Chapter 2 Navigating through spacetime

  Mathematics is the exquisitely perfect language needed for describing how the theory of relativity applies to the physical Universe and all of spacetime, and that description includes the strange behaviour that occurs near black holes. A mathematical description, while powerful and exact, even so can be something of a foreign and forbidding language for those without the appropriate technical training. Descriptive words, however eloquent, lack the rigour and power of a mathematical equation and can be imprecise and limiting. Pictures however, being(it is said) worth a thousand words, can be not only a useful compromise but a very helpful way to visualize what is going on. For this reason, it is well worth spending a little effort to understand a particular type of picture, called a spacetime diagram. This will help in understanding the nature of spacetime around black holes.

  Spacetime diagrams

  The cartoon in Figure 3 shows a simple spacetime diagram.Following tradition, the `time-like' axis is the one that is vertical on the page and the `space-like' axis is drawn perpendicular to this.Of course, we really need four axes to describe spacetime because there are three space-like axes (usually denoted x, y, and z) and one time-like axis. However, two axes will suffice for our purpose(and of course four mutually perpendicular axes are impossible to draw!).Where these two axes intersect is called the origin, and this may be regarded as the point of `here and now' for the observer who has constructed their spacetime diagram. An idealized instantaneous event, say the click of a camera shutter,occurs at a particular moment in time and at a particular location in space. Such an instantaneous event is represented by a dot on a spacetime diagram, appropriate to the time and spatial location in question. There are two dots in Figure 3, which are spatially separated (they do not occur at the same point on the space axis)but they are simultaneous (they have the identical coordinate on the time axis). You could imagine these two dots correspond to the simultaneous shutter presses of two photographers who are standing some distance apart from one another, photographing the same spectacle. If points represent events, what do lines in a spacetime diagram represent? A line simply shows a path of an object through spacetime. As we live our lives, we journey through spacetime and the path we leave behind us (somewhat as a snail leaves a glistening trail of slime behind it) is a line in spacetime,and in the jargon this is called a worldline. If you stay at home all day, your worldline is a vertical path through spacetime (with space coordinate = `22 Acacia Avenue', for example). You move forward in time but are fixed in space. If on the other hand you made a long journey, your worldline slants over because your distance changes with time, because you move in space as well as time.

  3. Asimplespacetime diagram.

  For example, look at the worldline shown in Figure 3, the line which is part vertical, then further up becomes slanting. This corresponds to the worldline of some other entity, which is stationary for the time indicated by the vertical extent of the line. An example might be a camera belonging to one of the photographers, left on a chair (so that its worldline is vertical because its position isn't changing), before it was stolen and whisked away (when the spatial location changes continuously).Where this line becomes slanting is where its spatial location is changing with time. The slope of this line tells you about the rate of change of distance with time, which is more commonly called the speed. In this case this is the speed at which the thief is whisking away the stolen camera. The faster the thief is making off with the camera, in other words the more ground he is covering in a given time, the less vertical and themore slanting this part of the line will be. There is of course a robust upper limit to the speed at which the thief can run off with his illegally gotten gains and this,as discussed in Chapter 1, is the speed of light. The trajectory of a beam of light would be represented by a maximally slanting line(commonly represented in spacetime diagrams as being at45 degrees to the time axis by using cleverly crafted units).Because nothing can go faster than that speed, no worldline can be at a greater angle to the time axis than this.

