Find the list of unreachable events from a given event

(i.e. events not in the future light cone of a given event)
In[]:=
getCausallyDisconnectedRegions[{"AB""BAAB","A""BA"},"AB",4]
Out[]=
In[]:=
HighlightGraph[ResourceFunction["SubstitutionSystemCausalGraph"][{"AB""BAAB","A""BA"},"AB",4],{26,27,28,29,38,39,40,41,42,43,44,45}]
Out[]=

An event A is causally disconnected from B if the future light cones of A and B do not intersect

“A is causally connected to B” means there is a path from A to B in the causal graph
In[]:=
ImageReflect
,TopBottom
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Black hole test: does every “final event” lie in the future light cone of all events

In[]:=
getCausallyDisconnectedRegions[{"AB""BAAB","A""BA"},"AB",4]
Out[]=
In[]:=
getCausallyDisconnectedRegions[{"AB""BAB","A""BA"},"AB",4]
Out[]=
{}
If there no black holes in evidence, there would be no unreachable elements, i.e. this list of disconnected regions would be empty
{ node  list of final nodes unreachable from this node , ... }
Currently looking at nodes at step t, and at what nodes at step 2 t are unreachable.

Code


Black hole finding

In[]:=
getCausallyDisconnectedRegions[{"AB""BAAB","A""BA"},"AB",1,4]
Out[]=
{2{26,27,28,29,38,39,40,41,42,43,44,45}}
In[]:=
getCausallyDisconnectedRegions[{"AB""BAAB","A""BA"},"AB",1,4]
Out[]=
{2{26,27,28,29,38,39,40,41,42,43,44,45}}
In[]:=
Graph[ResourceFunction["SubstitutionSystemCausalGraph"][{"AB""BAAB","A""BA"},"AB",4],VertexLabelsAutomatic]
Out[]=
In[]:=
getCausallyDisconnectedRegions[{"AB""BAAB","A""BA"},"AB",2,5]
Out[]=
In[]:=
First/@%
Out[]=
{2,4,5,6,7,8,9}
The LHS events are ones whose future light cones are not “complete”
What is the “causal connection graph” of nodes 2, 4, 5, 6, 8?
Case 1: “final null set” of A and B are identical
Case 2: “final null set” of A contains final null set of B
Case 3: intersect (A  X and B  X)
Case 4: disjoint
In[]:=
Clear[IntersectionRule]
In[]:=
IntersectionRule[s1_->list1_,s2_->list2_]/;!IntersectingQ[list1,list2]:={}
In[]:=
IntersectionRule[s1_->list1_,s2_->list2_]/;(SubsetQ[list1,list2]&&!SubsetQ[list2,list1]):={DirectedEdge[s1,s2]}
In[]:=
IntersectionRule[s1_->list1_,s2_->list2_]/;(SubsetQ[list2,list1]&&!SubsetQ[list1,list2]):={DirectedEdge[s2,s1]}
In[]:=
IntersectionRule[s1_->list1_,s2_->list2_]/;Sort[list1]===Sort[list2]:={DirectedEdge[s1,s2],DirectedEdge[s2,s1]}
In[]:=
IntersectionRule[s1_->list1_,s2_->list2_]/;(IntersectingQ[list1,list2]&&!SubsetQ[list1,list2]&&!SubsetQ[list2,list1]):={UndirectedEdge[s1,s2]}
In[]:=
getCausallyConnectedRegions[{"AB""BAAB","A""BA"},"AB",2,5]
Out[]=
In[]:=
CausalConnectionGraph[rule_,init_,ti_,tf_]:=With[{u=getCausallyConnectedRegions[rule,init,ti,tf]},SimpleGraph[Flatten[Outer[IntersectionRule,u,u,1]]]]
In[]:=
CausalConnectionGraphAt[rule_,init_,ti_,tf_]:=With[{u=getCausallyConnectedRegionsAt[rule,init,ti,tf]},SimpleGraph[Flatten[Outer[IntersectionRule,u,u,1]]]]
In[]:=
CausalConnectionGraph[{"AB""BAAB","A""BA"},"AB",2,5]
Out[]=

All rules

Failure of global hyperbolicity:

WolframModel case

Minimal Black Hole?

This has an apparent horizon, that survives for a certain amount of time....

Cosmological event horizon

Two black holes ?

Buggy...