## Space Graphs

Space Graphs

Hypergraph volume

X is a hypergraph point

V

r

V

r

Sqrt[Det[g(X+δ)]] = Sqrt[Det[g(X)]] ( 1 - 1/6 (X)+O()+.... [[[ ?? the 6 ]]]

R

ij

δ

i

δ

j

3

δ

Integrate this over all directions:

∑(X+)

V

r

δ

i

Summing over all neighboring balls : ?? how much overlap

On a grid graph, indices on the tensors are equivalent to edges

V(X+δ) = V(X) + δ V’(X) + O(δ^2)

To get the standard volume of ball, you average over all directions δ

“Generalized tensor indices” that index the hyperedges from a particular point

[[ Have to have directed hyperedges ]]

For constant dimension, same formula for V

V

r

V

r

V

r

d(X+δ) = d(X) + δ d’(X) + ....

V(X+δ) = r^d(X) (1 + log(r) δ d’(X) + ...)

Consider a dimension field d(X): can have a dimension change tensor

δ is an infinitesimal distance along a geodesic [in the limit, this corresponds to many hyperedges]

#### What Tensors Are

What Tensors Are

Put a scalar field on a base space [i.e. assign a number to every hypervertex]

Associate a number with every hyperedge from every hypervertex [for a grid graph, this limits to a Cartesian tensor]

#### What are the δi?

What are the ?

δ

i

At the lowest level, imagine they are single hyperedges

R

ii

[[ For any given vertex, we can compute the change of going along that vertex ]]

V

r

For any node, we can compute the change of V going along each hyperedge

gtest=UndirectedGraph[Rule@@@WolframModel[{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},10,"FinalState"]]

In[]:=

Out[]=

vrall=GraphNeighborhoodVolumes[gtest,All,Automatic];

In[]:=

Take[vrall,5]

In[]:=

2{1,2,5,15,35,53,75,100,137,189},4{1,5,15,35,53,75,100,137,189,244},31{1,7,17,30,48,70,93,116,162,215},100{1,4,15,27,45,70,93,119,158,208},55{1,7,17,32,52,75,99,135,179,235}

Out[]=

NeighborhoodGraph[gtest,1,1,VertexLabelsAutomatic]

In[]:=

Out[]=

vrall[1]

In[]:=

{1,3,7,15,26,43,67,88,114,147}

Out[]=

vrall[97]

In[]:=

{1,6,13,24,39,60,81,105,131,187}

Out[]=

vrall[178]

In[]:=

{1,4,11,21,35,58,78,104,138,181}

Out[]=

vrall[97]-vrall[1]

In[]:=

{0,3,6,9,13,17,14,17,17,40}

Out[]=

vrall[178]-vrall[1]

In[]:=

{0,1,4,6,9,15,11,16,24,34}

Out[]=

1/2((vrall[97]-vrall[1])+(vrall[178]-vrall[1]))//N

In[]:=

{0.,2.,5.,7.5,11.,16.,12.5,16.5,20.5,37.}

Out[]=

#### Variation of Vr(X) with X

Variation of (X) with X

V

r

Flux of change of V

Claim: V(X) is related to the average of V(X) for all its neighbors

## Light Cones

Light Cones

Causal volume

T is a vector from one spacetime event to another (T + δT) [T is base point, δT is the “transporting vector”]

Unlike the case of need to define a foliation

V

r

C

t

WolframModel[{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},5,"LayeredCausalGraph"]

In[]:=

Out[]=

We can define another layering : which is another foliation

Valid foliations always have every arrow pointing down

Can in practice just make slices at any angle [[[ just rewrite the vertex coordinates ]]]

Valid foliations always have every arrow pointing down

Can in practice just make slices at any angle [[[ just rewrite the vertex coordinates ]]]

GraphEmbedding

In[]:=

{{0.,4.},{0.,3.},{0.,2.},{1.,2.},{0.,1.},{-1.,1.},{3.,1.},{1.,1.},{0.,0.},{1.,0.},{-2.,0.},{-1.,0.},{4.,0.},{2.,0.},{3.,0.}}

Out[]=

RotationMatrix[30°].#&/@%193

In[]:=

{{-2.,3.4641},{-1.5,2.59808},{-1.,1.73205},{-0.133975,2.23205},{-0.5,0.866025},{-1.36603,0.366025},{2.09808,2.36603},{0.366025,1.36603},{0.,0.},{0.866025,0.5},{-1.73205,-1.},{-0.866025,-0.5},{3.4641,2.},{1.73205,1.},{2.59808,1.5}}

Out[]=

VertexList

In[]:=

{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}

Out[]=

Thread[%%%]

In[]:=

Out[]=

Graph,VertexCoordinates%197

In[]:=

Out[]=

This is rotated too much....

Can we lay out the causal graph so every edge is at 45° ?

C

Σ:t

Spacetime volume = ( 1 - 1/6 R t t ) X flat space volume

At every point, average over ti, tj : get scalar curvature

Average over a region of nodes in the causal graph of for a given t

C

t

Every edge in the causal growth as an elementary interval; spacetime volume element for an elementary light cone is 1

Averaging for every node in the graph t^d (1 - 1/(6 d) R t^2 )

C

t

de Sitter space or something as a model non-flat spacetime....

https://www.frontiersin.org/articles/10.3389/fphy.2019.00032/full

https://www.frontiersin.org/articles/10.3389/fphy.2019.00032/full

[[[ Higher order terms would disappear because they are higher order in the ti, which are small ]]]

## Tμν

T

μν

Put a marker on the graph, and assume that it propagates through edges of the causal graph.

There is a continuity equation for this marker ... assume at a bifurcation that the markers splits in each direction

Image you want to compute R ignoring marked edges