It is just that space-time is not constructed from a unique set. The analogy with a pack of cards is again quite apt. Newtonian space-time is an ordinary pack; Minkowski space-time is a magical pack. Look at it one way, and cards run through the block with one inclination. Look at it another way, and different cards run with a different inclination. But whichever way you look, cards are there.
CHAPTER 10
The Discovery of General Relativity
FUNNY GEOMETRY
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.
These three basic relationships between events – being time-like, spacelike or light-like – are the same in all Lorentz frames. This is because the three types are determined by the light cones, which are real features in space-time, just as rivers are real features of a continent. In contrast, the coordinate axes are like lines ‘painted’ on space-time – they are no more real than the grid lines on a map. Moreover, in a change from one frame to another, the coordinate axes never cross the light cones. The time axis moves but stays within the light cone, while the space axes stay within the ‘present’ as defined above. This is illustrated in Figure 28 for space with two dimensions, which shows how the light cone gets its name. It also highlights the great difference between the Newtonian and Einsteinian worlds. In the former, past, present and future are defined throughout the universe, and the present is a single simultaneity hyperplane. In the latter, they are defined separately for each event in space-time, and the present is much larger.
Now we can talk about distance. In ordinary space it is always positive. The distance relationships are reflected in Pythagoras’ theorem: the square of the hypotenuse in any right-angled triangle is equal to the sum of the squares of the other two sides: H2 = A2+B2.
Minkowski was led to introduce a ‘distance’ in space-time by noting a curious fact. For observers who use the xy frame in Figures 27 and 28, event A is separated from O by the space-like interval EA and by the time-like interval DA. For observers who use the starred frame, however, O and A are at the same space point and are merely separated by the time-like interval OA. The xy observers measure EA with a rod and DA with a clock, obtaining results X and T, respectively. With their clock, observers in the starred frame can measure only the time-like interval OA. Now, their clock runs at a different rate to the xy clock, so they will find that OA is not T but Tstarred. Using Einstein’s results, Minkowski found that (Tstarred)2 = T2−X2. This is just like Pythagoras’ theorem, except for the minus sign.
There are several important things about this result. Einstein had shown that observers moving relative to each other would not agree about distances and times between pairs of events. However, Minkowski found something on which they will always agree. Measurements of the space-like separation (by a rod) and the time-like separation (by a clock) of the same two events O and A can be made by observers moving at any speed. They will all disagree about the results of the separate measurements, but they will all find the same value for the square of the time-like separation minus the square of the space-like separation. It will always be equal to the square of the time-like separation, called the proper time, of the unique observer for whom O and A are at the same space position. This result created a sensation. Space and time, like rods and clocks, seem to have completely different natures, but Einstein and Minkowski showed that they are inseparably linked.