mburns's Journal: When Geometry Fails

Mathematics is contingent. Even mathematics with the best expressiveness and possessing the most thorough of observer lock-ins can go out of range. Its premises can fail so that even locked-in observers can see some consequences of the different mathematics that can perform contrary to the original in its absence.

Spacetime geometry seems to be the most expressive of mathematical systems that is yet not self symbolic. It has the most elegant of observer lock-ins in its Bianchi identities, which lead directly to the conservation "laws", that are actually corollaries of the identities.

But still there are visible failures to the idea of spacetime, namely the existence of solid surfaces and colored light, that point to the relevance of some other system of mathematics where the premise of spacetime fails on the small scale.

On the very largest scale, consider that these particular failures become much less notable. On the smallest scale, there is a literal failure of spacetime; the logically contrary system of quantum mechanics takes over where it can. But on the intermediate scale, quantum mechanics can not contradict spacetime where the premise of spacetime holds firm. It is just the symptoms of quantum mechanics that are not contrary - solid surfaces and colored light - that can appear to an otherwise locked-in observer.

I grudgingly concede to Niels Bohr the importance of his complementarity principle. The concession is grudging because his interpretation of this principle and of the uncertainty principle is abusive, an attack on the utility to humanity of analysis and understanding that does not follow from the data, and apparently comes from a motive of a religious style of mystification.

So spacetime's premise, that observers can be made to agree on the magnitude of any given spacetime interval, must give way to some alternative mathematical system when this magnitude can not be established. On the small scale is one instance; tachyonic observers and observers beyond event horizons are more.

It is best to set up a particular duality and uncertainty principle by identifying another mathematical system that comes into range on its own when spacetime fails. But I have no special ideas on how to do this. I do think that the categorical proposition interpretation of quantum states, studied by Joseph Jauch, is the most relevant of the other systems. It is not obvious that a wave equation will serve as a dual theory here because it would not be free from the presumption of a metric; its necessity should be proved from the intersection of spacetime and quantum mechanics. The original uncertainty principle is not obvious here, neither is any modification of it.

Let's doubt the need for double relativity here. Side effects on the intermediate and large scale that actually break spacetime by varying the speed of light should be proved. I have already criticized inflation and the cosmological constant because they break the premise of spacetime, the metric.

So, without establishing a duality, let me expound on how the premise of spacetime can fail on the small scale. It is just there, on the small scale, where observers can not catch failures of the metric, the essence of spacetime. You can think of these failures as little tabs torn out of or inserted into the fabric of spacetime. These tabs represent an increased or decreased interval, in one or both directions and in parallel or orthogonal directions, as compared to their surrounding and parallel intervals.

These inserted or deleted tabs violate the Bianchi identities of spacetime, so they are not conserved like ordinary dilations and contractions are; the tabs do not persist in the relativistic direction of their alignment. If they did persist, then they would not be failures of spacetime. The Bianchi identities, after all, have the conservation "laws" as easy corollaries, and the existence of the spacetime metric is more than enough to prove the Bianchi identities.

So how do these tabs relate to the uncertainty principle, especially over long ranges? It seems that the original uncertainty principle and the correspondence principle imply that these tab failures actively balance out over large spacetime intervals. There is no need or mechanism for the spacetime metric to fail when it is within its natural domain.

Do I even have the right dimensionality? Planck's constant has the geometric dimensionality of L^2, so there is no standard size for these tabs, of dimensionality L. I must mention Lee Smolin's writing on quantized areas here, with their correspondence to quantum transactions.

And there are systems that might be made to emulate spacetime on the intermediate scale, but that are built on different premises, and may differ from spacetime on the small scale. By Spinoza's principle of noncensorship, these would coexist with spacetime where they agree. The logical agreement of quantum states that are in spacelike relations and the logical disagreement of quantum states that are in timelike relations makes one such system. I would like to retrieve a spacetime metric from this system.

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Michael J. Burns