  Worldlines on a spacetime diagram having this maximally slanting angle, corresponding to this maximal speed, the speed of light,give rise to an important concept called a light cone. The idea of this is very simple: you can only have an effect on the Universe in the future by some prior cause and that causal sequence cannot propagate faster than the speed of light. Therefore your `sphere of influence' right now is contained in a restricted range of spacetime, namely that part which is within a 45-degree angle to the positive time axis as shown in Figure 4.Moreover, you can only have been influenced by a causal chain of events that could not have propagated faster than the speed of light. Therefore only events within a 45-degree angle to the backwards time axis can influence you now. If we now draw a spacetime diagram with two space-like axes and one time-like axis, then the triangles in Figure 4 become cones and these are what we mean by light cones,as shown in Figure 5. The light cone in Figure 5 delineates regions of space within which an observer (deemed to be located at the origin, their `here and now') could in principle reach (or have reached in the past) without having to invoke breaking the cosmic speed limit and travelling faster than the speed of light. The region centred on the positive (future) time axis is known as the future light cone while the cone centred on the negative time axis (i.e.past times) is known as the past light cone.

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  4. Asimplelightconediagram.

  Thus the assassination of Julius Caesar in 44 BC is part of your past, because there is a conceivable causal link between that event and you. (If you had to learn about it at school, that demonstrates the existence of a causal link!) Because light from the Andromeda Galaxy can reach a telescope on Earth, it too is part of your past.However, the light takes 6 million years to get to us, so it is the

  5. A spacetime diagramshowing the light cone of a particular observer.

  Andromeda Galaxy of 6 million years ago that is part of your past and sits on your light cone. The Andromeda Galaxy of today, or even the Andromeda Galaxy of 44 BC, is outside your light cone.Events happening on Andromeda, either now or even back in44 BC, cannot influence you right nowbecause any causal link would have had to travel faster than the speed of light.

  The three spacetime diagrams that we have seen in this chapter so far have their axes labelled as time and space. In fact, professionals wouldn't normally include axis labels or even the axes in spacetime diagrams. This isn't simply that it is so routine that time goes up and space goes across that professional astrophysicists get sloppy (though that's not an unknown phenomenon) but it is because the exact positions in spacetime cannot be agreed upon by all observers. In the world of special relativity, the notion of simultaneity breaks down. Just because two events are seen to be simultaneous for one observer doesn't at all mean that they are simultaneous for other observers.

  Thus the two photographers pressing the shutters of their cameras『simultaneously' will not be what an observer travelling in a spacecraft very fast relative to the cameras sees. That observer will deduce one camera click occurring substantially before the other.The two points in Figure 3 which I drew at the same vertical height (since I claimed the events occurred at the same time)would appear at different vertical positions on the spacetime diagram of the rapidly travelling observer. Einstein's relativity insists her diagram is just as valid as mine. So if the points on a spacetime diagram depend on an observer's point of view, i.e. their frame of reference, what's the reason for drawing them?

  To understand this, it is helpful to focus on the worldline of a moving particle and so we will now draw a new spacetime diagram in which a particle moves through spacetime, taking its light cone with it (this trick is known as working within the co-moving frame). Notice that in Figure 6 the particle's path (i.e.its worldline) always stays within the light cone as it cannot travel faster than the speed of light.

  Einstein's Special Theory of Relativity, which is a subset of his General Theory, pertains to a restricted set of physical situations.A different conceptual framework beyond Special Relativity is needed in the context of spacetime which is expanding, the pre-eminent example of which is the expanding Universe. In this context, themanifestation of causality is such that you cannotmove faster than the speed of light with respect to your local bit of space.

  How do objects know where to go?

  Although photons have no mass, it turns out that they are still influenced by gravity. It is best not to think of this as due to a force, but rather that this comes about because of the curvature of spacetime. A photon is usually thought to travel in a straight line,which is where we get the notion of a `light ray'. However, through a curved spacetime itwill follow a path known as a geodesic.

