Spacetime null cones metrics conformal geometry

When, in 1908, the distinguished mathematician Hermann Minkowski— who had coincidentally been one of Einstein's teachers at the Zurich Polytechnic—demonstrated that he could encapsulate the basics of special relativity in terms of an unusual type of 4-dimensional geometry, Einstein was less than enthusiastic about the idea. But later he realized the crucial importance of Minkowski's geometric notion of space-time. Indeed, it formed an essential ingredient of his own generalization of Minkowski's proposal to provide the curved space-time basis of his general theory of relativity.

Minkowski's 4-space incorporated the standard three dimensions of space with a fourth dimension to describe the passage of time. Accordingly, the points of this 4-space are frequently referred to as events, since any such point has a temporal as well as a spatial specification. There is not really anything very revolutionary about this, just in itself. But the key point of Minkowski's idea—which was revolutionary—is that the geometry of his 4-space does not separate out naturally into a time dimension and (more significantly) a family of ordinary Euclidean 3-spaces, one for each given time. Instead, Minkowski's space-time has a different kind of geometric structure, giving a curious twist to Euclid's ancient idea of geometry. It provides an overall geometry to space-time, making it one indivisible whole, which completely encodes the structure of Einstein's special relativity.

Thus, in Minkowski's 4-geometry, we are not now to think of the space-time as being simply built out of a succession of 3-surfaces, each representing what we think of as 'space' at various different times (Fig. 2.10). For that interpretation, each of these 3-surfaces would describe a family of events all of which would be taken to be simultaneous with one another. In special relativity, the notion of 'simultaneous' for spatially separated events does not have an absolute meaning. Instead, 'simultaneity' would depend upon some arbitrarily chosen observer's velocity.

High Deformation Spacetime
Fig. 2.10 Space-time before Minkowski.

This, of course, is at odds with common experience, for we do seem to have a notion of simultaneity for distant events that is independent of our velocity. But (according to Einstein's special relativity) if we were to move at a speed that is comparable with that of light, then events that seem to us to be simultaneous would generally not seem to be simultaneous to some other such observer, with a different velocity. Moreover, the velocities would not even have to be very large if we are concerned with very distant events. For example, if two people stroll past each other in opposite directions along a path, then the events on the Andromeda Galaxy that they would each individually consider to be simultaneous with that particular event at which they pass one another would be likely to differ by several weeks,[225] see Fig. 2.11!

High Deformation Spacetime
Fig. 2.11 Two walkers amble past one another, but the event X of their passing is judged by each to be simultaneous with events on Andromeda differing by several weeks.

According to relativity, the notion of 'simultaneous', for distant events, is not an absolute thing, but depends upon some observer's velocity to be specified, so the slicing of space-time into a family of simultaneous 3-spaces is subjective in the sense that for a different observer velocity we get a different slicing. What Minkowski's space-time achieves is to provide an objective geometry, that is not dependent on some arbitrary observer's view of the world, and which does not have to change when one observer is replaced by another. In a certain sense, what Minkowski did was to take the 'relativity' out of special relativity theory, and to present us with an absolute picture of spatio-temporal activity.

But for this to give us a firm picture, we need a kind of structure for the 4-space to replace the idea of a temporal succession of 3-spaces. What structure is this? I shall use the letter M to denote Minkowski's

4-space. The most basic geometrical structure that Minkowski assigned to M is the notion of a null cone,[2 26] which describes how light propagates at any particular event p in M. The null cone—which is a double cone, with common vertex at p—tells us what the 'speed of light' is in any direction, at the event p (see Fig. 2.12(a)). The intuitive picture of a null cone is provided by a flash of light, initially focusing itself inwards precisely towards the event p (past null cone), and immediately afterwards spreading itself out from p (future null cone), like the flash of an explosion at p, so the spatial description (Fig. 2.12(b)) following the explosion becomes an expanding succession of concentric spheres. In my diagrams, I shall tend to draw null cones with their surfaces tilted at roughly 45° to the vertical, which is what we get if we choose space and time units so that the speed of light c = 1. Thus if we choose seconds for our time scale, then we choose a light-second (= 299 792 458 metres) for our unit of distance; if we choose years for our time scale, then we choose a light-year (=9.46 x 1012 kilometres) for our unit of distance, etc.[2.27]

Fig. 2.12 (a) Null cone at p in Minkowski's 4-space; (b) 3-space description of the future cone as an expanding succession of concentric spheres originating at p.

