Aaa MdVf

{¿~ (7,/'"6c-<ir°«/ + (a11 uPPer indices)

respectively. As with L(J(), n* maps these horizontal subspaces one-one onto thus again n* can be inverted to give a unique horizontal lift X eTu of any vector X e TM. In the particular case of, T(f£), u itself corresponds to a unique vector Wei1lKll)(J')1 and so there is an intrinsic horizontal vector field W defined on T(Ji) by the connection. In terms of local coordinates {x°, Vb},

This vector field may be interpreted as follows: the integral curve of W through u = (p, X) e is the horizontal lift of the geodesic in with tangent vector X at p. Thus the vector field W represents all geodesies on . In particular, the family of all geodesies through is the family of integral curves of W through the fibre 7r~1(jP)c \ the curves in have self intersections at least at p, but the curves in Ti^Ji) are non-intersecting everywhere.

General Relativity

In order to discuss the occurrence of singularities and the possible breakdown of General Relativity, it is important to have a precise statement of the theory and to indicate to what extent it is unique. Wo nluill llioi'tiforti |irom<nl. t lin t.lioory iifl a intmhot' of |irm(,ulni<>H itlwitit a mathematical model for space-time.

In § 3.1 we introduce the mathematical model and in § 3.2 the first two postulates, local causality and local energy conservation. These postulates are common to both Special and General Relativity, and thus may be regarded as tested by the many experiments that have been performed to check the former. In § 3.3 we derive the equations of the matter fields and obtain the energy-momentum tensor from a Lagrangian.

The third postulate, the field equations, is given in § 3.4. This is not so well established experimentally as the first two postulates, but we shall see that any alternative equations would seem to have one or more undesirable properties, or else require the existence of extra fields which have not yet been detected experimentally,

3.1 The space-time manifold

The mathematical model we shall use for space-time, i.e. the collection of all events, is a pair {~4t, g) where is a connected four-dimensional HausdorfF C00 manifold and g is a Lorentz metric (i.e. a metric of signature + 2) on

Two models (-#,g) and (^",g') will be taken to be equivalent if they are isometric, that is if there is a diffeomorphism 8: which carries the metric g into the metric g', i.e. g = g'. Strictly speaking then, the model for space-time is not just one pair (UK, g) ' but a whole equivalence class of all pairs (■JC, ,g') which are equivalent to g). We shall normally work with just one representative member (UK, g) of the equivalence class, but the fact that this pair is defined only up to equivalence is important in some situations, in particular in the discussion of the Cauchy problem in chapter 7.

The manifold is taken to be connected since we would have no knowledge of any disconnected component. It is taken to be Hausdorff since this seems to accord with normal experience. However in chapter 5 we shall consider an example in which one might dispense with this condition. Together with the existence of a Lorentz metric, the Hausdorff condition implies that is paracompact (Geroch (1968c)).

A manifold corresponds naturally to our intuitive ideas of the continuity of space and time. So far this continuity has been established for distances down to about 10-16cm by experiments on pion scattering (Foley el at. (1967)). It may be difficult to extend this to much hmiillur I (Mirths an l«> <lo ho would require a partido of «uoh high energy that several other particles might be created and confuse the experiment. Thus it may be that a manifold model for space-time is inappropriate for distances less than 10-16cm and that we should use theories in which space-time has some other structure on this scale. However such breakdowns of the manifold picture would not be expected to affect General Relativity until the typical gravitational length scale became of that order. This would happen when the density became about 10S8gm cm-3, which is a condition so extreme as to be completely beyond our present knowledge. Nevertheless, by adopting a manifold model for space-time, and making certain other reasonable assumptions, we shall show in chapters 8-10 that some breakdowns of General Relativity must occur. It may be the field equations that go wrong, or it may be that quantization of the metric is needed, or it may be a breakdown of the manifold structure itself that occurs.

The metric g enables the non-zero vectors at a point to be divided into three classes: a non-zero vector XeTp being said to be timelike, spacelike or null according to whether gr(X, X) is negative, positive or zero respectively (cf. figure 5).

