The End of Time: The Next Revolution in Physics

This chapter is about how Einstein progressed from special relativity, which does not incorporate gravity, to general relativity, which does. Einstein believed that he was simultaneously incorporating Mach’s principle as its deepest foundation, but later, as I said, he changed his mind and left this topic in a great muddle. My view is that, nevertheless, without being aware of it, Einstein did incorporate the principle. This has important implications for time. We start with a bit more about Minkowski’s discoveries, which is necessary if we are to understand the way Einstein set about things.

One of the most important concepts in physics and geometry is distance, which is measured with rods. Distances can be measured in a space of any number of dimensions. You can measure them along a line or curve, on a flat or curved surface, or in space. In Part 2 we saw how an abstract ‘distance’, the action, can be introduced in multidimensional configuration spaces like Platonia. Minkowski showed that a remarkable kind of four-dimensional distance exists in space-time. Its existence is a consequence of the experimental facts that underlie special relativity. These things are most easily explained if we assume that space has just one dimension, not three; space-time then has two dimensions. Such a space-time is shown in Figure 27. We must first of all learn about past, present, and future in space-time.

One of the distinguished coordinate systems that exists in space-time is shown in Figure 27, in which the x axis is for space and the t axis for time, which increases upward. This is the Lorentz frame of Alice in Figure 25. Her world line is the vertical t axis. The units of time and distance are chosen to make the speed of light unity. Light pulses that pass through event O at t = 0 in opposite directions in space travel in space-time along the two lines marked future light cone. Their continuations backward (the light’s motion before it reaches O) define the past light cone.

Figure 27 Past and future light cones and the division of space-time in time-like and space-like regions, as described in the text.

Each event has a light cone, but only O’s is shown. Relativity differs from Newtonian theory mainly through the light cone and its associated distinguished speed c, which is a limiting speed for all processes. Light plays a distinguished role in relativity simply because it has that speed. No material object can travel at or faster than it. If a material object passes through O, its world line must lie somewhere inside the light cone, for example OA in Figure 27.

The light cone divides space-time into qualitatively different regions. An event like A can be reached from O by a material object travelling slower than light. Two such events are time-like with respect to each other. For two such events there exists a Lorentz frame in which they have the same space coordinates but different time coordinates. For the points O and A this frame is shown in the upper right of Figure 28.

Next we consider events like B and C in Figure 27, outside the light cone of O. They are space-like with respect to O. No material body can reach them from O, since to do so it would have to travel faster than light. For two events that are mutually space-like there exists a Lorentz frame in which they have the same time coordinate but different space coordinates. For two space-like events, it is impossible to say which is the earlier in any absolute sense. In some Lorentz frames one will be earlier than the other (thus O is earlier than both B and C in Alice’s frame in Figure 27), but in others the temporal order will be reversed.

Figure 28 Past, present and future in a space-time with two dimensions of space. The object that moves along OA (bottom left) is at rest in the starred frame (top right). Its world line is O*A (O and O* are the same event).

Finally, two events that can be connected by a light ray have a light-like relationship. All points on the light cone of event 0, for example the point F, are light-like with respect to O.