Einstein's theory tells us that the speed of any massive particle must always be less than that of light. In space-time terms, this means that the world-line of such a particle—this being the locus of all the events that constitute the particle's history—must be directed within the null

Fig. 2.12 (a) Null cone at p in Minkowski's 4-space; (b) 3-space description of the future cone as an expanding succession of concentric spheres originating at p.

cone at each of its events. See Fig. 2.13. A particle may have a motion that is accelerated at some places along its world-line, whence its world-line need not be straight, the acceleration being expressed, in space-time terms, as a curvature of the world-line. Where the world-line is curved, it is the tangent vector to the world-line that must lie within the null cone. If the particle is massless,[228] such as a photon, then its world-line must lie along the null cone at each of its events, since its speed at every one of its events is indeed taken to be light speed.

The null cones also tell us about causality, which is the issue of determining which events are to be regarded as being able to influence which other events. One of the tenets of (special) relativity theory is the assertion that signals shall not be allowed to propagate faster than light. Accordingly, in terms of the geometry of M, we say that an event p would be permitted to have a causal influence on event q if there is a world-line connecting p to q, that is a (smooth) path from p to q lying on or within the null cones. For this, we need to specify an orientation to the path (indicated by attaching an 'arrow' to the path), that proceeds uniformly from past to future. This requires that M's geometry be assigned a time orientation, which amounts to a consistent continuous separate assign ment of 'past' and 'future' to the two components of each null cone. I have labelled a past component with a ' - ' sign and a future component with a ' +' sign. This is illustrated in Figs. 2.12(a) and 2.13, where the past null cone is distinguished in my drawings by the use of broken lines. The normal terminology of 'causation' takes causal influences to proceed in the past-to-future direction, i.e. along world-lines whose oriented tangent vectors point on or within future null cones.[2 29]

M's geometry is completely uniform, where each event is on an equal footing with every other event. But when we pass to Einstein's general theory of relativity, this uniformity is generally lost. Nonetheless, we again have a continuous assignment of time-oriented null cones, and again it is true that any massive particle has a world-line whose (future-oriented) tangent vectors all lie within these future null cones. And, as before, a massless particle (photon) has a world-line whose tangent vectors all lie along null cones. In Fig. 2.14 I have depicted the kind of situation which occurs in general relativity, where the null cones are not now arranged in a uniform fashion.

High Deformation Spacetime

We have to try to think of these cones being drawn on some kind of ideal 'rubber sheet' with the null cones printed on it. We can move the rubber sheet around and distort it in any way we like, so long as the deformation is done in a smooth way, where the null cones are carried around with the rubber sheet. Our null cones determine the 'causality structure' between events, and this is not altered by any such deformation provided that the cones are thought of as being carried around with the sheet.

A somewhat analogous situation is provided by Escher's depiction of the hyperbolic plane shown in Fig. 2.3(c), in §2.1, where we can imagine Escher's picture to be printed on such an ideal rubber sheet. We might choose one of the devils that appears to be close to the boundary, and move it, by such a smooth deformation of the sheet, so that it comes into the location previously occupied by one near the centre. This motion can be made to carry all the devils into locations previously occupied by other devils, and such a motion would describe a symmetry of the underlying hyperbolic geometry illustrated by Escher's picture. In general relativity, symmetries of this kind can occur (as with the Friedmann models described in §2.1), but this is rather exceptional. However, the possibility of carrying out such 'rubber-sheet' deformations is very much part of the general theory, these being referred to as 'diffeomorphisms' (or 'general coordinate transformations'). The idea is that such deformations do not alter the physical situation at all. The principle of 'general covariance', which is a cornerstone of Einstein's general relativity, is that we formulate physical laws in such a way that such 'rubber-sheet deformations' (diffeomorphisms) do not alter the physically meaningful properties of the space and its contents.

