Angular Momentum

?? circulation of causal edges around a timelike vector

Black hole properties

This is not a BH singularity ... : this is a cosmological horizon
​
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 ]

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 ]
ResourceFunction["SubstitutionSystemCausalGraph"][{"A""AB","XABABX""XXXX"},"XAAX",6]
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ResourceFunction["SubstitutionSystemCausalGraph"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},"XAAX",6]
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evo=ResourceFunction["SubstitutionSystemCausalEvolution"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},"XAAX",10]
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{{XAAX},{XABABX,{{2,2}{2,3},{3,3}{4,5}}},{XXXX,{{1,6}{1,4}}},{XXXXX,{{1,4}{1,5}}},{XXXXXX,{{1,4}{1,5}}},{XXXXXXX,{{1,4}{1,5}}},{XXXXXXXX,{{1,4}{1,5}}},{XXXXXXXXXX,{{1,4}{1,5},{5,8}{6,10}}},{XXXXXXXXXXXX,{{1,4}{1,5},{5,8}{6,10}}},{XXXXXXXXXXXXXXX,{{1,4}{1,5},{5,8}{6,10},{9,12}{11,15}}},{XXXXXXXXXXXXXXXXXX,{{1,4}{1,5},{5,8}{6,10},{9,12}{11,15}}}}
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ResourceFunction["SubstitutionSystemCausalPlot"][evo,"CausalGraph"True,"EventLabels"True]
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evo=ResourceFunction["SubstitutionSystemCausalEvolution"][{"A""AB","XABABX""XXXX","XXXX""XXXXX"},"AXAAX",10]
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{{AXAAX},{ABXABABX,{{1,1}{1,2},{3,3}{4,5},{4,4}{6,7}}},{ABBXXXX,{{1,1}{1,2},{3,8}{4,7}}},{ABBBXXXXX,{{1,1}{1,2},{4,7}{5,9}}},{ABBBBXXXXXX,{{1,1}{1,2},{5,8}{6,10}}},{ABBBBBXXXXXXX,{{1,1}{1,2},{6,9}{7,11}}},{ABBBBBBXXXXXXXX,{{1,1}{1,2},{7,10}{8,12}}},{ABBBBBBBXXXXXXXXXX,{{1,1}{1,2},{8,11}{9,13},{12,15}{14,18}}},{ABBBBBBBBXXXXXXXXXXXX,{{1,1}{1,2},{9,12}{10,14},{13,16}{15,19}}},{ABBBBBBBBBXXXXXXXXXXXXXXX,{{1,1}{1,2},{10,13}{11,15},{14,17}{16,20},{18,21}{21,25}}},{ABBBBBBBBBBXXXXXXXXXXXXXXXXXX,{{1,1}{1,2},{11,14}{12,16},{15,18}{17,21},{19,22}{22,26}}}}
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ResourceFunction["SubstitutionSystemCausalPlot"][evo,"CausalGraph"True,"EventLabels"True]
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Construction for Angular Momentum

Set up a timelike vector [AKA geodesic]

timelike vector: particle momentum vector
Pauli-Lubanski vector: spacelike vector
VectorPlot[{y,-x},{x,-3,3},{y,-3,3}]
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Graph3D[GridGraph[{5,5,5}]]
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VertexList[%]
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{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125}
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gg=Graph3D[GridGraph[{5,5,5}]]
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HighlightGraph[gg,PathGraph[FindShortestPath[gg,44,117]]]
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HighlightGraph[gg,{PathGraph[FindShortestPath[gg,44,117]],PathGraph[FindShortestPath[gg,44,70]]}]
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