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
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?]
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
I.e. Vacuum Einstein’s equations is equivalent to the statement that the bundle of geodesics do not change size
n as distribution function
is the analog of a one-particle distribution function
MFT: n depends only on n’s Roughly a uniformity/randomness statement about the graph
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
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
Equilibrium of spacetime
is statistically “stable”
(Don’t need particle randomness to get Euler equations)
Euler equations: SR
Analog is looking at the higher-order tensor shape of the geodesic cross-section [cf spherical harmonics]
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?
Is dimension a scalar??
We could take apart
and look at different directions for r
What is the dimension tensor? How many degrees of freedom does it have?
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
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
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
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?
The number of geodesic paths of length L is 2^L here....
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?)