The End of Time: The Next Revolution in Physics

Minkowski’s real discovery was that, in an analogous construction using Maxwell’s electromagnetic equations instead of Newton’s laws, the resulting space-time structure, now called Minkowski space-time, has no special ‘lamination’. It is more like a loaf of bread, through which you can slice in any way. The cut surface always looks the same. The way this shows up in changes of the coordinate grid is especially striking. Time becomes very like space but not quite identical.

The difference can be illustrated by the climbing frame. Here too a vertically held frame can be shifted, rotated and raised or lowered as a rigid unit. Maxwell’s laws still take the same form with respect to the displaced grid. But you can also tilt it from the vertical provided you do something else as well. For this, you need an ‘articulated’ grid, which we have in fact already encountered, in Figure 25 in the discussion of simultaneity. It is a typical example of the space-time diagrams that Minkowski introduced. Figure 26, with its remarkable demonstration that two families of observers moving relative to each other each see the rods of the others as contracted relative to their own, is one of Minkowski’s actual diagrams, slightly modified (merely to conform with the context of this book – the physical content remains unchanged).

In Figure 25, the original grid is ‘painted’ onto space-time with the dashes, while the dots show an alternative. As we saw, the law of nature that describes the behaviour of light pulses allows them to travel along the diagonals of either grid. A transformation of this law from one coordinate grid to another is called a Lorentz transformation, and the grids themselves are called Lorentz frames. I have already mentioned that you should not think in terms of there being one rectangular coordinate grid, and all the others oblique. Alice thinks Alice* has an oblique system compared to her, but Alice* thinks the same about Alice’s system. This is a consequence of the relativity principle, and a special property of space-time that we shall come to shortly. Minkowski pointed out that the transformation shown in Figure 25 is a kind of rotation in four dimensions. The possibility of making rotations in ordinary space is a deep reflection of its unitary, block-like nature. Minkowski saw the possibility of making a kind of rotation in space-time, which is impossible in Newtonian space-time, as the clearest evidence for the intimate fusion of space and time, even though the need for ‘articulation’ showed that time was still somewhat different in nature from space.

Einstein, Minkowski and others were able to show that all the laws of nature known in their time (except initially for gravitation) either already had a form that was exactly the same in all Lorentz frames or could be relatively easily modified so that they did. Even though the modifications were relatively easy once the idea was clear, their implications, including Einstein’s famous equation E = mc² (a prediction at that time), were mostly very startling. Minkowski, like Einstein and Poincaré, made a strong prediction that all laws of nature found in future would accord with the relativity principle, and emphasized that the guiding principle for finding such laws was to treat time exactly as if it were space.

Except for the intermingling of space and time and the distinguished role played by light, Minkowski’s space-time strongly resembles Newtonian space-time. Matter neither creates nor changes its rigid and absolute structure. It is like a football field, complete with markings, on which the players must abide by rules they cannot change.

ARE THERE NOWS IN RELATIVITY?

It is often said that relativity destroyed the concept of Now. In Newtonian physics the axes can never be tilted as they are in Figure 25. The simultaneity levels stay level, and there is a unique sequence of instants of time, each of which applies to the complete universe. This is overthrown in relativity, where each event belongs to a multitude of Nows. This has important implications for the way we think about past, present and future.

Even in Newtonian theory we can picture world history laid out before us. In this ‘God’s-eye’ view, the instants of time are all ‘there’ simultaneously. The alternative idea of a ‘moving present’ passing through the instants from the past to the future is theoretically possible but impossible to verify. It adds nothing to the scientific notion of time. Special relativity makes a ‘moving present’ pretty well untenable, even as a logical possibility.