† Sir Roger introduced his diagrammatic methods in the early 1960s and they were later used to great effect by Australian theorist Brandon Carter. Today, conformal spacetime diagrams are often referred to as Carter–Penrose diagrams.
‡ Hermann Minkowski, in his 1908 address ‘Raum und Zeit’ to the Society of German Natural Scientists and Physicians.
§ Distortion in the sense used here has nothing to do with the distortion of spacetime due to matter. We are talking about the need to distort the representation to fit it on a sheet of paper.
¶ Future lightlike infinity is often labelled ℑ+ (pronounced ‘scri plus’) and past lightlike infinity is labelled ℑ– (‘scri minus’).
** In three dimensions, the clocks populate all of space, not just a line.
†† This picture can be extended to three dimensions of space. The clocks would form a three-dimensional lattice spanning all of space, and one could imagine a latticework of rulers connecting them together. In the terminology of relativity, such a latticework of clocks and rulers is known as an inertial reference frame.
‡‡ Shakespeare’s Macbeth.
§§ To be precise, we should say anything that is not massless.
¶¶ Actually, Grey moves at 48.4 per cent the speed of light relative to Black, which you can just about see if you look really carefully at the figure. We chose 48.4 per cent for reasons that become clear in the next footnote.
*** If you know a bit of special relativity (which you will if you have read our earlier book Why does E = mc2?) you might like to note that the factor 8/7 = 1.14 is equal to 1/ √(1 – v2) for v = 0.484.
††† Admittedly difficult when he himself is standing on the shoulders of giants.
‡‡‡ Incidentally, you might like to note that the straight-line path between events in spacetime is longer than any non-straight line. This is because the geometry of spacetime is not the geometry of Euclid. If it were, the interval would be given by Pythagoras’ theorem: (Δτ)2 = (Δt)2 + (Δx)2. But the interval contains a minus sign: (Δτ)2 = (Δt)2 – (Δx)2, and that makes all the difference. The geometry of flat spacetime is what mathematicians call hyperbolic geometry.
§§§ Another way to see this is to note that, because Rindler starts at the bottom of region 1, all of region 1 lies within his future light cone.
¶¶¶ We say freefall is locally indistinguishable because the Earth’s gravitational field is not uniform, and this is detectable over large-enough distances. For example, objects fall towards the centre of the Earth, and this means that two objects which begin falling parallel to each other at some height will get closer together as they head towards the ground. This is called a tidal effect. We’ll meet these effects, which in the context of black holes lead to spaghettification, later in the book.
**** 1g is an acceleration equal to that of an object falling in the vicinity of the Earth’s surface, i.e. 9.8 metres per second squared.
4
Warping Spacetime
Less than five months before his death in 1916, while serving in the German army calculating the trajectories of artillery shells on the eastern front, the eminent astrophysicist Karl Schwarzschild discovered the first exact solution to the equations of Einstein’s General Theory of Relativity. Schwarzschild’s achievement was no less remarkable for the fact that he derived the solution and sent it to Einstein just a few weeks after the theory had been published. Einstein was impressed, writing in return, ‘I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way.’ Schwarzschild had found the equation describing, to very high accuracy, the geometry of spacetime around a star. Recall John Wheeler’s maxim: ‘Spacetime tells matter how to move; matter tells spacetime how to curve.’ Schwarzschild’s solution describes the curve of spacetime, and it’s then a reasonably straightforward task to work out how things move over it. Today, Schwarzschild’s solution is one of the first things taught in an undergraduate course on general relativity and, in most circumstances, it corresponds to a tiny improvement over the simpler Newtonian predictions for planetary orbits. But not in all circumstances, because the Schwarzschild solution, unbeknown to him and to Einstein in 1916, also describes black holes.
What does Schwarzschild’s solution to Einstein’s equations look like? We caught a glimpse of the answer when we thought about the warped tabletop of flatland. Flat Albert and his flat mathematician friends discovered that the geometry of Euclid no longer applies because the table’s surface is warped. The angles of triangles do not quite add up to 180 degrees and distances between points will not be described by the familiar form of Pythagoras’ theorem. If Flat Albert wanted to calculate the distance between two places in warped flatland, he’d have to find a way of representing the warping mathematically.
To warm up, let’s forget spacetime for the moment and return to Earth. The Earth’s surface is curved, and this means we can’t simply use Pythagoras’ theorem to calculate the distance between widely spaced cities such as Buenos Aires and Beijing. We can easily appreciate this because we are three-dimensional beings, and we know what a sphere looks like. For example, if we were to procure a 19,267-kilometre-long ruler and place it in downtown Buenos Aires, its tip would not land in Beijing. The reason is that the ruler is flat and the Earth isn’t. The ruler would stick out into the third dimension – its tip would end up off the surface, out in space. We could, however, imagine purchasing 20 million one-metre rulers and laying them end to end along the great circle route between the two cities. The rulers could be made to follow the curved contour of the Earth’s surface (in reality we’d have to tunnel through any mountains we encountered to keep our chain of rulers at sea level, so we are imagining a perfectly smooth spherical Earth). Give or take a few metres, we could measure the distance over the curved surface of the Earth this way. If we wanted to do better, we could make each ruler one centimetre long or even smaller. Smaller rulers have the advantage that they better track the curving surface of the Earth.
The idea that we can build up a curved surface using lots of little flat pieces is nicely illustrated by the Montreal Biosphere, designed by Buckminster Fuller for the 1967 World’s Fair. When viewed from afar, the biosphere is a perfect-looking sphere, but close-up we can see it’s made up of lots of small flat triangles, ‘sewn together’ but slightly tilted with respect to each other. Any shape could be constructed in this way; the geometry is determined by how the flat pieces are assembled.
Figure 4.1. The Montreal Biosphere, constructed for ‘Expo 67’. (meunierd/Shutterstock)
In general relativity, curved spacetime can be built up in the same way. Lots of little pieces of flat spacetime can be sewn together to make curved spacetime, and Schwarzschild’s solution to Einstein’s equations describes how they are sewn together in the vicinity of a star. The distance in spacetime between events on each little flat piece could be determined by an arrangement of clocks and rulers, i.e. the interval (Δτ)2 = (Δt)2 – (Δx)2. We can get a very accurate description by choosing the spacetime patches to be sufficiently small that the flat space formula for the interval is a good approximation over each patch. This is just like saying the distance between two points on one of the little triangles on Buckminster Fuller’s biosphere can be determined using Pythagoras’ theorem, despite the fact that the distance between two points separated by many little triangles requires a more difficult calculation because the surface is curved.