n

r

A function at every point on the causal graph and/or on the space graph

In causal graph, r is a time vector, which is the generator of a cone

For a given updating order, there is a definite light cone for every point

Assuming causal invariance, there is a unique causal graph, and then updating orders are just layerings of the graph....

There is a default layered embedding

Downward planar embedding : is a non-crossing layered embedding

n

r

d

r

2

r

Series[r^d[r],{r,0,3}]

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Plot[{rLog[r],r},{r,0,1}]

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#### Jonathan’s Tμν

Jonathan’s

T

μν

Non-interacting lumps just propagate unchanged through “time” : for any given foliation

Continuity equation for geodesics : how does a bundle of geodesics get thicker and not

This is also a continuity equation for vertices because each geodesic is basically a necklace of vertices (where the graph distance between vertices is the proper time) [is this in fact conformal time?]

This is also a continuity equation for vertices because each geodesic is basically a necklace of vertices (where the graph distance between vertices is the proper time) [is this in fact conformal time?]

Geodesic deviation equation: for a given slice of the geodesic bundle (parametrized by proper time), we can ask what the cross-sectional shape of the bundle is.

Sectional curvature ??? is Gaussian curvature of cross-section.

Divergence is Ricci tensor (dotted into the direction of the geodesic bundle)

Shape of the bundle cross-section gives the Riemann tensor

Sectional curvature ??? is Gaussian curvature of cross-section.

Divergence is Ricci tensor (dotted into the direction of the geodesic bundle)

Shape of the bundle cross-section gives the Riemann tensor

I.e. Vacuum Einstein’s equations is equivalent to the statement that the bundle of geodesics do not change size

#### n as distribution function

n as distribution function

n

r

Liouville equation:

MFT: n depends only on n’s

Roughly a uniformity/randomness statement about the graph

Roughly a uniformity/randomness statement about the graph

#### Special vs General Relativity

Special vs General Relativity

Consider a bundle (aka cone) of geodesics

Special relativity: the bundle is unchanged through spacetime

Which is the essentially the statement of the invariance of the speed of light

Which is the essentially the statement of the invariance of the speed of light

GR: the bundle has a fixed cross-sectional area

Two directions:

1. Additional constraint on the bundle

2. Corrections to e.g. the GR interpretation

1. Additional constraint on the bundle

2. Corrections to e.g. the GR interpretation

#### Equilibrium of spacetime

Equilibrium of spacetime

n

r

(Don’t need particle randomness to get Euler equations)

Euler equations: SR

#### Chapman-Enskog

Chapman-Enskog

Analog is looking at the higher-order tensor shape of the geodesic cross-section [cf spherical harmonics]

#### Higher-order corrections

Higher-order corrections

The next moment of the shape of the geodesic bundle

Are cubic terms possible? Are r log r terms possible?

#### What is the signature of dimension change?

What is the signature of dimension change?

Is dimension a scalar??

We could take apart and look at different directions for r

n

r

What is the dimension tensor? How many degrees of freedom does it have?

### Sectional curvature

Sectional curvature

Two orthonormal vectors: define a 2D plane in the manifold

Imagine fitting a sphere at a particular point (in general an ellipsoid)

Given two geodesics emanating from a point .... that defines a plane

Jonathan’s claim: from a given point, any set of non-parallel hyperedges can be thought of as orthonormal tangent vectors

#### Defining sectional curvature

Defining sectional curvature

Given points p and q, consider edges coming out of p and out of q

Ask how geodesically far apart are p and q, and how far apart are endpoints of following edges from them

Ask how geodesically far apart are p and q, and how far apart are endpoints of following edges from them

Curvature is the ratio of the distance between the endpoints of the links from p and q, and the distance between p and q themselves

d(p’,q’)/d(p,q)

Averaging over all possible “orthonormal vectors” gives a factor D

From p define a geodesic to p’

For all geodesics of a given length, what is the ratio d(p’,q’)/d(p,q)

( 1 - L^2/2 K + .... )

Dimensions enters because you average this quantity over the possible relative directions of p-p’ vs. q-q’

Given a particular p’, q’

Comparison between the “one-point function” that defines the total area of the bounding region, vs. the “two-point function” which counts the number of lines of a given length on the area.

In a n×n region of a square grid, there are n^2 points. How many geodesics of length L are there?

GridGraph[{6,6}]

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The number of geodesic paths of length L is 2^L here....

### Poincare group

Poincare group

SO(d)

Imagine a grid graph: it has certain symmetries, which are a subgroup of SO(n)

Is there a particle associated with quantized changes in dimension? What spin does it have? (Continuous?)

Invariance groups of fractals

#### Dimension of paths in QM?

Dimension of paths in QM?

WolframModel[{{1,2,1},{3,1,4}}{{4,5,4},{5,1,2},{6,5,2}},{{0,0,0},{0,0,0}},200,"CausalGraph"]

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Graph3D[%]

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