This is not to say that all geometrical structure is lost, where the only kind of geometry that remains for our space might be something merely of the nature of its topology (indeed sometimes referred to as 'rubber-sheet geometry', in which the surface of a teacup would be identical to that of a ring, etc.). But we must be careful to specify what structure is needed. The term manifold is frequently used for such a space, of some definite finite number of dimensions (where we may refer to a manifold of n dimensions as an n-manifold), a manifold being smooth but not necessarily assigned any further structure beyond its smoothness and topology. In the case of hyperbolic geometry, there is actually a notion of metric assigned to the manifold—a mathematical 'tensor' quantity

(see also §2.6), usually denoted by the letter g—which may be thought of as providing an assignment of a length[2 30] to any finite smooth curve in the space. Any deformation of the 'rubber sheet' constituting this manifold would carry with it any curve C connecting a pair of points p, q (where p and q are also carried by the deformation) and the length of the segment of C joining p to q assigned by g is deemed to be unaffected by this deformation (and, in this sense, g is also 'carried around' by the deformation).

This length notion also implies a notion of straight line, referred to as a geodesic, such a line l being characterized by the fact that for any two points p and q on l, not too far apart, the shortest curve (in the sense of length provided by g) from p to q is in fact the portion pq of l. See Fig. 2.15. (In this sense, a geodesic provides the 'shortest route between two points'.) We can also define angles between two smooth curves (this also being determined once g is given), so that the ordinary notions of geometry are available to us once g has been assigned. Nevertheless, this geometry would usually differ from the familiar Euclidean geometry.

Fig. 2.15 The metric g assigns lengths to curves and angles between them. The geodesic l provides the 'shortest route between p and q' in the metric g.

The hyperbolic geometry of Escher's picture (Fig. 2.3(c), Beltrami-Poincare conformal representation) thus also has its straight lines (geodesics). These can be understood in terms of the background length of curvi measured by g by g angle between curves determ length of curvi measured by g angle between curves determ

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Fig. 2.15 The metric g assigns lengths to curves and angles between them. The geodesic l provides the 'shortest route between p and q' in the metric g.

Euclidean geometry in which this figure is represented, as circular arcs meeting the boundary circle at right angles (see Fig. 2.16). Taking a and b to be the endpoints of the arc through two given points p and q, the hyperbolic g-distance between p and q turns out to be

C log

I qbWpal where the 'log' used here is a natural logarithm (2.302585. . . times the 'logio' of §1.2), '|qa|' etc. being the ordinary Euclidean distances in the background space, and C is a positive constant called the pseudo-radius of the hyperbolic space.

Fig. 2.16 'Straight lines' (geodesies) in conformal representation of hyperbolic geometry are circular ares meeting the boundary circle at right angles.

But rather than specifying the structure provided by such a g, one may assign some other type of geometry instead. The kind that will be of most concern for us here is the geometry known as conformal geometry. This is the structure that provides a measure to the angle between two smooth curves, at any point where they meet, but a notion of 'distance' or 'length' is not specified. As mentioned above, the concept of angle is actually determined by g, but g itself is not fixed by the angle notion. While the conformal structure does not fix the length measure, it does fix the ratios of the length measures in different directions at any point—

so it determines infinitesimal shapes. We can rescale this length measure up or down at different points without affecting the conformal structure (see Fig. 2.17). We express this rescaling as g ^ n2 g where n is a positive real number defined at each point, which varies smoothly over the space. Thus g and n2g give us the same conformal structure whatever positive n we choose, but g and n2g give us different metric structures (if n^1), where n is the factor of the scale change. (The reason for n appearing in 'squared' form in the expression 'n2g' is that the expressions for the direct measures of spatial—or temporal— separation, as provided by g, arise from the taking of a square root (see Note 2.30).) Returning to Escher's Fig. 2.3(c), we find that the conformal structure of the hyperbolic plane (though not its metric structure) is actually identical to that of the Euclidean space interior to the bounding circle (yet differing from the conformal structure of the entire Euclidean plane).

lengths differ, but angles agree

Fig. 2.17 Conformal structure does not fix length measure, but it does fix angles via the ratio of length measures in different directions at any point. Length measure can be rescaled up or down at different points without affecting the conformal structure.

lengths differ, but angles agree

Fig. 2.17 Conformal structure does not fix length measure, but it does fix angles via the ratio of length measures in different directions at any point. Length measure can be rescaled up or down at different points without affecting the conformal structure.

