Figure 4.3. Left: The distance between A and B is related to the coordinates of A and B via Pythagoras’ theorem. Right: The same two points, A and B, can be located using a wavy grid but the distance between them is not related to the coordinates via Pythagoras’ theorem.
What is true is that there is always a way to compute distances using any grid. It’s just that the formula is not the one due to Pythagoras. If X and Y are the coordinates labelling the wavy grid on the right of Figure 4.3 then, for any two points that are sufficiently close together the distance between them can always be written:
(dz)2 = a(dX)2 + b(dY)2 + c(dX)(dY)
where a, b and c are numbers that vary from place to place on the grid. We changed the notation and wrote dz instead of Δz. The two quantities have the same meaning – the difference between two coordinates – but we are going to reserve using the d’s for the special case in which these distances are small. This formula is true for any grid and a similar formula can be written down in more than two dimensions. For any curved surface, the set of numbers like a, b and c provide the rule for how to compute distances. Collectively, that set of numbers (like a, b and c) is called the ‘metric’ for the surface. Once we know the metric for our chosen coordinate grid, we can compute distances. A big part of general relativity is deriving the metric for a particular situation. This is what Schwarzschild did for the spacetime around a (non-rotating) star.
The Schwarzschild solution
We can now step back into spacetime and return to general relativity. Schwarzschild’s solution tells us the metric in the vicinity of any spherically symmetric distribution of matter like a star or black hole. Using the spacetime grid that Schwarzschild used (more on that in a moment), the corresponding interval (distance) between two nearby events outside of a star or black hole is:
As in Chapter 1, the Schwarzschild radius is given by the formula:
where G is Newton’s gravitational constant, M is the mass of the star and c is the speed of light. Pretty much everything that we want to know about non-spinning black holes in general relativity is contained within this one line of mathematics. We see that Schwarzschild chose a grid labelled with a time coordinate, t, and a distance coordinate R. Ignore the dΩ2 term for now, it won’t be important and we’ll explain why in a moment.
Because the spacetime is curved, it is not possible to find a single coordinate grid such that the flat (Minkowski) formula for the interval holds everywhere. That’s the reason for the factors in front of dt2 and dR2. Conceptually, these factors are no different from those we had to introduce in the simpler two-dimensional case above: they encode the information about the curvature. The formula is telling us that the warping of spacetime at a particular location depends on how close that location is to the star and how massive the star is. The t and R coordinate grid chosen by Schwarzschild doesn’t have to correspond directly to anything that can be measured using clocks or rulers. However, the R and t coordinates do have a physical interpretation, which will allow us to develop an intuitive picture of Schwarzschild’s spacetime.
We’ve sketched what we might term a ‘space diagram’ of Schwarzschild’s spacetime in Figure 4.4. The star sits at what we’ll call the ‘centre of attraction’. We’ve drawn two shells surrounding the star. Each shell is at a fixed Schwarzschild coordinate R from the centre of attraction (at R = 0). The R coordinate is defined in terms of the surface area of these shells. In flat space, the surface area of a sphere A = 4πR2, and R is the distance to the centre of the sphere as measured by a ruler. In the distorted space around a star (or black hole) this is no longer true (for a black hole it’s not even possible to lay a ruler down starting from the centre of attraction – the singularity). We can, however, always measure the area of spherical shells like those on the diagram and R is the radius a shell would have had if spacetime were flat. That’s how Schwarzschild chose this coordinate.
No matter how the spacetime is distorted, the distortion must be the same at every point on these spherical shells. This is because Schwarzschild assumed perfect spherical symmetry when he derived his equation. Think of a perfect sphere; every point on the surface is the same as every other point. The dΩ2 piece in the equation deals with the distance between events on a particular shell. It is precisely the same as the piece we find in the metric used to calculate distances on the surface of the (spherical) Earth. We’ve discussed this in a bit more detail in Box 4.1. We would need this piece if we wanted to calculate the details of orbits around a star or black hole, but in what follows we’ll always consider things moving only inwards or outwards. This will simplify matters while still capturing the important physics.
The Schwarzschild time coordinate also has a simple definition: t corresponds to the time as measured on a clock far away from the centre of attraction, where spacetime is almost flat. As we move inwards towards the star, spacetime becomes more curved and that’s why we need the factors in front of dR2 and dt2. These factors are more important as we move inwards and are close to 1 far away. This makes sense because it means that far away from the star the interval is the same as it is in flat space.
Figure 4.4. Schwarzschild ‘space diagram’. The star sits at the centre. Two imaginary spherical shells surround the star.
The fact that Schwarzschild’s coordinates have a simple interpretation far from the star allows us to understand what the curvature of spacetime means for the passage of time and the measurement of lengths closer in. Let’s imagine we have access to a small laboratory that we can position anywhere in spacetime. Our laboratory has no rockets attached and is therefore in freefall. Inside the laboratory there is a watch to measure the passage of time and a ruler to measure distance. Our lab is small in both space and time, which means we can assume spacetime is flat inside the lab. Let’s now locate the laboratory in the vicinity of the outer shell of Figure 4.4 and observe the watch ticking. If the ticks are so short that our laboratory stays at roughly the same R coordinate for the duration of the tick, Schwarzschild’s equation tells us that the spacetime interval between ticks is:
We’ve used an ‘approximately equals’ sign to emphasise that we’re making an approximation. In this case, dR ≈ 0 between the ticks so we can ignore the dR piece of Schwarzschild’s equation.
This equation is the origin of our claim in Chapter 1 that time passes more slowly for an astronaut when they are close to a star or black hole. dt2 is the time interval (squared) we would measure between the ticks of our laboratory watch as measured by a clock at rest far away from the star, where spacetime is not curved. The (1 – RS/R) factor corrects for the fact that this does not correspond to time in our laboratory, as measured by the laboratory watch. The curvature has distorted time and so we need more ticks of the distant clock for one tick of the laboratory watch. Time has slowed down at the location of the laboratory relative to far away. On the inner shell in Figure 4.4, R is smaller still. If we place our laboratory there, the number in front of dt2 will be even smaller, and therefore watches on the inner shell will run even slower.