Julian Barbour - The End of Time - The Next Revolution in Physics

Historical accidents(p. 123) Poincaré’s paper can be found in his T he Value of Science , Chapter 2. Pais’s book is in the Bibliography.

Background to the Crisis(p. 124) The best (moderately technical) historical background to the relativity revolution that I know of is the book by Max Born. It is available in paperback.

The Forgotten Aspects of Time(p. 135) My claims about the topics that somehow escaped Einstein’s attention are spelled out in detail in Barbour (1999, forthcoming). I have tried to make good the gap in the literature on the theory of clocks and duration in Barbour (1994a).

Hermann Minkowski’s ideas have penetrated deep into the psyche of modern physicists. They find it hard to contemplate any alternative to his grand vision, presented in a famous lecture at Cologne on 21 September 1908. Its opening words, a magical incantation if ever there was one, are etched on their souls:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

The branch of knowledge that considers what exists is ontology. These three sentences changed the ontology of the world – for physicists at least.

For most physicists in the nineteenth century, space was the most fundamental thing. It persisted in time and constituted the deepest level in ontology. Space, in turn, was made up of points. They were the ground of being, conceived as identical, infinitesimal grains of sand close-packed in a block. Space was like glass. It was, of course, three-dimensional. However, alerted by Einstein’s work to how the relativity principle mixed up space and time, Minkowski commented that ‘Nobody has ever noticed a place except at a time, or a time except at a place.’ He had the idea that space and time belonged together in a far deeper sense than anyone had hitherto suspected. He fused them into space-time and called the points of this four-dimensional entity events . They became the new ground of being.

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.

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.

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).