Ag

or, in the Planck units of §3.2, simply

E = 8nT+Ag, where A is the cosmological constant, and where the energy [0]-tensor T represents the mass-energy density and other quantities related to it via requirements of relativity; in other words, E (or equivalently, the energy tensor T) is the gravitational analogue of J. The Weyl tensor C is then the gravitational analogue of Maxwell's F.

We may ask what directly observable effects C and E might have, like magnetic fields being shown up by patterns of iron filings or by the pointing of a compass needle, and like electric fields being revealed by their effect on pith balls, etc. In fact, in an almost literal sense, we can actually see the effects of E and more particularly C, since these tensors have a direct and distinguishing effect on light rays—and in this respect E and T are completely equivalent, since Ag has no effect on light rays. It may justly be said that the first clear evidence in support of general relativity was such a direct observation—which came from (Sir) Arthur Eddington's expedition to the Island of Principe in order to view, during the solar eclipse of 1919, the apparent displacement of stars' locations due to the Sun's gravitational field.

Basically, E acts as a magnifying lens, whereas C acts as a purely astigmatic lens. These effects are well described if we imagine how light rays are affected as they pass near or through a massive body, such as the Sun. Of course ordinary light will not actually propagate through the body of the Sun (or of the obscuring Moon, during the eclipse, for that matter), so we do not directly observe those particular rays in this case. But we can imagine that if we could actually see the star field through the Sun, then that field would be magnified slightly, owing to the presence of E, where the gravitating material of the Sun's actual body resides. The pure effect of E would be simply to magnify one's 'view' of what lies behind, without distortion.[2 59] However, when it comes to the distortion of the image of the distant star field outside the Sun's apparent disc (and this is what is actually observed) we find a gradual reduction of the outward displacement the farther out we look, and this leads to an astigmatic distortion of the distant star field. These effects are illustrated in Fig. 2.47. The distortion of the field outside the Sun's limb makes a small circular pattern in the distant star field appear elliptical, and this ellipticity is a measure of the amount of Weyl curvature C intercepted by the line of sight.

Fig. 2.47 The presence of Weyl curvature surrounding a gravitating body (here the Sun) can be seen in the distorting (non-conformal) effect that it has on the background field.

In fact, this gravitational lensing effect, originally predicted by Einstein, has become an extremely important tool in modern astronomy and cosmology, since it provides a means of measuring mass distributions

Fig. 2.47 The presence of Weyl curvature surrounding a gravitating body (here the Sun) can be seen in the distorting (non-conformal) effect that it has on the background field.

that might even be otherwise completely invisible. In most of these cases, the distant background field consists of large numbers of very distant galaxies. The objective is to ascertain whether significant ellipticity has been introduced into the appearance of this background field, and to use this to estimate the actual intervening mass distribution whose gravitational field has caused the pattern of ellipticities. A snag, however, is that galaxies themselves tend to be rather elliptical, so one cannot usually tell whether or not an individual galaxy's image has been distorted. However, with large numbers of background field galaxies, statistics can be brought in, and often some very impressive estimates of mass distributions can be obtained in this way. On occasion, it is even possible to judge these things by eye, and some impressive examples are provided in Fig. 2.48, where the patterns of ellipticity make the presence of lensing sources particularly evident. One important application of this technique is in the mapping of dark matter distributions (see §2.1), since these are otherwise invisible.[2 60]

The fact that C introduces ellipticity into the images along light rays is indicative of its role as the quantity describing conformal curvature. At the end of §2.3 it was remarked that the conformal structure of space-time is in fact its null-cone structure. The conformal curvature of space-time, namely C, therefore measures the deviation of this null-cone structure from that of Minkowski space M. We see that the nature of this deviation is that it introduces ellipticity into bundles of light rays.

Let us now come to the condition that we require, in order to characterize the very special nature of the Big Bang. Basically, we require a statement that gravitational degrees of freedom were unexcited at the Big Bang, which means saying something like 'the Weyl curvature C vanished there'. For many years, I have indeed been proposing that some such condition 'C = 0' holds at initial-type singularities, as opposed to what evidently happens in the 'final-type' singularities occurring in black holes for which C is likely to become infinite, as it does towards the singularity in the Oppenheimer-Snyder collapse, and perhaps diverging extremely wildly as in BKL singularities.[261] In general terms, this condition of the vanishing of C at initial-type

Fig. 2.48 Gravitational lensing: (a) galaxy cluster Abell 1689; (b) galaxy cluster Abell 2218.

singularities—which I have termed the Weyl curvature hypothesis (or WCH)—seems appropriate, but it is a little awkward that there are in fact numerous different versions of such a statement. The trouble is, basically, that C is a tensor quantity and it is hard to make unambiguous mathematical assertions about how such quantities behave at space-time singularities, where the very notion of a tensor, in any ordinary sense, loses its meaning.

It is fortunate, therefore, that my Oxford colleague Paul Tod has made a detailed study of a quite different, and mathematically much more satisfactory way of formulating a 'WCH'. This is to say, more or less, that there is a Big Bang 3-surface U-, which acts as a smooth past boundary to the space-time M, when M is considered as a conformal manifold, just as happens in the exactly symmetrical FLRW models as is exhibited in the strict conformal diagrams of Fig. 2.34 and Fig. 2.35, but where the FLRW symmetry of these particular models is now not assumed. See Fig. 2.49. Tod's proposal at least constrains C to be finite at the Big Bang (since the conformal structure at U-, is assumed to be smooth), rather than C diverging wildly, and this statement might be taken to be sufficient for what is required.

Weyl Curvature Hypothesis
Fig. 2.49 Schematic conformal diagram of Paul Tod's proposal for a form of 'Weyl curvature hypothesis'; asserting that the Big Bang provides a smooth boundary B to the space-time M.

To make this condition mathematically clearer, it is convenient to assert it in the form that the space-time can be continued smoothly, as a conformal manifold, a little way prior to the hypersurface U-. To before the Big Bang? Surely not: the Big Bang is supposed to represent the beginning of all things, so there can be no 'before'. Never fear— this is just a mathematical trick. The extension is not supposed to have any physical meaning! Or might it . . .?

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