Thursday, July 26, 2012

Relativistic automata continued

Previously I covered some ideas about fractal automata and extending it to a kind of relativistic version. I'll summarise this then describe some thoughts on extending the system to be more relativistic.

translation

Starting with simple cellular automata, like conway's game of life, we generalise it by saying that the function acting on each cell (which chooses whether to turn the cell on or off) can be any function of n nearest neighbours. Typically we pick some small set, like the 9 neighbours in a 2d grid (including the centre one). Giving 2^n possible neighbour patterns and 2^2^n possible functions. The choice of function isn't too important here, what is interesting is the type of results you get by searching through the set of functions as applied to initially random data.

scale

We can expand this system to fractal cellular automata by using a grid tree rather than a single grid, the nearest neighbours for each cell include those on the parent and child grids as shown here: The coloured cells being example nearest neighbours of the red cell. Typically you watch the highest detail grid evolve. This system effectively gives you a continuous euclidean space, which is more physically real than a discrete grid.

non-uniform scale

These grids can be 3d too, or even 4d. But to deal with time properly we need to use a Minkowski 4d space rather than a euclidean grid. It can be approximated by making the time axis the long diagonal of the 4d grid and allowing non-uniform scales, i.e. cuboid grids. So just as scale becomes like an extra degree of freedom in fractal automata, we let scale in x, y and z separately become three extra degrees of freedom. As discussed in my previous post. The non-uniform scale allows the same rules to apply at different velocities, which are represented by the different non-uniform scales. At this point there are actually two ways that you can work with time here, you can make the cellular automata only depend on cells that are further back in time, which means that the automata spreads forwards along the time direction:
Or you can keep the cells dependent on all the neighbours (both forwards and backwards in the time direction). This means that a whole span of time evolves as you iterate the automata. This second method has three possible outcomes:
1. The evolving automata converges on a fixed result. This can then be 'played' as a single time line.
2. The automata ends up in a cycle, so no single time line exists, but you could consider each result as being parallel time lines.
3. The automata doesn't settle into a cycle (so is a strange attractor). This has some similarities with the many-universe interpretation of quantum theory, since there are infinite time lines, describable by a probability density function. So any of the time lines is a valid result, after it has converged.

symmetric sheer

The next step is to allow a diagonal stretch of the grid. This corresponds to information which changes more or less slowly with time:
This feature is part of general relativity, time is more dense close to high mass objects. This change in the 'speed of the physics' is what causes objects to fall under gravity. It hasn't been tested, but the idea is that moving points will deflect into the higher density areas, causing something vaguely like gravitation.
This extension has some problems which require a further addition to solve; the stretched grids have a different apparent maximum velocity, inconsistent with the real world.

I have headed the sections in this post to show the different symmetric transforms that each algorithm adds to the system. Continuing this theme and generalising, a square cell can be transformed by any transformation matrix. So there are a few remaining degrees of freedom that can be added as extra symmetries:

rotation

If the path of massive objects is allowed to rotate with time (as opposed to a boost, which is the non-uniform scale above) then the object will be able to attain just as high speed as a light object, which is physically real. However it doesn't appear to obey special relativity since the rotation could make any object go faster than some maximum (light) speed.
I think the way this is avoided is that the frame that we visualise is the square (not uniformly scaled) frame, and the frame of heavier objects is 'ghosted' onto this square frame, just as low resolution objects 'ghost' onto the high resolution mesh that we view. The ghosting process prevents the object going faster than light since it can't ghost beyond this speed.
This is again a bit speculative, but it seems that the movement of a high mass (stretched) grid can only change direction over a longer period of time, in other words its acceleration should be less in general, and would be symmetric to higher acceleration in a lower mass object.

non-symmetric sheer

The final degrees of freedom are non-symmetric sheering of the cell. This relates to symmetric sheer much as pressure relates to mass or energy in general relativity I think. So for a 4d space-time, the general transform would be a 4x5 matrix, giving 20 dimensions to the cellular automata.

dealing with scale problems

The problem with using symmetric scale for mass is that the stretch only jumps by some fixed factor, like two.
Similarly, the problem with using rotation with time to give a mass a velocity is that grid-based cellular automata can only rotate by 90 degrees.
The solution to such problems might be to treat the scales (which are the rows of the 4x4 matrix) the same way as position is treated (the 5th row), i.e. use the overall scale to allow each row to be continuous.
I'm not sure how well this idea works, so this problem leads to a completely different formulation that I will discuss in the next post.

Summary

Space-time is usually thought of as a 4d continuum of points, something which could be naively modelled as a 4d grid cellular automata.
Instead, if we think of space-time as built of 4d volumes then the description of such volumes which is independent of any reference frame is one which is symmetric to any 4x5 transform. This is quite similar to the principles of special and general relativity. Such a 20 dimensional automata could potentially have the following extra realisms over basic cellular automata:

1. It is continuous
2. parts can have velocity
3. nothing goes faster than light, time dilation etc of special relativity
4. many-world timelines, consistent with some quantum effects
5. mass affecting how fast something accelerates
6. gravitational wells (general relativity)
7. attraction based on pressure (also part of general relativity)