## 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 ]

In[]:=

ResourceFunction["SubstitutionSystemCausalGraph"][{"A""AB","XABABX""XXXX"},"XAAX",6]

## 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