## Then

The equality holding if and only if (8 8t)a 0 and hence if and only if A is a geodesic curve. Thus The vector (d dw)a u_0 will be called the variation vector Z. Conversely, given a continuous, piecewise C2 vector field Z along y(t) vanishing at q and p, we may define a variation a for which Z will be the variation vector by a(u,t) expr (wZ r), where ue(-c,c) for some e > 0 and r y(t). The variation of the length from qtop under a is BL where 2 g(S 8t, 8f8t) is the magnitude of the tangent...

## Preface

The subject of this book is the structure of space-time on length-scales from 10 13cm, the radius of an elementary particle, up to 1028cm, the radius of the universe. For reasons explained in chapters 1 and 3, we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable predictions about the universe first, that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity and secondly,...

## T T exp rr w

Where n is an integer > 2(r+)2 (r_) 2. In these coordinates, the metric is analytic everywhere except at r r, where it is degenerate. Ttio coordinates v' and w' are analytic functions of v and w for r 4 r+ or r_. Thus the manifold can be covered by an analytic atlas, consisting of local coordinate neighbourhoods defined by coordinates v and w for r+r, and by local coordinate neighbourhoods defined by if and w' for r 4 r+. The metric is analytic in this atlas. The case e2 m2 can be extended...

## Info

R* r + m oe((r m)2)----if e2 m2, r* r + mlog(r2 2mr + e2) + - arc tan I -if e2 > m2. Defining advanced and retarded coordinates v, w by the metric (5.26) takes the double null form 1 +- dv d > + r2 (dfl2 + sin2 6 d 2). (5.27) In the case c2 < m2, define new coordinates ', to by t> arctan exp I jj w arctan exp ' Then the metric (5.27) takes the form 1--+ -5164,-cosec 2v cosec 2w & v dw 4- 2(d02 + sin2 6 d 2), (5.28) tan v tan w - exp t 2 ) r) r-) '2 and a (r+)-2 (r_)2. The maximal...

## Jl

A space-time satisfying the causality, future and past distinguishing conditions, but not satisfying the strong causality condition at p. Two strips have been removed from a cylinder light cones are at 46 . If conditions (a) to (c) of proposition 6.4.5 hold and if in addition, (d) is null geodesically complete, then the strong causality condition holds on Suppose the strong causality condition did not hold at peJ(. Let be a convex normal neighbourhood oip and let Vn < be an...

## Differential geometry

The space-time structure discussed in the next chapter, and assumed through the rest of this book, is that of a manifold with a Lorentz metric and associated affine connection. In this chapter, we introduce in 2.1 the concept of a manifold and in 2.2 vectors and tensors, which are the natural geometric objects defined on the manifold. A discussion of maps of manifolds in 2.3 leads to the definitions of the induced maps of tensors, and of sub-manifolds. The derivative of the induced maps defined...

## Exact solutions

Any space-time metric can in a sense be regarded as satisfying Einstein's field equations (where we use the units of chapter 3), because, having determined the left-hand side of (5.1) from the metric tensor of the space-time ( .g), one can define T as the right-hand side of (5.1). The matter tensor so defined will in general have unreasonable physical properties the solution will be reasonable only if the matter content is reasonable. We shall mean by an exact solution of Einstein's equations,...

## A

KSbjtK i - (K< K d) 1 (log Q). e. (3.3) Knowing the conformal structure, one can choose a metric g which represents the conformal equivalence class of metrics and can evaluate the left-hand side of (3.3) for any test particle. Then the right-hand side of (3.3) determines (log i2) b up to the addition of a multiple of Ka ab. By considering another curve y'(t) whose tangent vector K'a is not parallel to Ka, one can find (log 0).b and so can determine Q everywhere up to a constant multiplying...

## Edge iy edge y

Let be a sequence of neighbourhoods of a point geedge (H+( f)) such that any neighbourhood of q encloses all the akn for n sufficiently large. In each there will be points pn e I (q, and rn e I+(q, which can be joined by a timelike curve A which does not intersect H+( f). This means that A cannot intersect D+( f). By proposition 6.5.1, qeD+(S ) and so I (q) < I (D+(y)) < J-(y) y D+(y). Thus pn must lie in I SP). Also every timelike curve from q which is inextend-ible in the past direction...

## The physical significance of curvature

In this chapter we consider the effect of space-time curvature on families of timelike and null curves. These could represent flow lines of fluids or the histories of photons. In 4.1 and 4.2 we derive the formulae for the rate of change of vorticity, shear and expansion of such families of curves the equation for the rate of change of expansion (Raychaudhuri's equation) plays a central role in the proofs of the singularity theorems of chapter 8. In 4.3 we discuss the general inequalities on the...

## 1

A space satisfying the strong causality condition, but in which the slightest variation of the metric would permit there to be closed timelike lines through p. Three strips have been removed from a cylinder light cones are at 45 . Figube 40. A space satisfying the strong causality condition, but in which the slightest variation of the metric would permit there to be closed timelike lines through p. Three strips have been removed from a cylinder light cones are at 45 . Even imposition...

## SRab hgabR gaigbSRii WagbAj gbA l8g SgVt g6Ri

+ (terms in SgciK and Sgtf), (7.5) g - (det g)1 ((det g) )le (det g (7.6) The plan is now as follows. We choose some suitable background metric and express the Einstein equations in the form Rab- Rgab 8 Rab- Rgab)+ & < *> - 8nTab. (7.7) One regards this as a second order non-linear set of differential equations to determine g in terms of the values of it and its first derivatives on some initial surface. Of course to complete the system one has to specify the equations governing the...

## GahQtc K 8ac7i

It will be convenient to take the contravariant form gah of the metric to be more fundamental and the covariant form gab as derived from it by (7.1). Using the alternating tensor tfabcd defined by the background metric, this relation can be expressed explicitly as where (det g)1 1 ff W L i is the determinant of the components of g in a basis which is orthonormal with respect to the metric The difference between the connection T defined by g and the connection defined by is a tensor, and can be...

## Causal structure

By postulate (a) of 3.2, a signal can be sent between two points of only if they can be joined by a non-spacelike curve. In this chapter we shall investigate further the properties of such causal relationships, establishing a number of results which will be used in chapter 8 to prove the existence of singularities. By 3.2, the study of causal relationships is equivalent to thatof the conformal geometry of i.e. of the set of all metrics g conformal to the physical metric g (g fi2g, where fi is a...