## Correspondence between NDTMs and causal invariance

Correspondence between NDTMs and causal invariance

Could simulate NDTM with string rewriting

The tape has A,B ... The head has X,Y

TuringMachine

This is a deterministic TM:

RulePlot[TuringMachine[{596440,2,3}]]

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We need {A,B,C} ⊗ {X,Y,_}, where _ means a tape state, and X,Y means the head is there

Rules will preserve single-headedness [need contingent causal invariance for the subspace of a single head]

With a single head, system is obviously causal invariant (“causant”).

#### Multiway NDTM = QTM

Multiway NDTM = QTM

NDTM : one path at a time

QTM : maintain amplitudes for all paths

QTM : maintain amplitudes for all paths

## Motion

Motion

Translation of a structure unchanged to another place

Different lumps of hypergraph : with particle-like local stability, can move around

### Existence of Particles

Existence of Particles

Non-ergodicity of hypergraph rewriting

Locally conserved subspaces of the graph space

### Interstellar travel

Interstellar travel

Pull a few hyperedges far away ??

Can you create a bridge that persists [[ in effect, that is a local dimension perturbation ]]

When you move something somewhere, mostly it leaves no tail/tale in spacetime

If normally a dimension would diffuse diffusively ... you could potentially prevent by having its structure be full of conserved things

Exotic “matter” : features of the graph that maintain perturbed dimension as other things maintain perturbed curvature ..... [[ analogy of Einstein-Rosen bridge maintained by ρ < 0 matter ]]

When you move something somewhere, mostly it leaves no tail/tale in spacetime

If normally a dimension would diffuse diffusively ... you could potentially prevent by having its structure be full of conserved things

Exotic “matter” : features of the graph that maintain perturbed dimension as other things maintain perturbed curvature ..... [[ analogy of Einstein-Rosen bridge maintained by ρ < 0 matter ]]

### Negative etc. masses

Negative etc. masses

Consider energy density

Think you’re getting ρ L^d

Imagine you have a lump; you break it in two

Chop a tube; how do you maintain energy conservation?

The dimension you don’t think about you can imagine are curled in balls

GridGraph[{6,6,6}]

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Single object: mass ρ L^2

GIven that you think you’re in 1D, expect ρ L/2 when you split in two

{GridGraph[{10}],GridGraph[{5,5}]}

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Graph,VertexLabelsAutomatic

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PathGraph[Range[100,110]]

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GraphUnion,PathGraph[Range[100,110]],PathGraph[Range[200,210]],Graph[{{110,3}}],Graph[{{23,200}}]

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In the lump, there is higher mass density than you’d expect (“it’s mass ‘hidden’ in extra dimensions”)

Regions of un-dimension-decayed spacetime from “inflation period” end up being the dark matter.....

Could the cosmic web be a reflection of a piece of hypergraph

Could the cosmic web be a reflection of a piece of hypergraph

### Cosmic Web

Cosmic Web

## Cosmic Censorship / Cauchy Problem

Cosmic Censorship / Cauchy Problem

When does our causal graph define a satisfactory Cauchy problem.

Are CTCs avoided in our causal graphs?

What about a “confluence diamond”?

What about a “confluence diamond”?

## Origin of G

Origin of G

Proportionality between node count in space graph (~mass) and actual [physical] volume of cones in causal graph

TL^3/M

( would suggest (spacetime volume)/mass as units of G )

G : T L^d / M [assuming mass, not mass density]

Assume mass density wrt time: M/T^3 (mass you can get to in a certain geodesic time) [time is the correct measure of graph distance]

Node density T^3 is the geodesic ball’s content

Compared to volume of spacetime cone: T L^d

T L^d = G nd T^d

G = L^d T^(1-d) / nd

I.e. nd has units of mass..... [i.e. every node has a certain amount of mass]

Fundamental point:

Assume mass density wrt time: M/T^3 (mass you can get to in a certain geodesic time) [time is the correct measure of graph distance]

Node density T^3 is the geodesic ball’s content

Compared to volume of spacetime cone: T L^d

T L^d = G nd T^d

G = L^d T^(1-d) / nd

I.e. nd has units of mass..... [i.e. every node has a certain amount of mass]

Fundamental point:

mass density : M/L^d or M T^d/c^d

G c^d has units of M T^d

Actual units of G: L^3 M^-1 T^-2

In Planck length: has c^3

Assuming cone measurement is wrt space: c^d T is cone volume

G scaling might be changing all the time as hypergraph for universe refines....

