By a spacelike two-surface we mean an imbedded two-dimensional submanifold defined locally by ft = 0, /2 = 0 where fi and /2 are C2 functions such that when fx = 0, /2 = 0 then f1;a and /2;o are non-vanishing and not parallel and ifi;a+^f2-.a)(fi:b+MU.b)9ab = 0

for two distinct real values /i1 and fi2 °f A- Then any vector lying in t he two-surface is necessarily spacelike. We shall define Nf and N2a, the two null vectors normal to as proportional to gab(f1.7b+¿¿1f2.6) and S^i/i;b +tlzfi\b) respectively, and normalize them so that

One can complete the pseudo-orthonormal basis by introducing two spacelike unit vectors and Yta orthogonal to each other and to and N2a. We define the two null second fundamental tensors of Sf as:

nXab = - Nnc; d(YfYla+Y2<Y2a) (Y/Ylb+Y2*Y2b), where n takes the values 1, 2. The tensors and 2Xab arc symmetric.

There will be two families of null geodesies normal to £f corresponding to the two null normals Nxa and N2a. Consider the family whose tangent vector K equals N2 at £?. We may fix our pseudoorthogonal basis E1( E2, E3j E4 by taking Ej = Y1( E2 = Y2, E3 = Nj, E4 = N2 at and parallelly propagating along the null geodesies. The projection into the space SQ of the vector Z representing the separation of neighbouring null geodesies from the null geodesic y(v) will satisfy (4.30) and the initial conditions

at q on y(v) at £?. As before the vorticity of these fields will be zero. The initial value of the expansion 6 will be 2Xab9ah- Analogous to proposition 4.4.3 we have:

Proposition 4.4.6

If RdbKaKb ^ 0 everywhere and 2xabgab is negative there will be a point conjugate to £f along y(v) within an affine distance 2/(- 2xabgab) from y. □

From their definition, the existence of conjugate points implies the existence of self-intersections or caustics in families of geodesies. A further significance of conjugate points will be discussed in the next section.

4.5 Variation of arc-length

In this section we consider timelike and non-spacelike curves which are piecewise C3 but which may have points at which their tangent vector is discontinuous. We shall require that at such points the two tangent vectors that is, they point into the same half of the null cone. Proposition 4.5.1

Let ^ be a convcx normal coordinate neighbourhood about q. Then the points which can be reached from q by timelike (respectively non-spacelike) curves in & are those of the form exp5 (X), XeTq where g(X, X) < 0 (respectively < 0). (Here, and for the rest of this section, we consider the map exp to be restricted to the neighbourhood of the origin in Tg which is diffeomorphic to % under expff.)

In other words, the null geodesies from q form the boundary of the region in which can be reached from q by timelike or non-spacelike curves in <?/. This is fairly obvious intuitively but because it is fundamental to the concept of causality we shall prove it rigorously. We first establish the following lemma:

Lemma 4.5.2

In 9/ the timelike geodesies through q are orthogonal to the three-surfaces of constant cr (<r < 0) where the value of <r atpe^is defined to be fffcxp^p, expfl-1p).

The proof is based on the fact that the vector representing the separation of points equal distances along neighbouring geodesies remains orthogonal to the geodesies if it is so initially. More precisely, let X(i) denote a curve in Tg, where g(X(t), X(i)) = — 1. One must show that the corresponding curves A(t) = expQ(s0X(t)) (sQ constant) in where defined, are orthogonal to the timelike geodesies y(s) = expe(sX(i0)) (/0 constant). Thus- in terms of the two-surface a defined by x(s, t) = exp5($X(0), one must prove that t. and ~ satisfy t. and ~ satisfy

(1 l\ - 1 (1 \as' it) ~ 9 [es 8s' 8t)+g\8s' 8s 8t)'

Geodesic

Geodesic

The first term on the right is zero as 8/8s is the unit tangent vector to the timelike geodesies from q. In the second term one has from the definition of the Lie derivative that

8 18 8\ (8 T> 8\ 18 18 8\ n ThuS 8sg(&• 8t) = g\8s'8iFsj = 2 8tg[^^) = 0-

Therefore g(8l8s, 8f8t) is independent of s. But at s = 0, (8f8t)a = 0. Thus g(8[8s, 8j8t) is identically zero. □

Proof of proposition 4.5.1. Let CQ denote the set of all timelike vectors at q. These constitute the interior of a solid cone in Tg with vertex at the origin. Let y(t) be a timelike curve in from qtop and let y(t) be the piecewise C2 curve in Tq defined by y(t) = expff-1(y(i)). Then identifying the tangent space to Tg with Tq itself, one has

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