We’ve sketched this patchwork view of spacetime in Figure 4.2. If we were five-dimensional beings with an innate sense of hyperbolic geometry, we would be able to visualise how the little pieces sew together to make a ‘surface’ curved into a fifth dimension. Good luck with that, but the basic idea is quite simple. We are to think of curved spacetime as being tiled by lots of little flat pieces of spacetime, each slightly tilted with respect to their neighbours and adorned with their own grids of clocks and rulers. The challenge in general relativity is to specify how the pieces are sewn together. If we know that, we can calculate the interval between widely separated events on the curved surface by adding up all the intervals on each patch, just as we laid down the little rulers to measure the distance between Buenos Aires and Beijing.
Figure 4.2. Building up spacetime by sewing together a patchwork of tiny regions each of which is flat.
The idea that curved spacetime is well approximated by flat spacetime over sufficiently small distances and intervals of time, just as the Earth is flat over sufficiently small distances, is precisely what Einstein had in mind when he had his happiest thought:
‘At that moment there came to me the happiest thought of my life … for an observer falling freely from the roof of a house no gravitational field exists during his fall – at least not in his immediate vicinity. That is, if the observer releases any objects, they remain in a state of rest or uniform motion relative to him, respectively, independent of their unique chemical or physical nature. Therefore, the observer is entitled to interpret his state as that of rest.’
This quote is wonderful because it illuminates Einstein’s thinking. He didn’t think mathematically, at least initially. He thought in simple pictures and asked simple questions. What does the fact that gravity can be removed by falling tell me? If gravity can’t be detected in a freely falling observer’s immediate vicinity, spacetime must be flat in their immediate vicinity. Don’t get confused about what it feels like to actually fall off a roof by the way – we’re considering an idealised fall in a vacuum and ignoring air resistance. Simplify the problem down to its essence. All cows are spherical to a theoretical physicist, which is why they are clear thinkers but shit farmers. Gravity is a strange force because it can be removed by falling. Einstein’s genius was to see the connection between this idea and a geometric picture of gravity as curved spacetime. Gravity appears not because there is a fundamental force of attraction between things, as we learn at school, but because small patches of spacetime are tilted relative to their neighbours in the vicinity of massive objects.
If there is no force of gravity, why does a person fall off a roof and hit the ground, or the Moon orbit the Earth? The answer is that the person and the Moon are both following straight lines over curved spacetime. We can be more specific if we recall the Twin Paradox in the previous chapter. There, we encountered the Principle of Maximum Ageing. An astronaut that does not accelerate takes a path over spacetime between two events that maximises the time they measure on their wristwatch between those events. In general relativity, the Principle of Maximum Ageing is placed centre stage as a fundamental law of Nature that determines a freely falling object’s worldline over curved spacetime. As Einstein says in his quote, an observer in freefall ‘is entitled to interpret his state as that of rest’. This implies that the path a freely falling object takes over curved spacetime must be the path that maximises the time on a wristwatch carried by the object. On each little flat patch, this path will be a straight line across the patch, but in curved spacetime the patches sew together to make a curve. The result is entirely analogous to the case of the little flat rulers on Earth. The straight lines must match up with each other end-to-end, but the resulting path is curved. The result in spacetime is what we see as an orbit – the paths of the planets around the Sun. Or, for that matter, the fall as someone slips unfortunately off a roof. In a way, the path the unfortunate faller takes on their way to the ground is entirely logical – they are maximising the time they have left.
Schwarzschild’s solution for the curvature of spacetime, when paired with the Principle of Maximum Ageing, is all we need to calculate the worldlines of anything falling in the vicinity of a planet, star or black hole.
The opportunity to acquire a deeper understanding of general relativity and Schwarzschild’s solution lies within our grasp, and it would be a shame not to go all the way when we’ve come this far. So, over the next few pages there is a little more mathematics than in the rest of the book. There is nothing much more complicated than Pythagoras’ theorem, but if you really don’t like mathematics then don’t worry; normal diagrammatic service will be resumed shortly.
The metric: calculating distances on curved surfaces
By 1908, Einstein had the basic idea of gravity as curved spacetime, but it took him a further seven years to achieve its mathematical realisation in the form of general relativity. His challenge was to find a way of calculating the distance between two events if spacetime is curved. When asked why it took so long, here is what he said: ‘The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning.’17
To understand what Einstein meant, we’ll leave spacetime for a moment and return to two-dimensional Euclidean geometry. Let’s choose two points A and B and draw a straight line between them, as shown in the left-hand picture in Figure 4.3. The line will have a length that we could measure with a ruler. Call that length Δz. The line of length Δz is also the hypotenuse of a right-angled triangle with sides of length Δx and Δy. Pythagoras’ theorem relates these three lengths:
(Δz)2 = (Δx)2 + (Δy)2
In this equation, all the quantities are distances that can be measured by rulers. They also happen to be the difference in coordinates using the grid that you can see in the background. Specifically, A is at x = 3 and y = 2, which we write (3,2), and B is at (9,7), so Δx = 9 – 3 = 6 and Δy = 7 – 2 = 5. Using Pythagoras gives us Δz = √61.
Now look at the right-hand picture in Figure 4.3. It is the same pair of points, A and B, but now with a different grid. The new grid is perfectly good for labelling points: A is at (5,3) and B is at (7,9). But the differences in these coordinates cannot be used in Pythagoras’ theorem. This is a nice illustration of the arbitrariness of grids. Any grid will do for labelling points, but some grids are more useful than others. Here, the square grid makes it easier to calculate the length from A to B. On a flat surface, we can always choose a rectangular grid to make life easier, but on a curved surface, it is impossible to pick a single grid such that the Pythagorean equation works for the distance between every pair of points. The distinction between the coordinate grid as a mesh for labelling points and as a device to help calculate distances between points is what Einstein was referring to when he said it is ‘not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning’. We need to follow his words of warning and not get too attached to the grids we lay down over spacetime.