Sqrt[g(X)]
Cone is an integral of Sqrt[g] over the tube
Small t wrt radius of graph; large t wrt 1
Source of change in Sqrt[g] is divergence or convergence of geodesics
Ricci tensor: area of geodesic bundle
Weyl tensor: [trace free part of Riemann tensor]
1. Individual hyperedge
2. Space seems continuous
3.
4. Size of t he universe
2. Space seems continuous
3.
4. Size of t he universe
Claim is that between scales 1 and 2, you have to have a perfect “geodesic monopole” to achieve correct dimensional space.
Nothing can cancel R t t. But the higher order terms could cancel in various ways.
Action verion:
R Sqrt[g]
R Sqrt[g]
Einstein-Hilbert action is:
bulk geodesic volume
bulk geodesic volume
Dimension extremization
Dimension extremization
Limits
Limits
Graph metric: distance along a hyperedge = 1
Physical metric: whole hypergraph is unit size (or 10^26 m), and the edges are scaled accordingly
[[ We could rescale our plots
res=;
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Computing the spacetime average of Orientations give us RIntegrating over the spacetime volume gives
R
μν
g
With each new t (or r) you want the growth rate to not differ from r^d
For large t, growth rate has to be t^(d+1)
For large t, growth rate has to be t^(d+1)
Paths
Paths
If no refinement
[[Transitivity of geodesics ??]]
[[Transitivity of geodesics ??]]
JG argument
JG argument
t is in graph coordinates
t
Let be small enough that we can series expand
It is the length of the cone that we are studying
The R has units of L^-2, measured with a scaled distance
What is the global dimension, which goes way above size; this is at the “t scale” (i.e. at the size of the whole graph, i.e. where T~1)?
If you average over directions, you are only left with the Ricci part (?)
t
It is the length of the cone that we are studying
C()~(1-R)
t
d+1
t
2
t
The R has units of L^-2, measured with a scaled distance
What is the global dimension, which goes way above
t
t
If you average over directions, you are only left with the Ricci part (?)
T
ijkl
x
i
x
j
x
k
x
l
tt=Array[a,{3,3,3,3}]
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Sum[tt[[i,j,k,l]]xx[[n,i]]xx[[n,j]]xx[[n,k]]xx[[n,l]],{n,Length[xx]},{i,3},{j,3},{k,3},{l,3}]//FullSimplify
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Momentum
Momentum
Pick a hypersurface
+ neighbors
Number of edges of causal graph slicing a hypersurface
[If the hypersurface is energy]
+ neighbors
Number of edges of causal graph slicing a hypersurface
[If the hypersurface is energy]
Momentum + energy are frame dependent.
Given a structure in causal graph:
purely timelike
Excess of edges in CG
[Derivative of excess of edges ???] <energy conservation?>
There are different ways to increase the number of hyperedges
Think about a buckyball....
Excess of edges due to energy is compensated by a decrease in edges due to positive curvature
Fixed energy density : 1 + δρ
Given a structure in causal graph:
purely timelike
Excess of edges in CG
[Derivative of excess of edges ???] <energy conservation?>
There are different ways to increase the number of hyperedges
Think about a buckyball....
Excess of edges due to energy is compensated by a decrease in edges due to positive curvature
Fixed energy density : 1 + δρ
Energy momentum
Energy momentum
vacuum energy +
E = m 2c
E = m
2
c
Projecting edges through spacelike surface
Continuum Limit
Continuum Limit
Foliations are the way we idealize the observer
[foliations are we turn reality into the subjective experience of the observer]
[foliations are we turn reality into the subjective experience of the observer]
If we want a simple description of physics, need to have simple foliations
Quantum analog
Quantum analog
Non-interacting branchial graph [ bunch of separated pieces without entanglement ] : free field case
Boost is like a transformation between observers
Boost is like a transformation between observers
Basic indeterminancy: you can go either way on a critical pair. But entanglement in branchial graph.
Curvature measures entanglement because it is a sign of non-commutation
[x, p] = i ℏ
Go across the spatial graph and down the causal graph
Operators vs. states ....
Basic indeterminancy: you can go either way on a critical pair. But entanglement in branchial graph.
Curvature measures entanglement because it is a sign of non-commutation
[x, p] = i ℏ
Go across the spatial graph and down the causal graph
Operators vs. states ....