The End of Time: The Next Revolution in Physics

If you go to any point in a real landscape, you get a view. Except for special and artificial landscapes, the view is different from each point. If you wanted to meet someone, you could give them a snapshot taken from your preferred meeting point. Your friend could then identify it. Thus, points in a real country can be identified by pictures. In a somewhat similar way, I should like you to imagine Triangle Land. Each point in Triangle Land stands for a triangle, which is a real thing you can see or imagine. However, whereas you view a landscape by standing at a point and looking around you, Triangle Land is more like a surface that seems featureless until you touch a point on it. When you do this, a picture lights up on a screen in front of you. Each point you touch gives a different picture. In Triangle Land, which is actually three-dimensional, the pictures you see are triangles. A convenient way of representing Triangle Land is portrayed in Figures 3 and 4.

I have gone to some trouble to describe Triangle Land because it can be used to model the totality of possible Nows. Like real countries, and unlike absolute space, which extends to infinity in all directions, it has frontiers. There are the sheets, ribs and apex of Figure 4. They are there by logical necessity. If Nows were as simple as triangles, the pyramid in Figure 4 could be seen as a model of eternity, for one notion of eternity is surely that it is simply all the Nows that can be, laid out before us so that we can survey them all.

Figure 3 The seven triangles represent several possible arrangements of a model universe of three particles A, B, C. Each triangle is a possible Now. Each Now is associated with a point (black diamond) in the ‘room’ formed by the three grid axes AB, BC, CA, which meet at the corner of the ‘room’ farthest from you. The black diamond that represents a given triangle ABC is situated where the distance to the ‘floor’ is the length of the side AB (measured along the vertical axis), and the distances to the two ‘walls’ are equal to the other two sides, BC and CA. The dash-dotted lines show the grid coordinates. In this way, each model Now is associated with a unique point in the ‘room’. As explained in the text, if you ‘touched’ one of the black diamonds, the corresponding triangle would light up. However, not every point in the ‘room’ corresponds to a possible triangle – see Figure 4.

A three-particle model universe is, of course, unrealistic, but it conveys the idea. In a universe of four particles, the Nows are tetrahedrons. Whatever the number of particles, they form some structure, a configuration. Plastic balls joined by struts to form a rigid structure are often used to model molecules, including macromolecules such as DNA, which are ‘megamolecules’. You can move such a structure around without changing its shape. For any chosen number of balls, many different structures can be formed. That is how I should like you to think about the instants of time. Each Now is a structure.

Figure 4 This shows the same ‘room’ and axes as in Figure 3, but without the walls shaded. Something more important is illustrated here. In any triangle, no one side can be longer than the sum of the other two. Therefore, points in the ‘room’ in Figure 3 for which one coordinate is larger than the sum of the other two do not correspond to possible triangles. All triangles must have coordinates inside the ‘sheets’ spanned between the three ‘ribs’ that run (towards you) at 45° between the three pairs of axes AB, BC (up to the left), AB, CA (up to the right) and BC, CA (along the ‘floor’, almost towards you). Points outside the sheets do not correspond to possible triangles. However, points on the sheets, the ribs and the apex of the pyramid formed by them correspond to special triangles. If vertex A in the thin triangle at the bottom right of Figure 3 is moved until it lies on BC, the triangle becomes a line, which is still just a triangle, because BC is now equal to (but not greater than) the sum of CA and AB. Such a triangle is represented by a point on one of the ‘sheets’ in Figure 4. If point A is then moved, say, towards 8, the point representing the corresponding triangle in Figure 4 moves along the ‘sheet’ to the corresponding ‘rib’, which represents the even more special ‘triangles’ for which two points coincide. Finally, the apex, where the three ribs meet in the far corner of the ‘room’, corresponds to the unique and most special case in which all three particles coincide. Thus, Triangle Land has a ‘shape’ which arises from the rules that triangles must satisfy. The unique point at which the three particles coincide I call Alpha.