WOLFRAM NOTEBOOK

Angular Momentum

?? circulation of causal edges around a timelike vector

Black hole properties

This is not a BH singularity ... : this is a cosmological horizon
In[]:=
ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX"},{"XAAX"},6,"StatesGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX"},{"XAAX"},6,"CausalGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX"},{"XAAX"},6,"CausalGraphStructure"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},{"XAAX"},8,"CausalGraphStructure"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},{"XAAX"},8,"CausalGraphStructure"]//LayeredGraphPlot
Out[]=
In a better case, there would be additional causal edges going into the sink ... but each causal edge, as it falls in, can generate a branch pair.
Need to mark branchlike and spacelike edges

[ to do: annotated multiway causal graph ]

Everything that goes into the BH region ends up with the other member of its branch pair not going in
Why can’t both members of a branch pair go into the event horizon? [ Either both members are inside the BH; both are outside; or it straddles the event horizon ]
In[]:=
ResourceFunction["SubstitutionSystemCausalGraph"][{"A""AB","XABABX""XXXX"},"XAAX",6]

Construction for Angular Momentum

Set up a timelike vector [AKA geodesic]

timelike vector: particle momentum vector
Pauli-Lubanski vector: spacelike vector
2D generalization of geodesic : ??? string action

Jonathan’s tube idea

v . ds
(d+1 - dimensional spacetime)
timelike momentum vector
spacelike surface defining our time slice : d dimensional
timelike hypersurface : d-1 dimensional
Step 1: project onto a spacelike hypersurface
Step 2: define a 2D plane
Rotation is defined by starting at a node: pick one geodesic to another node
Then look at other geodesic
integrate over all geodesics of length r that go through a particular point
take pairs of geodesics through a point, and integrate over them
[XXX] dr1 dr2
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.