## Dead ends

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?

Trapped surfaces around dead ends?

Past light cone of a dead end tends to be trivial

### Event horizon is a cusp

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

Cosmic Censorship

#### Weak Cosmic Censorship

Weak Cosmic Censorship

No naked singularity without a trapped surface

#### Strong Cosmic Censorship

Strong Cosmic Censorship

Deterministic future

## GR

GR

#### Trapped surface without singularity

Trapped surface without singularity

Can happen? [[ Avoid singularities in theorems .... ]] [ eternally collapsing ]

Singularity theorems: specifically 3+1 ?? Not 1+1

## Spacelike Singularity

Spacelike Singularity

Could lead into a CTC

[[Time stops with everything led to a particular place in space]]

Funnel in causal graph

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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LayeredGraphPlot[%342,AspectRatio1/2]

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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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LayeredGraphPlot[%,AspectRatio1/2]

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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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LayeredGraphPlot[%,AspectRatio1/2]

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ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y,z},{u,x}}{{x,u,v},{z,y},{z,u}}},{{{0,0,0},{0,0}}},9,"CausalGraphStructure"]

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LayeredGraphPlot[%,AspectRatio1/2]

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Graph3D

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## Timelike Singularity

Timelike Singularity

You keep going down a causal “line”, but you never get anywhere else.

ResourceFunction["SubstitutionSystemCausalGraph"]["A""A","A",5]

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ResourceFunction["SubstitutionSystemCausalGraph"]["AB""AB","AB",5]

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ResourceFunction["SubstitutionSystemCausalGraph"][{"A""A","B""B"},"AB",5]

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ResourceFunction["SubstitutionSystemCausalGraph"][{"A""A","A""B","B""BB"},"A",5]

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ResourceFunction["MultiwaySystem"][{"A""A","A""B","B""BB"},{"A"},5,"CausalGraph"]

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ResourceFunction["MultiwaySystem"][{"A""A","A""B","B""BB"},{"A"},5,"CausalGraph","IncludeStepNumber"True]

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ResourceFunction["MultiwaySystem"][{"A""A"},{"A"},5,"CausalGraph"]

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ResourceFunction["SubstitutionSystemCausalGraph"]["A""AAA","A",5]

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LayeredGraphPlot[ResourceFunction["SubstitutionSystemCausalGraph"]["A""AAA","A",5],AspectRatio1/2]

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Spacelike dominates timelike?

Behavior does not depend on initial conditions; i.e. many behaviors are all consistent with initial conditions

#### weak solutions to PDEs

weak solutions to PDEs

Integral form is solvable, but not the differential form

#### “going horizontal”

“going horizontal”

time function

In LayeredCausalGraph are things horizontal?

#### In a CTC-like curve

In a CTC-like curve

### White hole

White hole

Initially trapped; later not trapped

There is a future light cone, but no past light cone

Out[]=

RHS looks like white hole from within the LHS

## Hyperbolicity

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

GR hyperbolicity fails when spacelike hypersurfaces intersect

[ Fails for ADM ]

## Multiway case

Multiway case

LayeredGraphPlot[ResourceFunction["MultiwaySystem"]["WolframModel"{{{1,2},{2,3}}{{2,3},{3,1},{4,1}}},{{{0,0},{0,0}}},5,"CausalGraphStructure"],AspectRatio1/2]

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LayeredGraphPlot[ResourceFunction["MultiwaySystem"]["WolframModel"{{{1,2},{2,3}}{{2,3},{3,1},{4,1}}},{{{0,0},{0,0}}},6,"CausalGraphStructure"],AspectRatio1/2]

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$Aborted

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### Disconnection

Disconnection

LayeredGraphPlot[ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y}}{{y,z},{y,z}}},{{{0,0}}},6,"CausalGraphStructure"],AspectRatio1/2]

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LayeredGraphPlot[ResourceFunction["MultiwaySystem"]["WolframModel"{{{x,y}}{{y,z},{y,z}}},{{{0,0}}},7,"CausalGraphStructure"],AspectRatio1/2]

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ResourceFunction["WolframModel"][{{x,y}}{{y,z},{y,z}},Automatic,6,"CausalGraph"]

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#### Branchlike singularity = preserved quantum state

Branchlike singularity = preserved quantum state

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

## Hawking Radiation

Hawking Radiation

#### Branchial case

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

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 ]

[ 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