## Definitions

Definitions

### Both these functions return expression indices

Both these functions return expression indices

#### Function to deduce the effective spatial ball from the genuine light (causal) cone:

Function to deduce the effective spatial ball from the genuine light (causal) cone:

CausalConeSpatialBall[wmo_,expr0_]:=Module[{t=wmo["CompleteGenerationsCount"],fexprs},fexprs=wmo["StateEdgeIndicesAfterEvent",-1];Intersection[Cases[VertexOutComponent[wmo["ExpressionsEventsGraph"],{expr0}],{"Expression",n_}:>n],fexprs]]

In[]:=

CausalConeSpatialBall[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},4],{"Expression",8}]

In[]:=

{27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42}

Out[]=

#### Function to construct balls in the effective expression-deduced spatial hypergraph:

Function to construct balls in the effective expression-deduced spatial hypergraph:

Construct the effective expressions hypergraph:

ExpressionsSpatialHypergraph[wmo_]:=Module[{ix=wmo["StateEdgeIndicesAfterEvent",-1],es},Values[Merge[Association@@@(Thread/@Thread[wmo["AllExpressions"][[ix]]ix]),Identity]]]

In[]:=

ExpressionsSpatialBall[wmo_,center_,radius_]:=With[{esh=ExpressionsSpatialHypergraph[wmo]},VertexList[Last[ResourceFunction["HypergraphNeighborhoods"][esh,center,radius]]]]

In[]:=

ExpressionsSpatialBall[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},4],31,1]

In[]:=

{31,35,24,32,38,43,44,34,37,41}

Out[]=

#### Function to find the center of the effective spatial ball from the genuine light cone:

Function to find the center of the effective spatial ball from the genuine light cone:

Subhypergraph[h_,vertices_]:=Select[h,AllTrue[#,MemberQ[vertices,#]&]&]

In[]:=

HypergraphToCompleteGraph[h_]:=Graph[UndirectedEdge@@@Catenate[Subsets[#,{2}]&/@h]]

In[]:=

CausalConeSpatialBallCenter[wmo_,expr0_]:=GraphCenter[First[TakeLargestBy[ConnectedGraphComponents[HypergraphToCompleteGraph[Subhypergraph[ExpressionsSpatialHypergraph[wmo],CausalConeSpatialBall[wmo,expr0]]]],VertexCount,1]]]

In[]:=

CausalConeSpatialBallCenter[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},4],{"Expression",8}]

In[]:=

{37,31,34,41}

Out[]=

### These functions take expression indices and make plots

These functions take expression indices and make plots

UnorderedHypergraphPlot[h_,opts___]:=ResourceFunction["WolframModelPlot"][h,opts,"ArrowheadLength"0,EdgeStyle<|{_,_,_..}Transparent|>,"EdgePolygonStyle"<|{_,_,_..}Directive[Hue[0.63,0.66,0.81],Opacity[0.1],EdgeForm[Directive[Hue[0.63,0.7,0.5],Opacity[0.7]]]]|>]

In[]:=

HighlightedUnorderedHypergraphPlot[h_,vertices_,opts___]:=UnorderedHypergraphPlot[h,GraphHighlightJoin[vertices,Subhypergraph[h,vertices]]]

In[]:=

ExpressionsSpatialHypergraph[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},4]]

In[]:=

{{20,27,28,35,36},{20,39,42,45},{24,31,32,35,38,43,44},{24,25,26},{25,27,30,39,40},{26,29,33,43,46},{28,29,30},{31,34,37,41},{32,33,34},{36,37,38},{40,41,42},{44,45,46}}

Out[]=

UnorderedHypergraphPlot[%]

In[]:=

Out[]=

Genuine light cone intersecting the reconstructed spatial slice:

With[{wmo=ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},4]},HighlightedUnorderedHypergraphPlot[ExpressionsSpatialHypergraph[wmo],CausalConeSpatialBall[wmo,{"Expression",8}]]]

In[]:=

Out[]=

Deduced geodesic ball in the reconstructed spatial slice:

With[{wmo=ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},4]},HighlightedUnorderedHypergraphPlot[ExpressionsSpatialHypergraph[wmo],ExpressionsSpatialBall[wmo,First[CausalConeSpatialBallCenter[wmo,{"Expression",8}]],1]]]

In[]:=

Out[]=

Genuine light cone:

With[{wmo=ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},7]},HighlightedUnorderedHypergraphPlot[ExpressionsSpatialHypergraph[wmo],CausalConeSpatialBall[wmo,{"Expression",30}]]]

In[]:=

Out[]=

Effective reconstructions:

With[{wmo=ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},7]},HighlightedUnorderedHypergraphPlot[ExpressionsSpatialHypergraph[wmo],ExpressionsSpatialBall[wmo,First[CausalConeSpatialBallCenter[wmo,{"Expression",30}]],1]]]

In[]:=

Out[]=

### Workflow: pick an expression at a particular generation ... then see how its genuine light cone expands, and compare it with the sizes of reconstructed spatial balls with certain radii

Workflow: pick an expression at a particular generation ... then see how its genuine light cone expands, and compare it with the sizes of reconstructed spatial balls with certain radii

“subspace navigator” / “subspace demon”

# Types of Graphs

Types of Graphs

## The Components of a Hypergraph

The Components of a Hypergraph

### Type of Vertices

Type of Vertices

Elements (aka atoms)

Relations (aka expressions)

Events

( Whole states )

### Type of Edges

Type of Edges

Relations (i.e. what elements are in the same relation)

Elements (i.e. what relations contain the same elements)

Events [e.g. in states graph]

Common history / common future [will lead to the same event]

[one object causes both objects]

[one object causes both objects]

Causal [connects objects which directly cause each other, by virtue of sharing components]

[one object causes another objects]

[one object causes another objects]

### Selection of Objects to Put in the Graph

Selection of Objects to Put in the Graph

We are starting from a poset

We can either select chains, or antichains

[ E.g. spacelike antichains, branchlike antichains, etc. ]

### System Type

System Type

Single-way

Multiway

## Basic Spatial Hypergraph

Basic Spatial Hypergraph

#### Vertices: Elements

Edges: Relations

Selection: Spacelike separated subset of relations

Vertices: Elements

Edges: Relations

Selection: Spacelike separated subset of relations

Edges: Relations

Selection: Spacelike separated subset of relations

[[[ Need: labels for edges: which in this case are just the list of elements at the ends of the edge ]]]

## Dual Spatial Hypergraphs

Dual Spatial Hypergraphs

#### Vertices: Relations

Edges: Elements

Selection: (same as basic spatial hypergraph)

Vertices: Relations

Edges: Elements

Selection: (same as basic spatial hypergraph)

Edges: Elements

Selection: (same as basic spatial hypergraph)

[[[ We should flip vertex and edge styles ]]]

[[[ Hyperedges should be labeled with atoms ]]]

[[[ cf Chris Pratt’s WSS project ]]]

## Events Common-History Spatial Graph

Events Common-History Spatial Graph

#### Vertices: Events

Edges: Common-History [need new color: somewhat branchial]

Selection: (same as basic spatial hypergraph, but for events)

Vertices: Events

Edges: Common-History [need new color: somewhat branchial]

Selection: (same as basic spatial hypergraph, but for events)

Edges: Common-History [need new color: somewhat branchial]

Selection: (same as basic spatial hypergraph, but for events)

1 step of history:

2 steps of history:

[[[ If we go all the way back to the big bang, then we get a complete graph ]]]

## Events Common-Future Spatial Graph

Events Common-Future Spatial Graph

## Basic Causal Graph (AKA Events Shared-Relations Graph)

Basic Causal Graph (AKA Events Shared-Relations Graph)

#### Vertices: Events

Edges: Relations (i.e. events that share a relation are joined)

Selection: All (for a particular updating order) [unless you run a multiway system]

Vertices: Events

Edges: Relations (i.e. events that share a relation are joined)

Selection: All (for a particular updating order) [unless you run a multiway system]