The order of differentiability, r, of the metric ought to be sufficient for the field equations to be defined. They can be defined in a distributional sense if the metric coordinate components gah and gab are continuous and have locally square integrable generalized first derivatives with respect to the local coordinates. (A set of functions /. a on Rn are said to be the generalized derivatives of a function/on Rn if, for any C00 function i¡r on Rn with compact support,

However this condition is too weak, since it guarantees neither the existence nor the uniqueness of geodesies, for which a C2- metric is required. (A C2- metric is one for which the first coordinate derivatives of the metric coordinate components satisfy a local Lipschitz condition, see § 2.1.) We shall in fact assume for most of the book that the metric is at least Ca. This allows the field equations (which involve the second derivatives of the metric) to be defined at every point. In § 8.4 we shall weaken the condition on the metric to C2- and show that this does not affect the results on the occurrence of singularities.

In chapter 7, we use a different kind of differentiability condition in order to show that the time development of the field equations is determined by suitable initial conditions. We require there that the metric components and their generalized first derivatives up to order m(m ^ 4) are locally square integrable. This would certainly be true if the metric were C*.

In fact, the order of differentiability of the metric is probably not physically significant. Since one can never measure the metric exactly, but only with some margin of error, one could never determine that there was an actual discontinuity in its derivatives of any order. Thus one can always represent one's measurements by a C® metric.

If the metric is assumed to be Cr, the atlas of the manifold must be Cr+1. However, one can always find an analytic subatlas in any C* atlas (s 1) (Whitney (1636), cf. Munkres (1954)). Thus it is no restriction to assume from the start that the atlas is analytic, even though one could physically determine only a Cr+1 atlas if the metric were Cr.

We have to impose some condition on our model (Ut, g) to ensure that it includes all the non-singular points of space-time. We shall say that the Cr pair is a Cr-extension of (uÉ'.g) if there is an iso metric Cr imbedding fi: UÍ-+UC. If there were such an extension {Jt', g') we should have to regard points of UÉ" as also being points of space-time. We therefore require that the model (U¡f,g) is Cr-inextendible, that is there is no Cr extension (uÉ", g') of (UK, g) where ¡i(UÍ) does not equal UÍ'.

As an example of a pair (U(lt gx) which is not inextendible, consider two-dimensional Euclidean space with the ar-axis removed between x1 = — 1 and xt = +1. The obvious way to extend this would simply be to replace the missing points, but one could also extend it by taking another copy (^2> 62) of the space, and identifying the bottom side of the x1-axis for ¡a^l < 1 with the top side of the a:2-axis for \x2\ < 1, and also identifying the top side of the a^-axis for jor^l < 1 with the bottom side of the ar2-axis for \x2\ < 1. The resultant space g3) is inextendible but not complete as we have left out the points x1 = + 1, yl = 0. We cannot put these points back in because we were perverse enough to extend the top and bottom sides of the ar-axis on different sheets. If however one takes the subset % of defined by 1 < x1 < 2, — 1 <yx< 1) then one could extend the pair 63and put back the point x1 = 1,2/! = 0. This motivates a rather stronger definition of inextendibility: a pair {J(, g) is said to be Cr-locally inextendible if there is no open set °ll ^ with non-compact closure in ¿K, such that the pair has an extension (W, g') in which the closure of the image of is compact.

3.2 The matter fields

There will be various fields on , such as the electromagnetic field, the neutrino field, etc., which describe the matter content of space-time. These fields will obey equations which can be expressed as relations between tensors on in which all derivatives with respect to position are covariant derivatives with respect to the symmetric connection defined by the metric g. This is so because the only relations defined by a manifold structure are tensor relations, and the only connection defined so far is that given by the metric. If there were another connection on > the difference between the two connections would be a tensor and could be regarded as another physical field. Similarly another metric on could be regarded as a further physical field. (The equations of the matter fields are sometimes expressed as relations between spinors on We do not deal with such relations in this book, as they are not needed for the problems we wish to consider. In fact, all spinor equations can be replaced by rather more complicated tensor equations; see e.g. Ruse (1937).)