  6. Aspacetimediagramofaparticlemovingalongitsworldline,thatis always containedwithinits futurelight cone.

  Despite its Earth-based connotations, a geodesic (whose name comes from geodesy, i.e. measuring the lie of the land of our planet's surface) is an important concept describing the nature of spacetime throughout the Universe. If space were not curved(meaning entirely consistent with everyday geometry that we may have learned at school from Euclid or one of his successors), then a geodesic would be the `straight line path' that a light ray would travel. But the shortest distance between two points, which is the route that a light ray `wants' to take, is known by the term `null geodesic'. In curved space the shortest distance between two points isn't what we think of as straight, but `geodesics are straight lines in curved spaces'. A straight line can also be characterized as the path you follow by keeping moving in the same direction. An example of how geometry is seriously different on a curved surface comes from considering lines of longitude on a sphere. Two adjacent lines of longitude (which are parallel to one another at the equator) will meet at a point at the pole, as shown in Figure 7.However, in flat space parallel lines will meet only at infinity (as per Euclid's last axiom).

  Actually, where spacetime is curved, for example because of the presence of mass, that curvature is manifested in the path that a light ray or (a mental device used by physicists) a `test particle' freely able to move with no influence of any external force, would move along between two events. Two events should be regarded as two points in 4-D space time, each denoted in the form (t, x, y, z).

  7. Lines oflongitudeon asphere areparallel atthe equator, andmeetat a point at the poles.

  A rule called a metric tells us how clocks and rulers measure the separations between events in space and time and provide the basis for working out problems in geometry. A very simple example of a metric is Pythagoras' theorem, which tells us how to compute the distance between two points that lie in a plane. The solutions to Einstein's field equations tell us how to calculate the metric of spacetime when the distribution of matter is known.We use this to construct the geodesics for the real Universe. For example, one of the first pieces of observational evidence for General Relativity was the bending of starlight by the Sun,measured during a solar eclipse (a good time to examine the apparent positions of stars close to the Sun's disc because light from the disc is blocked out by theMoon, an opportunity seized upon by Sir Arthur Eddington in 1919). The Sun's mass curves spacetime. Thus the shortest path (the geodesic) from a distant star to a telescope on Earth is not quite a straight line: it is bent round by the Sun's gravitational field, as shown in Figure 8.

  The bending of starlight demonstrates that space is curved, but Einstein's General Theory tells us it is actually spacetime that is curved. Therefore we might expect that mass also has some strange effects on time. In fact, even the Earth's gravitational field is sufficient to make Earth-bound clocks tick a bit slower than they would do in deep space, although the effect is small (roughly one part in a billion) but measurable. The gravitational effects near the event horizon of a black hole are much stronger. Thus, even for the simplest case of a non-spinning black hole, time runs differently close to the black hole compared to how it runs at a huge distance from the black hole. This is a real effect and does not depend on how the time is measured (for example by an atomic clock, or by a digital watch). It follows directly from the curvature of spacetime induced by the mass which tips the light cones towards the mass.Figure 9 indicates the general effect.

  8. Amass suchasthe Suncausesdistortion, or curvature, inspacetime.

  Black holes profoundly affect the orientations of the light cones.As a particle approaches a black hole, its future light cone tilts more and more towards the black hole, so that the black hole becomes more and more a part of its inevitable future. When the particle crosses the event horizon, all of its possible future trajectories end inside the black hole. Just within the event horizon, the light cone tilting is so great that one side becomes parallel with the event horizon and the future lies entirely within the event horizon; escape from the black hole is not possible.Figure 9 also illustrates this point: it is essentially a representation of `local spacetime diagrams', because the assembly of light cones allows you to understand the local conditions experienced by a test particle located at different positions. In this figure, time increases up the page and so this diagram also gives a sense of how a black hole forms and grows due to infalling matter.

  9. Diagramofthe spacetimesurroundingablackhole showinghowthefuturelight conesforobjects onthe eventhorizonlie insidetheeventhorizon.

  Just as for the dark stars of Michell and Laplace discussed in Chapter 1 which could have sustained planetary systems in orbit around them much like our Solar System, so it is that we only know that a black hole is nearby due to its gravitational pull. This might lead you to think that the only property that characterizes a black hole is itsmass. In fact,whether or not a black hole is rotating has a dramatic effect on its properties, and I will explain how this comes about in Chapter 3.