Science of Discworld III

close to the vertical; particles that move very fast cover a lot of space in very little time, so their world-lines are nearly horizontal. In between, running diagonally, are the world-lines of particles that cover a given amount of space in the same amount of time – measured in the right units. Those units are chosen to correspond via the speed of light – say years for time and light-years for space. What covers one light-year of space in one year of time? Light, of course. So diagonal world-lines correspond to particles of light – photons – or anything else that can move at the same speed.
    Relativity forbids bodies that move faster than light. The world-lines that correspond to such bodies are called timelike curves, and the timelike curves passing through a given event form a cone, called its ‘light cone’. Actually, this is like two cones stuck together at their sharp tips, one pointing forward, the other backward. The forward-pointing cone contains the future of the event, all the points in spacetime that it could possibly influence. The backward-pointing cone contains its past, the events that could possibly influence it . Everything else is forbidden territory, elsewheres and elsewhens that have no possible causal connections to the chosen event.
    Minkowski spacetime is said to be ‘flat’ – it represents the motion of particles when no forces are acting on them. Forces change themotion, and the most important force is gravity. Einstein invented general relativity in order to incorporate gravity into special relativity. In Newtonian physics, gravity is a force: it pulls particles away from the straight lines that they would naturally follow if no force were acting. In general relativity, gravity is a geometric feature of the universe – a form of spacetime curvature.
    In Minkowski spacetime, points represent events, which have a location in both space and time. The ‘distance’ between two events must capture how far apart they are in space, and how far apart they are in time. It turns out that the way to do this is, roughly speaking, to take the distance between them in space and subtract the distance between them in time. This quantity is called the interval between the two events. If, instead, you did what seems obvious and added the time-distance to the space-distance, then space and time would be on exactly the same physical footing. However, there are clear differences: free motion in space is easy, but free motion in time is not. Subtracting the time-difference reflects this distinction; mathematically it amounts to considering time as imaginary space – space multiplied by the square root of minus one. And it has a remarkable effect: if a particle travels with the speed of light, then the interval between any two events along its world-line is zero.
    Think of a photon, a particle of light. It travels, of course, at the speed of light. As one year of time passes, it travels one light-year. The sum of 1 and 1 is 2, but that’s not how you get the interval. The interval is the difference 1 – 1, which is 0. So the interval is related to the apparent rate of passage of time for a moving observer. The faster an object moves, the slower time on it appears to pass. This effect is called time dilation . As you travel closer and closer to the speed of light, the passage of time, as you experience it, slows down. If you could travel at the speed of light, time would be frozen. No time passes on a photon.
    In Newtonian physics, particles that move when no forces are acting follow straight lines. Straight lines minimise the distance betweenpoints. In relativistic physics, freely moving particles minimise the interval, and follow geodesics . Finally, gravity is incorporated, not as an extra force, but as a distortion of the structure of spacetime, which changes the size of the interval and alters the shapes of geodesics. This variable interval between nearby events is called the metric of spacetime.
    The usual image is to say that spacetime becomes ‘curved’, though this term is easily misinterpreted. In particular, it doesn’t have to be curved round anything else. The curvature is interpreted physically as the force of gravity, and it causes light cones to deform.
    One result is ‘gravitational lensing’, the bending of light by massive objects, which Einstein discovered in 1911 and published in 1915. He predicted that gravity should bend light by twice the amount that Newton’s Laws