#### Summary

Summary

Going along an edge in the causal graph is unit of time

Projected onto the spatial graph, that is c τ

Projected onto the spatial graph, that is c τ

Geodesic spatial volume is basically τ^d [which gives the instantaneous mass]

## Hypergraph Isomorphism “Distance” / Fiber Bundles for Gauges

Hypergraph Isomorphism “Distance” / Fiber Bundles for Gauges

## What is Higgs field?

What is Higgs field?

## What is a massless particle

What is a massless particle

Deformation that has no nodes involved ?

How do gauge particles get mass?

Why are the gauge particles massless?

#### Gauge Field

Gauge Field

We have a base space; we have rules being applied; these rules have different configurations related by hypergraph isomorphism

At every point in the base space there is a rule that could be applied

E.g. dimers on a grid

At every point in the base space there is a rule that could be applied

E.g. dimers on a grid

GridGraph[{5,5}]

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Massless particles carry information about the permutation of indices in the rewritings

∂

μ

A

μ

2

∂

2

m

You have a function which is a value in a gauge group. f(X) Move to f(X+δ) The value will effectively change just because the coordinates at that new point in the fiber bundle are different.

What quantities are independent of this effect? (E.g. curvatures)

What can we measure wrt to the internal degrees of freedom of the hypergraphs?

What quantities are independent of this effect? (E.g. curvatures)

What can we measure wrt to the internal degrees of freedom of the hypergraphs?

Each local region of edges is the content of a fiber in a fiber bundle. The choice of which edge is picked for an update is locally arbitrary

Imagine perturbing the system by changing which edges got picked for a particular rewrite. The effect of that change propagates at the speed of rewriting (i.e. speed of light). What is the structure of the change?

Make a gauge-like perturbation; this will give a gauge boson propagating out.

Claim: long-range effects of gauge bosons are analogs to rewriting order artefacts

Imagine perturbing the system by changing which edges got picked for a particular rewrite. The effect of that change propagates at the speed of rewriting (i.e. speed of light). What is the structure of the change?

Make a gauge-like perturbation; this will give a gauge boson propagating out.

Claim: long-range effects of gauge bosons are analogs to rewriting order artefacts

#### Accelerated Frames / Uniform Gravitational Field

Accelerated Frames / Uniform Gravitational Field

Slice causal graph at successively steeper angles.

#### Gravitational Covariant Derivative

Gravitational Covariant Derivative

There is a function on a space that is deformed

Change of function just because of deformation of space

Change of function just because of deformation of space

## Steeper Foliations

Steeper Foliations

Fundamental point: there is one causal graph, but different foliations/different frame let you explore it in different orders.

Different inertial frame: angle on the grid

But you are always thinking of “time” as vertical wrt your spacelike hypersurface

But you are always thinking of “time” as vertical wrt your spacelike hypersurface

Rotate,45Degree

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To get to a particular “global generation”, you experience more updating events

More updating events means more nodes, which means more mass.........

Same γ factor for mass and time. [Covariant transformation of momentum]

Same γ factor for mass and time. [Covariant transformation of momentum]

Visiting order for the causal graph

There is a canonical/optimal visiting order: aka graph embedding as downward layered graph

#### Event Horizons in Accelerating Frames

Event Horizons in Accelerating Frames

Critical pair converges in one step (say) of the global clock, but can take arbitrarily many in an accelerating frame

## Hasse Diagrams

Hasse Diagrams