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and the vorticity vector as

The covariant derivative of the vector V may be expressed in terms of these quantities; ^ = ^ + ^ + ^_^^ (4.i7)

This decomposition of the gradient of the fluid velocity vector is directly analogous to that in Newtonian hydrodynamics.

In the Fermi-propagated orthonormal basis the vorticity and expansion can be expressed in terms of the matrix Aap and its inverse A~\fi- d

From the deviation equation (4.8) it follows that d2

This equation enables one to calculate the propagation of the vorticity, shear and expansion along the integral curves of V if one knows the Riemann tensor.

Multiplying by A~xfy and taking the antisymmetric part, one obtains ^

Thus the propagation of vorticity depends on the antisymmetric gradient of the acceleration but not the 'tidal force'. Another form of the above equation is

Therefore AyaojytAtpi& a constant matrix if the curves are geodesies; in particular, if the curves are geodesies and the vorticity vanishes at one point on a curve, it will vanish at all points on the curve. If the curves are the flow lines of a perfect fluid it follows from (4.1) that

This conservation law is the relativistic form of the Newtonian vorticity conservation law. In the geodesic or pressure-free case, this takes the usual form that the magnitude of the vorticity vector is inversely proportional to the area of a cross-section orthogonal to the vorticity vector of an element of the fluid. When the pressure is nonzero, there is an extra relativistic effect arising from the fact that compression of the fluid does work on the fluid and therefore increases the mass and so the inertia of an element of the fluid (cf. (3.9)). This means that the vorticity of a fluid increases less under compression than would otherwise be expected.

Multiplying (4.21) by A~xfy and taking the symmetric part, one finds ,

¿i6»A = - R«m ~ - 6ay 6yP + ?«.;/» + £ (4-25)

(This equation and (4.23) can be expressed in terms of a general, non-orthonormal, non-Fermi-propagated basis by replacing the ordinary derivatives with Fermi derivatives and projecting everything into the subspace orthogonal to V.)

If the fluid is isentropic, this implies the conservation law: WA co tAte = constant, ya yi ifl

where ya yi ifl where

-RabVaVb + 2toi — 2<ri — \6i+fa. 2co* = wo6w°6 > 0, 2(7« = <rabcr°b > 0.

This equation, which was discovered by Landau and independently by Raychaudhuri, will be of great importance later. From it one sees that vorticity induces expansion as might be expected by analogy with centrifugal force while shear induces contraction. By the field equations, the term RabVaVb = 4n(fi+3p) for a perfect fluid whose flow lines have tangent vectors Va. Thus one would expect this term also to induce contraction. We shall give a general discussion of the sign of this term in §4.3.

The trace-free part of (4.25) is j;<rab = - Cacta Vc Vd + hKc V Kd ~ ^ac - ^

where Ca6cd is the Weyl tensor. Since this tensor is trace-free it does not enter directly in the expansion equation (4.26). However since the term — 2<r2 occurs on the right of the expansion equation, the Weyl tensor produces convergence indirectly by inducing shear. The Riemann tensor can be expressed in terms of the Weyl tensor and the Ricci tensor:

Rabcd = Cdbcd ~ 9a[d^c] b ~ Shield) a ~ h^9alc9d]b-The Ricci tensor is given by the Einstein equations:

Kb-h9abR + Mab = 87TTab. Thus the Weyl tensor is that part of the curvature which is not determined locally by the matter distribution. However it cannot be entirely arbitrary as the Riemann tensor must satisfy the Bianchi identities: D

These can be rewritten as

These equations are rather similar to Maxwell's equations in electrodynamics: Fab ^ = JOj where Fab is the electromagnetic field tensor and Ja is the source current. Thus in a sense one could regard the Bianchi identities (4.28) as field equations for the Weyl tensor giving that part of the curvature at a point that depends on the matter distribution at other points. (This approach has been used to analyse the behaviour of gravitational radiation in papers by Newman and Penrose (1962), Newman and Unti (1962) and Hawking (1966a).)

4.2 Null curves

The Riemann tensor will affect the rate of change of separation of null curves as well as that of timelike curves. For simplicity, we shall consider only null geodesies. These could represent the histories of photons; the effect of the Riemann tensor will be to distort or focus small bundles of light rays.

To investigate this, we consider the deviation equation for a congruence of null geodesies with tangent vector K {g{K, K) = 0). There are two important differences between this case and that of the timelike curves considered in the previous section. First, one could normalize the tangent vector V to the timelike curves by requiring V, V) = — 1. In effect this means that one parametrized the curves by the arc-length s. However this is clearly impossible with null curves as they have zero arc-lengths. The best one can do is to choose an affine parameter v; then the tangent vector K will obey

However one could multiply v by a function / which was constant along each curve. Then fv would be another affine parameter and the corresponding tangent vector would be/-1K. Thus, given the curves as point sets in the manifold, the tangent vector is only really unique up to a constant factor along each curve. The second difference is that Qg, the quotient of Tg by K, is not now isomorphic to Hg, the subspace of Tq orthogonal to K, since HQ includes the vector K itself as p(K, K) = 0. In fact as will be shown below, one is not really interested in the whole of Qg but only in the subspace SQ consisting of equivalence classes of vectors in Hg which differ only by a multiple of K. In the case of light rays, one can regard an element of Sg as representing the separation between two neighbouring light rays which were emitted at the same time by a source.

