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WOLFRAM NOTEBOOK
Tμν is flux of hyperedges through a hypersurface
T
μν
Energy is roughly the number of transitions of hyperedges (number of times a hyperedge is updated)
Momentum is roughly the movement of hyperedges relative to the background
[How does this transform as we change the foliation]
[How does this transform as we change the foliation]
Rest mass is number of nodes (in the spatial graph)
Rest mass is number of nodes (in the spatial graph)
Lorentz transformations
Lorentz transformations
Imagine we have a time stack of spatial graphs
These are knitted together by the causal graph: which remains invariant in different foliations
Causal graph is an equivalence of spacetime stacks
These are knitted together by the causal graph: which remains invariant in different foliations
Causal graph is an equivalence of spacetime stacks
For space and time we are transforming the
Spatial distance is (roughly) geodesic distance on the spatial graph
Spatial distance is (roughly) geodesic distance on the spatial graph
Temporal distance is geodesic distance in the causal graph [proper time]
Temporal distance is geodesic distance in the causal graph [proper time]
Different frame: seeing a staggered spatial graph with events from different digraph layers
Measuring energy-momentum: how many hyperedge updates are in a certain spacetime volume [aka a cone with a certain proper time height] (a limiting elementary light cone)
Measuring energy-momentum: how many hyperedge updates are in a certain spacetime volume [aka a cone with a certain proper time height] (a limiting elementary light cone)
Transforming between energy and momentum depends on foliation
Multiplier of Ct is energy-momentum
Multiplier of is energy-momentum
C
t
Growth rate of is dimension/curvature
C
t
C
t
d
t
2
t
t^2 is ti tj
P[R] is the projection of in the timelike direction in the foliation
R
μ⋁
ρ=+titj(X)
ρ
0
ρ
1
X is a spacetime point
ρ vs node count : R is a pure node count measurement; if we interpret ρ as a mass then we need a conversion
1 unit of proper time
Dimensions of are nodes per spacetime volume (c^d)
C
t
ρ has dimensions M L^-d ; has dimensions T^-2 M L^-d
ρ
1
G has dimensions L^d M^-1 T^-2
E = m 2c
E = m
2
c
Rest mass is a count of the number of nodes/edges involved in a particle [characteristic of the particle]
Motion has to locally look like change of foliation
What does the causal graph look like with a particular moving through it?
In[]:=
WolframModel[{{{1},{1,2}}{{1,2},{2}},{{1,2},{2,2,2},{2,3}}{{1,2},{3,3,3,3},{2,3}},{{1,2},{2,2,2,2},{2,3}}{{1,2},{1,1,1},{2,3}}},Append[Catenate[Table[{{i,i+1},{1,1,1}+i,{i+1,i+2}},{i,1,19,2}]],{2}],20,"LayeredCausalGraph"]
Out[]=
In[]:=
WolframModel[{{{1},{1,2}}{{1,2},{2}},{{1,2},{2,2,2},{2,3}}{{1,2},{3,3,3,3},{2,3}},{{1,2},{2,2,2,2},{2,3}}{{1,2},{1,1,1},{2,3}}},Append[Catenate[Table[{{i,i+1},{1,1,1}+i,{i+1,i+2}},{i,1,19,2}]],{2}],20,"CausalGraph"]
Out[]=
Rest mass is a fixed number of nodes in the spatial graph
In motion, there must be a bunch of update events
E = m c^2 is a tradeoff between energy that be got from motion, and rest mass
[E energy is the hypotenuse distance; mass is vertical distance]
number of nodes involved is m t (t = height)
hypotenuse length is energy
E^2 = p^2 c^2 + m^2 c^4
E^2 = p^2 c^2 + m^2 c^4
Clocks vs. observer computation
Clocks vs. observer computation
Imagine that the clocks are set in a very complicated way. At the beginning, the observer assumes they are correct. They try to infer gravitational fields consistent with what the clocks tell them, and with GR.
Can the observer define a coordinate system so that the clocks (however their mechanism may work) remain synchronous?
Can you predict what the clocks will do, and therefore set up a coordinate system you understand?
Can you predict what the clocks will do, and therefore set up a coordinate system you understand?
There is a clock everywhere in space; they are all doing different computations, and tweeting every time they get a result
There is a clock everywhere in space; they are all doing different computations, and tweeting every time they get a result
Can the observer explain this with gravitational fields? Observer is making a model based on gravity
Can the observer explain this with gravitational fields? Observer is making a model based on gravity
Don’t make a neural net, make a configuration
T
μν
Can a set of gravitating masses emulate a Turing machine?
Can a set of gravitating masses emulate a Turing machine?
Vs make computation out of gravitational waves