When we come to space-time geometry, these ideas still apply, but there are some significant differences, owing to the 'twist' that Minkowski introduced into the ideas of Euclidean geometry. This twist is what mathematicians refer to as a change of the signature of the metric. In algebraic terms, this simply refers to a few + signs being changed to - signs, and it basically tells us how many of a set of n mutually orthogonal directions, for an n-dimensional space, are to be considered as 'timelike' (within the null cone) and how many 'spacelike' (outside the null cone). In Euclidean geometry, and its curved version known as Riemannian geometry, we think of all directions as being spacelike. The usual idea of 'space-time' involves only 1 of the directions being timelike, in such an orthogonal set, the rest being spacelike. We call it Minkowskian if it is flat and Lorentzian if it is curved. In the ordinary type of (Lorentzian) space-time that we are considering here, n = 4, and the signature is '1+3' separating our 4 mutually orthogonal directions into 1 timelike direction and 3 spacelike ones. 'Orthogonality' between spacelike directions (and between timelike ones, had we had more than 1 of them) means simply 'at right angles', whereas between a spacelike and a timelike direction it looks geometrically more like the situation depicted in Fig. 2.18, the orthogonal directions being symmetrically related to the null direction between them. Physically, an observer whose world-line is in the timelike direction regards events in an orthogonal spacelike direction to be simultaneous.

Fig. 2.18 'Orthogonality' of spacelike and timelike directions in Lorentzian spacetime, represented in a Euclidean picture for which the null cone is right-angled.

In ordinary (Euclidean or Riemannian) geometry, we tend to think of lengths in terms of spatial separation, which is something that we might

Lorentzian

Cable Tension Screw Drawing

Fig. 2.18 'Orthogonality' of spacelike and timelike directions in Lorentzian spacetime, represented in a Euclidean picture for which the null cone is right-angled.

Lorentzian perhaps use a ruler to measure. But what is a ruler, in (Minkowskian or Lorentzian) space-time terms? It is a strip, which is not immediately the most obvious gadget for measuring the spatial separation between two events p and q. See Fig. 2.19. We can put p on one edge of the strip and q on the other. We can also assume that the ruler is narrow and unaccelerated, so that the space-time curvature effects of Einstein's (Lorentzian) general relativity are not of relevance, and a treatment according to special relativity should be adequate. But according to special relativity theory, in order for the distance measure provided by the ruler to give the correct space-time separation between p and q, we require that these events be simultaneous in the rest-frame of the ruler. How can we ensure that these events are actually simultaneous in the ruler's rest-frame? Well, we can use Einstein's original type of argument for this, although he was thinking more in terms of a train in uniform motion, than a ruler—so let us now phrase things that way too.

ruler history

(or train)

Minkowski space M

ruler history

(or train)

Minkowski space M

not simultaneous in ruler's rest-frame so separation qp is not ruler's length ruler time ruler not simultaneous in ruler's rest-frame so separation qp is not ruler's length

Fig. 2.19 A spacelike separation between points p and q in M is not directly measured by a ruler that is a 2-dimensional strip.

Let us refer to the end of the train (ruler) containing the event p as the front, and the end containing q as the back. We imagine an observer situated at the front, sending a light signal from an event r to the back of the train, timed so as to arrive there precisely at the event q, whereupon the signal is immediately reflected back to the front, to be received by the observer at the event s. See Fig. 2.20. The observer then judges q to be simultaneous with p, in the train's rest-frame, if p occurs half-way between emission and final reception of the signal, i.e. if the time interval from r to p is precisely the same as that from p to s. The length of the train (i.e. of the ruler) then (and only then) would agree with the spatial interval between p and q.

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Fig. 2.20 The ruler (or train) measures the separation pq only when they are simultaneous, so light signals and clocks are needed instead.