The theory one obtains depends on what matter fields one incorporates in it. One should of course include all such fields which have been experimentally observed, but one might postulate the existence of as yet undetected fields. Thus for example Brans and Dicke (Dicke (1964), appendix 7) postulate the existence of a long range scalar field which is weakly coupled to the trace of the energy-momentum tensor. In the form given in Dicke (1964) appendix 2, the Brans-Dicke theory can be regarded simply as General Relativity with an extra scalar field. Whether this scalar field has been experimentally detected or not is at present under dispute.

We shall denote the matter fields included in the theory by x^uf '"bc...dt where the subscript (t) numbers the fields considered. The following two postulates on the nature of the equations obeyed by the x¥uf '"bc...d are common to both the Special and the General Theories of Relativity.

Postulate (a): Local causality

The equations governing the matter fields must be such that if is a convex normal neighbourhood and p and q are points in then a signal can be sent in % between p and q if and only if p and q can be joined by a C1 curve lying entirely in %, whose tangent vector is everywhere non-zero and is either timelike or null; we shall call such a curve, non-spacelike. (Our formulation of relativity excludes the possibility of particles such as tachyons, which move on spacelike curves.) Whether the signal is sent from p to q or from qtop will depend on the direction of time in The problem of whether a consistent direction of time can be assigned at all points of space-time will be considered in § 6.2.

A more precise statement of this postulate can be given in terms of the Cauchy problem of the matter fields. Let ps<% be such that every non-spacelike curve through p intersects the spacelike surface x* = 0 within Let ¿F be the set of points in the surface x* ~ 0 which can be reached by non-spacelike curves in % from p. Then we require that the values of the matter fields at p must be uniquely determined by the values of the fields and their derivatives up to some finite order on and that they are not uniquely determined by the values on any proper subset of & to which it can be continuously retracted. (For a fuller discussion of the Cauchy problem, see chapter 7.)

It is this postulate which sets the metric g apart from the other fields on and gives it its distinctive geometrical character. If {«"} are normal coordinates in about p, it is intuitively fairly obvious (and is proved in chapter 4) that the points which can be reached from p by non-spacelike curves in are those whose coordinates satisfy

The boundary of these points is formed by the image of the null cone of p under the exponential map, that is the set of all null geodesies through p. Thus by observing which points can communicate with p, one can determine the null cone Np in Tp. Once Np is known, the metric aty may be determined up to a conformal factor. This may be seen as follows: let X, Yei^, be respectively timelike and spacelike vectors. The equation g(X+AY, X+AY) = g(X, X) + 2Ag(X, Y) + A^(Y, Y)

will have two real roots Ax and Aa as gr(X, X) < 0 and gr(Y, Y) > 0. If Np is known, Ax and A2 may be determined. But

Thus the ratio of the magnitudes of a timelike vector and a spacelike vector may be found from the null cone. Then if W and Z are any two non-null vectors at p, g( W,Z) = ¿(?(W,W) + 0(Z,Z)-0(W + Z,W + Z)).

Each of the magnitudes on the right-hand side may be compared with the magnitude of either X or Y, and so gr(W, Z)/gr(X, X) may be found. (If W + Z is null, the corresponding expression involving W + 2Z could be used.) Thus observation of local causality enables one to measure the metric up to a conformal factor. In practice this measurement is performed most conveniently using the experimental fact that no signal has been observed to travel faster than electromagnetic radiation. This means that light must travel on null geodesies. This however is a consequence of the particular equations the electromagnetic field obeys, not of the theory of relativity itself. Causality will be considered further in chapter 6. Among other results, it will be shown that causal relations may be used to determine the topological structure The conformal factor in the metric may be determined using postulate (b) below; thus all the elements of the theory will be physically observable.

Postulate (b) : Local conservation of energy and momentum The equations governing the matter fields are such that there exists a symmetric tensor 7705, called the energy-momentum tensor, which depends on the fields, their covariant derivatives, and the metric, and which has the properties:

(i) Tab vanishes on an open set if and only if all the matter fields vanish on

(ii) Tab obeys the equation

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