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# Relativity

In mathematics one makes the distinction between a manifold and a metric. A manifold is a topological structure of points. A metric is a distance function that describes how far apart any two points are. One naturally thinks of a metric as being solely determined by the manifold. That is one thinks that two points have a fixed distance between them no matter how the universe changes in other ways.

Special relativity shows that this is not true. The distance we measure between two points is affected by our motion relative to those points. Special relativity suggests, but does not require, that there is no manifold of points or absolute frame of reference. There are only objects and relationships between those objects. This seems strange and unintuitive but it is certainly possible in a continuous universe. In a fully discrete universe it is not possible to do away with the underlying manifold and special relativity can only be approximately true in such a universe.

One way special relativity can come about is if all physical effects are electromagnetic. This is impossible in the exiting theory because electromagnetic waves travel at the speed of light. If the theory had a small element of the correct sort of nonlinearity one could create stable soliton wave structures. Time and distance within such a structure is affected by the motion of the structure. Both the linear motion of the system and any internal dynamics are described by the same partial differential equation. For example if the structure were moving at the velocity of light it could have no internal dynamics. The only dynamics would be the motion of the object in space. In effect time would stop on the object.

The faster an object moves the slower its internal dynamics relative to an object at rest.

General relativity extends so that the distance we measure is also affected by the gravitational field.

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