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
In[]:=
gtest=UndirectedGraph[Rule@@@WolframModel[{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},10,"FinalState"]]
Out[]=
In[]:=
vrall=GraphNeighborhoodVolumes[gtest,All,Automatic];
In[]:=
Take[vrall,5]
Out[]=
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}
In[]:=
NeighborhoodGraph[gtest,1,1,VertexLabelsAutomatic]
Out[]=
In[]:=
vrall[1]
Out[]=
{1,3,7,15,26,43,67,88,114,147}
In[]:=
vrall[97]
Out[]=
{1,6,13,24,39,60,81,105,131,187}
In[]:=
vrall[178]
Out[]=
{1,4,11,21,35,58,78,104,138,181}
In[]:=
vrall[97]-vrall[1]
Out[]=
{0,3,6,9,13,17,14,17,17,40}
In[]:=
vrall[178]-vrall[1]
Out[]=
{0,1,4,6,9,15,11,16,24,34}
In[]:=
1/2((vrall[97]-vrall[1])+(vrall[178]-vrall[1]))//N
Out[]=
{0.,2.,5.,7.5,11.,16.,12.5,16.5,20.5,37.}
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
In[]:=
WolframModel[{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},5,"LayeredCausalGraph"]
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 ]]]
This is rotated too much....
Can we lay out the causal graph so every edge is at 45° ?
Spacetime volume = ( 1 - 1/6 R t t ) X flat space volume
At every point, average over ti, tj : get scalar curvature
Every edge in the causal growth as an elementary interval; spacetime volume element for an elementary light cone is 1
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 ]]]
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