Friday, October 5, 2012

Mandelbrot physics idea

This is a skeleton of an idea about a possible way to generalise the Mandelbrot set into one that obeys Einstein's field equations and so would represent a 'universe' with gravity and time.

The Mandelbrot set is the constraint Z = Z^2 + C, this can be thought of as its 'law of physics' and it is invariant to any conformal transformation. This is similar to the theory of relativity which defines the laws of physics as invariant to any 4x4 transformation, this is called general covariance. The reason we want general covariance is that it means the laws of physics aren't dependent on any coordinate frame and therefore they are in some sense universal laws. For the same reason the Mandelbrot set is said to be universal, and this is why it appears an infinite number of times in certain other fractals. By the way, it isn't necessary that our laws of physics be universal, but it is massively more likely that we find ourself under universal laws than not.

Equally it is massively more likely that our universe is on the limit set (the invariant set) of its laws than not. All fractals, like the Mandelbrot set are limit sets.

I wonder, if we replaced the Mandelbrot set with M = M*M + C where M and C are 4x4 matrices, then would we have a rule which is invariant to any 4x4 transform? My guess is that we would.
This set of 4x4 matrices could be thought of as the set of energy-stress tensors in Einstein's field equations, or equally the set of tensors representing the curvature of space (since this is proportional to the energy-stress tensor).

Notice that these curvature tensors don't have an associated position in space-time. This is actually what we want because the space-time in general relativity can me many different topologies and we shouldn't expect it to fit inside a single coordinate system.

So we interpret our set of curvatures to be correct and must build our space-time such that it contains only these curvatures. The constraints on how the curvatures should be distributed in space is defined by Einstein's relativity. In particular, there should be no torsion in the space-time, and the space-time should be such that the curvature tensors satisfy the special type of Ricci curvature as defined in Einstein's field equations.

I don't understand exactly how to implement this, but the basic idea is to start at a random curvature tensor in the set and build up the space-time geometry in the neighbourhood based on the subset of neighbouring tensors that you want to use to satisfy the field equations. Each curvature tensor in the set has 16 axes of neighbours, whereas we only need 4 axes of neighbours to generate space-time, so possibly this will help in building the space-time. Additionally, the set should already obey general covariance.
The idea is that you therefore only calculate and see the space-time geometry in your vicinity, no need for a global coordinate system and we naturally get a space-time which is specific to the observer.

You may be thinking that this particular universe will appear to be almost full with matter since every point will have a curvature of some sort. I don't think this is necessarily true, the algorithm to generate a compliant space-time is free to generate as much space and time as it likes, so it could well generate large areas of space between nearby curvature tensors, in order to satisfy the field equations.
Additionally, each curvature tensor can appear many times in many different spots of the generated space-time, since the 16 dimensional set allows the relatively thin 4d space-time to overlap and pass through itself from many different angles.


So, IF we could write an algorithm to generate compliant space-time to satisfy the set of curvature tensors, then our simple 4x4 Mandelbrot set may be a very interesting 'Einstein universe' to study. Not least because, like the Mandelbrot set, it literally comes from nothing; you start from a zero matrix and simply apply the rule to itself.
This idea also is perhaps a slightly different way of looking at the universe. Space-time is simply the result of interpreting a set of curvatures under a few constraints, it isn't fundamental but an interpretation.