As before we introduce dual bases Ej, E2, Es, E4) and E1, E2, E3, E4 of Tg and T* at some point g on a curve y(v). However we will not choose them to be orthonormal. We take E4 equal to K, Es to be some other null vector L having unit negative scalar product with E4 (<7(E3, E3i) = 0, <7(E3, E4) = — 1) and Ex and E2 to be unit spacelike vectors, orthogonal to each other and to Es and E,

(p(Ei, Ej) = p(E2,E2) = 1, <7(Ej,E2) = stEj, Es) = g(Eu E4) = 0, etc.).

Note that because of the non-orthonormal character of the basis, the form E3 is in fact equal to the form — Kagab and E4 is — L"gab. It can be seen that Ej, E2 and E4 constitute a basis for Hg while the projections into Qg of Ex, E2 and Es form a basis of Qg, and the projections of Ej and E2 form a basis of Sg. We shall normally not distinguish between a vector Z and its projection into Qg or Sg. We shall call a basis having the properties of Ej, E2, Es, E4, above, pseudo-orthonormal. By parallelly transporting them along the geodesic y(v) one obtains a pseudo-orthonormal basis at each point of y{v).

Wo use this basis to analyse the deviation equation for null geodesies. If Z is the vector representing the separation of corresponding points on neighbouring curves, one has, as before:

In the pseudo-orthonormal basis Ka.t will be zero as K is geodesic. Therefore one can express the 1, 2 and 3 components of (4.30) as a system of ordinary differential equations:

where as before Greek indices take the values 1, 2, 3. This shows that the projection of Z into the space Qg obeys a propagation equation which involves only this projection, and not the component of Z parallel to K. Further Ks.c = 0 since (KagabKb).c = 0. This implies that Z3 = —ZaKa is constant along the geodesic y(v). This can be interpreted as saying that light rays emitted from the same source at different times maintain a constant separation in time. As this is the case, one is more interested in the behaviour of neighbouring null geodesies which have purely spatial separations, i.e. one is interested in vectors Z for which Za = 0. The projections of such vectors will then lie in the subspace Sg and will obey the equation

^-Zm = Km.nZn, where m, n take the values 1, 2 only. This is similar to (4.7) for the timelike case, except that now one is dealing only with a two-dimensional space of connecting vectors Z.

As in the previous section, one can express Zm in terms of their values at some point q:

where Amn(v) is a 2 x 2 matrix which satisfies

As before we call the antisymmetric part of Km. n the vorticity <£)mn, the symmetric part the rate of separation Bmn and the trace the expansion B. We also define the shear amn as the trace-free part of Bmn. They obey similar equations to the analogous quantities in the previous section: ^

Equation (4.35) is the analogue of the Raychaudhuri equation for timelike geodesies. One sees again that vorticity causes expansion while shear causes contraction. We shall show in the next section that the Ricci tensor term —RabKaKb will normally be negative, and so cause focussing. As before the Weyl tensor does not affect the expansion directly but causes distortion which in turn causes contraction (cf. Penrose (1966)).

4.3 Energy conditions

In the actual universe the energy-momentum tensor will be made up of contributions from a large number of different matter fields. It would therefore be impossibly complicated to describe the exact energy-momentum tensor even if one knew the precise form of the* contribution of each field and the equations of motion governing it. In fact, one has little idea of the behaviour of matter under extreme conditions of density and pressure. Thus it might seem that one has little hope of predicting the occurrence of singularities in the universe from the Einstein equations as one does not know the right-hand side of the equations. However there are certain inequalities which it is physically reasonable to assume for the energy-momentum tensor. These will be discussed in this section. It turns out that in many circumstances these are sufficient to prove the occurrence of singularities, independent of the exact form of the energy-momentum tensor.

The first of these inequalities is: The weak energy condition

The energy-momentum tensor at each obeys the inequality

T^ WaWb > 0 for any timelike vector WeTp. By continuity this will then also be true for any null vector W eTp.

To an observer whose world-line at p has unit tangent vector V, the local energy density appears to be Tab VaVb. Thus this assumption is equivalent to saying that the energy density as measured by any observer is non-negative. This would seem very reasonable physically. To investigate further the significance of this assumption we use the fact that one may express the components Tab of the energy-momentum tensor at p with respect to an orthonormal basis Ex, E2, E3, E4i (E4 timelike) in one of four canonical forms.

This is the general case in which the energy-momentum tensor has a timelike eigenvector E4. This eigenvector is unique unless [i= —pa (a =1,2,3). The eigenvalue [i represents the energy-density as measured by an observer whose world-line at p has unit tangent vector E4 and the eigenvalues pa (a = 1,2,3) represent the principal pressures in the three spacelike directions Ea. This is the form of the energy-momentum for all observed fields with non-zero rest mass and also for all zero rest mass fields except in special cases when it is type II.

Type I.

Type II.