We notice that not only is this a little more complicated than simply 'laying down a ruler' to measure the spatial separation between events, but what is actually measured by the observer would be the time intervals rp and ps. These (equal) time intervals directly provide the measure of the spatial interval pq that is being ascertained (in units where the speed of light c is taken to be unity). This illustrates the key fact about the metric of space-time, namely that it is really something that has much more directly to do with the measurement of time rather than distance. Instead of providing a length measurement for curves, it directly provides us with a time measurement. Moreover, it is not all curves that are assigned a time measure: it is for the curves referred to as causal, that could be the world-lines of particles, these curves being everywhere either timelike (with tangent vectors within the null cones, achieved by massive particles) or null (with tangent vectors along the null cones, achieved by massless particles). What the space-time metric g does is to assign a time measure to any finite segment of a causal curve (the contribution to the time measure being zero for any portion of the curve which is null). In this sense, the 'geometry' that the metric of space-time possesses should really be called 'chronometry', as the distinguished Irish relativity theorist John L. Synge has suggested.12311 It is important for the physical basis of general relativity theory that extremely precise clocks actually exist in Nature, at a fundamental level, since the whole theory depends upon a naturally defined metric g.[232] In fact, this time measure is something quite central to physics, for there is a clear sense in which any individual (stable) massive particle plays a role as a virtually perfect clock. If m is the particle's mass (assumed to be constant), then we find that it has a rest energy[2331 E given by Einstein's famous formula

E = mc2, which is fundamental to relativity theory. The other, almost equally famous formula—fundamental to quantum theory—is Max Planck's

E=hv

(h being Planck's constant), telling us that this particle's rest energy defines for it a particular frequency v of quantum oscillation (see Fig. 2.21). In other words, any stable massive particle behaves as a very precise quantum clock, which 'ticks away' with the specific frequency

V = m ©, in exact proportion to its mass, via the constant (fundamental) quantity c2/h.

Martini Glass Schematic

In fact the quantum frequency of a single particle is extremely high, and it cannot be directly harnessed to make a usable clock. For a clock that can be used in practice, we need a system containing many particles, combined together and acting appropriately in concert. But the key point is still that to build a clock we do need mass. Massless particles (e.g. photons) alone cannot be used to make a clock, because their frequencies would have to be zero; a photon would take until eternity before its internal 'clock' gets even to its first 'tick'! This fact will be of great significance for us later.

All this is in accordance with Fig. 2.22, where we see different identical clocks, all originating at the same eventp, but moving with different velocities which are allowed to be comparable with (but less than) the speed of light. The bowl-shaped 3-surfaces (hyperboloids, in ordinary geometry) mark off the successive 'ticks' of the identical clocks. (These 3-surfaces are analogues of spheres for Minkowski's geometry, being the surfaces of constant 'distance' from a fixed point.) We note that a massless particle, since its world-line runs along the light cone, never reaches even the first of the bowl-shaped surfaces, in agreement with what has been said above.

Fig. 2.22 Bowl-shaped 3-surfaces mark off the successive 'ticks' of identical clocks.

Finally, the notion of a geodesic, for a timelike curve, has the physical interpretation as the world-line of a massive particle in free motion clo sychri hi clo sychri hi

Fig. 2.22 Bowl-shaped 3-surfaces mark off the successive 'ticks' of identical clocks.

under gravity. Mathematically, a timelike geodesic line l is characterized by the fact that for any two points p and q on l, not too far apart, the longest curve (in the sense of length of time provided by g) from p to q is in fact a portion of l. See Fig. 2.23—a curious reversal of the length-minimizing property of geodesic Euclidean or Riemannian spaces. This notion of geodesic applies also to null geodesics, the 'length' being zero in this case, and the null-cone structure of the space-time alone is sufficient to determine them. This null-cone structure is actually equivalent to the space-time's conformal structure, a fact that will have importance for us later.

Fig. 2.23 A timelike geodesic line l is characterized by the fact that for any two points p and q on l, not too far apart, the longest local curve from p to q is in fact a portion of l.

timelike geodesic l timelike geodesic l

I \

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