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WOLFRAM NOTEBOOK
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
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Series[r^d[r],{r,0,3}]
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In[]:=
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?
In[]:=
GridGraph[{6,6}]
Out[]=
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?
In[]:=
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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In[]:=
Graph3D[%]
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