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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ListLinePlot[res]

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Function[u,MapIndexed[{First[#2]/Length[u],#1}&,u]]/@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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xx=%165;

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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 ....