Edges: Relations (i.e. events that share a relation are joined)

Selection: All (for a particular updating order) [unless you run a multiway system]

[[[ Edges need to be labeled by the expressions that are in common ]]]

## Events Shared-Atoms Graph

Events Shared-Atoms Graph

## Relations+Events Causal Graph

Relations+Events Causal Graph

# Key Question: Geodesic Size of Causal Ball in Spatial Slice

Key Question: Geodesic Size of Causal Ball in Spatial Slice

#### Is the boundary of the causal ball uniformly growing in geodesic distance from its “center”?

Is the boundary of the causal ball uniformly growing in geodesic distance from its “center”?

### Ordinary Spatial Hypergraph

Ordinary Spatial Hypergraph

Want to know the relations affected by a previous event (or relation)

Pick e.g. a relation at step t, and see what relations it affects at step t’

###

#### For non-expanding rules, can look at light cone independent of expansion of the universe....

For non-expanding rules, can look at light cone independent of expansion of the universe....

## Case 1: Causal Cone from the Past

Case 1: Causal Cone from the Past

Set up beacons in the early universe; know how long it takes then to between them; now compute speed based on current beacon positions : speed then measured using current distances will seem to be >c

## Case 2: Causal Cone to the Future

Case 2: Causal Cone to the Future

[Same story as with past if space contracts]

#### Set up a highway (a space tube, made of metal); then let space contract inside it

Set up a highway (a space tube, made of metal); then let space contract inside it

Need a sparser grid, then we need to glue it.....

#### Does the interior of the tunnel have to be ordinary space?

Does the interior of the tunnel have to be ordinary space?

Yes, if we ordinary matter to pass through it; but no if we just want “signals” made of something pass through.....

#### Idealized case: we just have markers moving on the space, and we are not updating the space

Idealized case: we just have markers moving on the space, and we are not updating the space

#### Then we want a sparser-edge interior to the space tunnel

Then we want a sparser-edge interior to the space tunnel

### If the space doesn’t change “under you” ... then you can’t exceed the rule diameter per elementary time

If the space doesn’t change “under you” ... then you can’t exceed the rule diameter per elementary time

### But it is not under your control whether the space “moves towards you” [except Alcubierre, which “feeds space into you”]

But it is not under your control whether the space “moves towards you” [except Alcubierre, which “feeds space into you”]

#### It could be that it depends on quantum fluctuations; and there is no classical version

It could be that it depends on quantum fluctuations; and there is no classical version

## Lay out a causal graph so that its nodes are “correct geodesic distances”

Lay out a causal graph so that its nodes are “correct geodesic distances”

If we lay out the events according to the geodesic distances of the events-common-history graph ... then any causal edge going at “more than 45°” is an FTL edge.

#### Compare: painting a light cone, and painting a geodesic ball......

Compare: painting a light cone, and painting a geodesic ball......

This is the geodesic ball:

#### [[ Need computational irreducibility to get isotropy and arbitrary boosts working ]]

[[ Need computational irreducibility to get isotropy and arbitrary boosts working ]]

#### How does a space demon (vs a gas demon) violate Lorentz invariance? [Edges it picks are not frame independent]

How does a space demon (vs a gas demon) violate Lorentz invariance? [Edges it picks are not frame independent]

#### Heat into mechanical work....

Heat into mechanical work....

## Effective Spatial Ball etc.

Effective Spatial Ball etc.

## Graph Gluing...

Graph Gluing...

# Can FTL Be Used to Create Closed Timelike Curves?

Can FTL Be Used to Create Closed Timelike Curves?

[[ Consider an FTL-boosted frame ]]

<< In current setup, there cannot be CTCs, because we don’t do isomorphism between timelike separated relations >>

<< In global multiway systems, there can be CTCs through global isomorphism >>

<< In global multiway systems, there can be CTCs through global isomorphism >>

# Casimir Effect

Casimir Effect

### For a 1D string, basically imagine there are markers that prevent interchanges in some place

For a 1D string, basically imagine there are markers that prevent interchanges in some place