Angular Momentum
Angular Momentum
?? circulation of causal edges around a timelike vector
?? circulation of causal edges around a timelike vector
Black hole properties
Black hole properties
This is not a BH singularity ... : this is a cosmological horizon
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ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX"},{"XAAX"},6,"StatesGraph"]
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ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX"},{"XAAX"},6,"CausalGraph"]
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ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX"},{"XAAX"},6,"CausalGraphStructure"]
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ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},{"XAAX"},8,"CausalGraphStructure"]
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ResourceFunction["MultiwaySystem"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},{"XAAX"},8,"CausalGraphStructure"]//LayeredGraphPlot
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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 ]
[ 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 ]
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 ]
Construction for Angular Momentum
Construction for Angular Momentum
Set up a timelike vector [AKA geodesic]
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
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
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
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
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