The goal is to introduce the class of “relativistically coherent” models - the ones that can viably be transformed from one reference frame to another. An example of implementation of moving particles will be introduced, and special relativity will be examined in the models in question.
Introduction
Introduction
Whether you’re sitting down or walking while reading this post, it should be the same post. The substance of what happens in the Universe should not depend on the reference frame. The same should hold true for Wolfram Models. You should not have a “privileged” reference frame, a particle that is standing in one should be moving in the other.Let us recall what the reference frame corresponds to all the way from the Technical Introduction:On the causal graph, we can introduce a foliation to turn the graph of update events into a sequence of states. A standing observer’s foliation is horizontal, like this:If we have a moving observer, that just corresponds to a tilted foliation:But, if we want our initial system not to be the special reference frame, we need to be able to produce a system with the evolution from the new reference frame directly:It is not at all a given that a Wolfram model will obey this property. It is especially non-obvious how to have subluminal particles in this formalism.Here is an example of a system that is not relativistically coherent - see how the value of the clock at the time of the collision changes when we pass to another reference frame, since we skip some clock ticks when taking the slice:
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startPos1D:=makeParticle[makeClock[generateSimple1DSpace[10],5],2,3]moveRule:={{a,b},{a,a,b},{b,c}}->{{a,b},{b,c},{b,b,c}}moveRuleND:={{a,a,b},{a,b,c}}->{{b,b,c},{a,b,c}}tickRule:={{a}}->{{a},{a}}GraphicsRow[{GraphicsColumn@Join[{"Original system"},WolframModel[{moveRule,tickRule},startPos1D,3,"StatesPlotsList"],{"N=16 at collision"}],WolframModel[{moveRule,tickRule},startPos1D,3,"CausalGraph"],GraphicsColumn@Join[{"After taking a slice"},WolframModel[{moveRule,tickRule},makeParticle[makeClock[generateSimple1DSpace[10],5],3,4],3,"StatesPlotsList"],{"N=4 at collision"}]}]
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In this project, I introduce a definition of the property called relativistic coherence, a test for it, and some examples of what particles should look like.
To gather more intuition, we shall start by attempting to construct some particles. After that, we will properly introduce relativistic coherence, and test some systems for it.
To gather more intuition, we shall start by attempting to construct some particles. After that, we will properly introduce relativistic coherence, and test some systems for it.
Initialization code
Initialization code
The basic particles and spaces
The basic particles and spaces
Here is what toy models for 1- and 2-dimensional spaces could look like:The 2D example doesn’t have the right symmetries, it is just a toy example that has the main properties for our usecase. For more information on what space could look like, check out my previous project: What should space look like in the Wolfram model?Here, there is a sample naive particle in both cases. It has a tip that indicates the direction of movement, and we have a rule to move it:for 1 dimension or for the higher-dimensional examples.However, no matter what we do to this particle, it will always move 1 step in space after a time step of our evolution:
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^Code for generating the spaces
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If we take any system with a particle like this, for example by having a foliation, it will always contain a particle moving 1 step per 1 step of evolution. There is no reason for panic: this is simply a photon. We know from Special Relativity that something does not change its speed after a Lorentz boost if and only if it moves with the speed of light.
However, it would be nice to have something else in our system. Let’s introduce a proper definition of relativistic coherence, and check some systems for it.
However, it would be nice to have something else in our system. Let’s introduce a proper definition of relativistic coherence, and check some systems for it.
Relativistic Coherence
Relativistic Coherence
Let’s quickly define two basic things for the Wolfram Model:
1. The model itself takes the initial state (e.g., a hypergraph) and the rules, and returns the causal graph:
2. Given a causal graph, we can obtain a state (a hypergraph) using a slice:
A slice can be represented a subset of the space of events (vertices of the causal graph), such that if an event is included in the slice, all its dependencies in the causal graph are included in the slice. On the first graph below, taking the slice means applying all the events above the red line to the initial state.
Now, here is the definition of relativistic coherence:
1. The model itself takes the initial state (e.g., a hypergraph) and the rules, and returns the causal graph:
WolframModel(s,r)c.g.
2. Given a causal graph, we can obtain a state (a hypergraph) using a slice:
getState(cg,slice)s∈H
A slice can be represented a subset of the space of events (vertices of the causal graph), such that if an event is included in the slice, all its dependencies in the causal graph are included in the slice. On the first graph below, taking the slice means applying all the events above the red line to the initial state.
Now, here is the definition of relativistic coherence:
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A computational test for relativistic coherence
A computational test for relativistic coherence
And so we can apply the small Lorentz boost and run the test:
This test was ran on the first 100 systems from the Registry of Notable Universes, and out of them 15 passed the test, and 20 timed out.
Here is another example of what a test output looks like:
Here is another example of what a test output looks like:
So, what should subluminal particles look like?
So, what should subluminal particles look like?
Concluding Remarks
Concluding Remarks
We have introduced a test for special relativity, applied it to a subset of the Registry of Notable Universes, and speculated on what moving particles could look like.
What remains to be investigated is how such systems can occur naturally. Perhaps, some particles resembling chaotic processes should behave in a relativistic way by default - at least, for some subset of systems. However, that remains to be investigated.
What remains to be investigated is how such systems can occur naturally. Perhaps, some particles resembling chaotic processes should behave in a relativistic way by default - at least, for some subset of systems. However, that remains to be investigated.
Acknowledgments
Acknowledgments
I thank my mentors Hatem Elshatlawy and Nik Murzin for help with this project, as well as Xerxes Arsiwalla, Max Piskunov, and Stephen Wolfram.
References
References
1
.S.Wolfram (2020), “Wolfram Physics Project: Technical Introduction.” https://wolframphysics.org/technical-introduction/.
Cite This Notebook
Cite This Notebook