Dead ends

Something could be a global normal form: whole universe stops
Local normal form: a piece of the universe stops
Spatially extended normal forms?
[ It could restart after an arbitrarily long time ]

Trapped surfaces around dead ends?

Past light cone of a dead end tends to be trivial

Event horizon is a cusp

Go to one side and you’re trapped; go to the other side and you’re not
Nodes can be trapped; some nodes can go either way

Cosmic Censorship

Weak Cosmic Censorship

No naked singularity without a trapped surface

Strong Cosmic Censorship

Deterministic future

GR

Trapped surface without singularity

Can happen? [[ Avoid singularities in theorems .... ]] [ eternally collapsing ]
Singularity theorems: specifically 3+1 ?? Not 1+1

Spacelike Singularity

Could lead into a CTC
[[Time stops with everything led to a particular place in space]]
Funnel in causal graph
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ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y,z},{x,u,v}}{{y,x,w},{w,u,s},{v,z,u}}},{{{0,0,0},{0,0,0}}},5,"CausalGraphStructure"]
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ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y,z},{u,x}}{{x,u,v},{z,y},{z,u}}},{{{0,0,0},{0,0}}},5,"CausalGraphStructure"]
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ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y,z},{u,x}}{{x,u,v},{z,y},{z,u}}},{{{0,0,0},{0,0}}},6,"CausalGraphStructure"]
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ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y,z},{u,x}}{{x,u,v},{z,y},{z,u}}},{{{0,0,0},{0,0}}},7,"CausalGraphStructure"]
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ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y,z},{u,x}}{{x,u,v},{z,y},{z,u}}},{{{0,0,0},{0,0}}},8,"CausalGraphStructure"]
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Timelike Singularity

You keep going down a causal “line”, but you never get anywhere else.
Spacelike dominates timelike?
Behavior does not depend on initial conditions; i.e. many behaviors are all consistent with initial conditions

weak solutions to PDEs

Integral form is solvable, but not the differential form

“going horizontal”

time function
In LayeredCausalGraph are things horizontal?

In a CTC-like curve

White hole

Initially trapped; later not trapped
There is a future light cone, but no past light cone
RHS looks like white hole from within the LHS

Hyperbolicity

Beyond max scan....
Intersections of characteristics...
Necessary condition: layers = “max scan” time steps
There exists a layering so that all edges point down. [[ Requires no cycles + ??? ]]
Global hyperbolicity: every point in space has to be updated

GR hyperbolicity fails when spacelike hypersurfaces intersect

[ Fails for ADM ]

Multiway case

Disconnection

Branchlike singularity = preserved quantum state

(i.e. it doesn’t move in branchial space) (i.e. it doesn’t entangle)

Hawking Radiation

Branchial case

One event can spawn two others, one of which is trapped, the other is not
? Could the black hole emit particles without losing mass

Spatial case

Minimal unit of causal disconnection is a spatial pair
There is an energy density (which is essentially vacuum energy)
There should be net momentum generated by uncanceled causal edge

[ Implies black hole wind ]

Potentially: around a black hole it is as if there was local expansion of the universe
As if there was a coat of momentum around the BH
Claim: effect is small because it is the area, compared to the volume