Therefore, a black hole in the multiway causal graph may be characterized by the presence of two distinct horizons: a standard event horizon corresponding to regular causal disconnection, and an entanglement event horizon corresponding to multiway disconnection, which always lies strictly on the exterior of the causal event horizon. As such, from the point of view of an external observer in the multiway causal graph watching an infalling object to a black hole, the object will appear to “freeze” (due to quantum Zeno effects that are the multiway analog of time dilation) at the entanglement horizon, and will never get close to the true causal event horizon. Since Hawking radiation (which occurs as a consequence of non-convergent branch pairs in the multiway evolution graph) is emitted from the entanglement horizon and not the causal event horizon, the particles that get radiated from the black hole may be perfectly correlated with the information contained within the infalling object, without any apparent or actual violation of special relativity (since no information ever crossed a spacetime event horizon), thus resolving the black hole information paradox.

This resolution is formally quite similar to the standard resolution to the black hole information paradox implied by the holographic principle and the AdS/CFT duality.

Recall that the multiway causal graph encodes both the structure of the (purely quantum mechanical) multiway evolution graph, as well as the structures of the (purely relativistic) causal graphs corresponding to each branch of multiway evolution. Therefore, one can imagine “walling off” a certain bundle of causal edges in the multiway causal graph corresponding to some particular branch of multiway evolution, such that all of the causal edges inside the boundary of the wall correspond to edges in a purely relativistic causal graph (i.e. they designate causal relations between events in spacetime), whilst all of the causal edges intersecting the boundary of the wall correspond to edges in a purely quantum mechanical multiway graph (i.e. they designate causal relations between events in branchtime). As such, one immediately obtains a duality between the bulk gravitational theory on the interior of the wall, and the boundary quantum mechanical theory on the surface of the wall, just as in AdS/CFT. ]]>

More precisely, a formal statement of the ER=EPR conjecture is that the Bekenstein–Hawking entropy of a pair of entangled black holes is equivalent to their entanglement entropy. If Hawking radiation effects occur as a result of branch pairs that fail to reconverge as a consequence of disconnections in the multiway causal graph, the ER=EPR conjecture is really just a rather elementary statement about the geometry of branchtime (in other words, it states that the natural distance metric in branchtime is the entanglement entropy of pairs of microstates, which one can prove directly from the properties of the Fubini–Study metric tensor).

]]>One begins by noting that a pair of abstract rewrite relations, R1 and R2, are said to “commute” if the state obtained by applying R1 and then R2 is identical to the state obtained by applying R2 and then R1. If a multiway Wolfram model evolution is not confluent, in the sense that there exist bifurcations in the multiway evolution graph that never re-converge, then this immediately implies the existence of non-commutative rewrite relations (since an abstract rewriting system is confluent if and only if it commutes with itself). Since each updating event in the multiway system can be thought of as being the application of an abstract rewrite relation, it follows that there must exist pairs of updating events that do not commute, in the sense that the final hypergraph obtained will depend upon the timelike-ordering of the application of those events.

If we now interpret the multiway system as being the discrete analog of a (complex) projective Hilbert space, with the rewrite relations being linear operators acting on this space, then this statement immediately reduces to the statement of the standard uncertainty principle regarding the timelike-orderings of measurement operations for pairs of non-commuting observables in quantum mechanics.

]]>A geodesic bundle propagating through an ordinary (i.e. purely relativistic) causal graph can be thought of as corresponding to the trajectory of a collection of test particles. On the other hand, a geodesic bundle propagating through a pure multiway evolution graph can be thought of as corresponding to the trajectory of a quantum wave packet (since each branch of the multiway evolution graph is analogous to the evolution history of a single quantum eigenstate, and therefore a collection of multiway branches corresponds to the evolution history of some linear superposition of those eigenstates).

Since the multiway causal graph reflects both the structure of the multiway evolution graph and the structure of all of its associated causal graphs, any valid foliation of the multiway causal graph will consist of a time-ordered sequence of hypersurfaces, each of which will contain pairs of updating events that can be either spacelike-separated, branchlike-separated, or some combination of the two, with the type of separation depending upon the particular choice of foliation. Therefore, an observer who is embedded within a particular multiway causal foliation will, in general, find it impossible to determine whether a geodesic bundle propagating through the multiway causal graph corresponds to the evolution of a collection of test particles, a wave packet, or both (since the geodesics themselves will either appear to be purely spacelike-separated, purely branchlike-separated, or a combination, depending on the observer). Thus, wave-particle duality is just one of many immediate consequences of the principle of multiway relativity—the preservation of timelike-orderings of updating events in the multiway causal graph, independent of the choice of foliation.

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