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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
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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
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 +
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 ....