The End of Time: The Next Revolution in Physics

Such atoms of existence – the constituent events of space-time – are very different from the entities that I suggested in Part 2 as the true atoms of existence. The main aim of Part 3 is to show that space-time can be conceived of in two ways – as a collection of events, but also as an assemblage of extended configurations put together by the principle of best matching and the introduction of a ‘time spacing’ through a distinguished simplifier, as explained for the Newtonian case at the end of Chapter 7. However, reflecting the relativity of simultaneity, the assemblage has an additional remarkable property that gives rise to the main dilemma we face in trying to establish the true nature of time.

FROM THREE TO FOUR DIMENSIONS

In itself, the fusion of space and time was not such a radical step. It can be done for Newtonian space and time. To picture this, we must suppose that ordinary space has only two dimensions and not three. We can then imagine space as a blank card, and the bodies in space as marks on it. Any relative arrangement of these marks defines an instant of time.

The solution of Tait’s problem showed how relative configurations can, if their bodies obey Newton’s laws, be placed in absolute space at their positions at corresponding absolute times. If space is pictured as two-dimensional, absolute space is modelled not by a room but by a flat surface. The solution of Tait’s problem places each card on the surface in positions determined by the marks on the cards. In these positions, in which the centre of mass can be fixed at one point, any body moving inertially moves along a straight line on the surface.

Keeping all the cards horizontal (parallel to the surface), we can put a vertical spacing between them which is proportional to the amount of absolute time between them. This is like imagining the amount of time between 11 o’clock and 12 o’clock as a distance, and is a very convenient way of visualizing things. The resulting structure can be called Newtonian space-time. The one dimension of time has been put together with the two of space. Newton’s laws can be expressed very beautifully in this three-dimensional structure, which is a kind of block. Whatever motion a body has, it must follow some path in this block. Minkowski called this path its world line. If the body does not move in space, which is a special case of inertial motion, its world line is vertically upwards. If it is moving inertially with some velocity, then it has a straight world line which is inclined to the vertical. The faster the motion, the larger the angle with the vertical.

In reality, ordinary space has three dimensions and Newtonian space-time four. Instead of cards placed at vertical positions representing different times, or simultaneity levels, we must imagine three-dimensional spaces fused into a four-dimensional block. This is impossible to visualize, but the model with only two space dimensions is a good substitute.

Newtonian space-time differs in an important respect from space, in which all directions are on an equal footing and none is distinguished from any other. In Newtonian space-time, one direction is singled out. This is reflected in its representation as a pack of cards. Directions that lie in a card, in a simultaneity level, are quite different from the time line that runs vertically through the cards. Newtonian space-time is ‘laminated’. If you were to ‘cut through it’ at an angle, the ‘lamination’ would be revealed. You would be ‘cutting through’ the simultaneity levels. The inequivalence of directions can be expressed in the language of coordinates.

Just as you can put a coordinate grid on a two-dimensional map, you can ‘paint’ a rectangular grid on Newtonian space-time with one of the axes perpendicular (parallel to the time line). The laws of motion can be formulated in terms of the grid. For example, bodies moving inertially travel along lines that are straight relative to the grid. You can then ‘move’ the grid around as a complete unit into different positions in space-time and see if the motions relative to the new grid satisfy the same laws as they did in the old. For Newton’s laws there is considerable but not complete freedom to move the grid. Provided it is maintained vertical, it can be shifted and rotated in ordinary space, just like a child’s climbing frame, and it can also be raised and lowered in the vertical time direction. However, tilting the vertical axis is not allowed. Newtonian forces (in gravity and electrostatics, for example) are transmitted instantaneously – horizontally in the model. If you tilt the grid from the original time axis, you leave the old simultaneity levels. The forces are not transmitted through the new levels.