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## Main Question or Discussion Point

Imagine a light source, double-slit, and a curved screen in vacuum, shaped so that all parts of the interference pattern are created simultaneously. Define distance as proportional to the time light requires to reach a point. Detectors at each slit can be operating or not. Call the source S, the slits A and B, a point of constructive interference at the screen C, and a point of destructive interference at the screen D. With the detectors at slits on, one can say that the photon traveled (for example) from S to A to D, destroying the destructive interference there. The distance from S to D is given then by d(S,A) + d(A, D). However, with the slit detectors off, no light arrives at D. This implies that d(S,D) is infinity (light does not arrive in any finite time.)

The above shows a violation of the triangle inequality. d(S,A) + d(A,D) is finite, thus less than d(S,D).

Therefore I conclude that spacetime is not metrizable with this metric. However, there seems to be no reason why this metric should be any less valid than many equivalent ones in use.

The idea of spacetime not having a metric is understandably uncomfortable. Obviously, something that acts much like a metric must arise on the large scale of classical objects. We do have a perception of distance. However, it cannot be a true metric. I think this idea is usually rejected on the suspicion that all of physics would collapse if it were true. There are two closely linked ideas here which I consider important to separate.

1. “We must be able to ignore the price of tea in China while measuring the mass of an electron.” The separability of the universe into distinct non-interacting parts is essential to the functioning of physics. We cannot consider everything in the universe, we have to be able to ignore most data in order to perform any effective analysis.

2. “Spacetime locality must not be violated.” This is I suspect justified using #1, but is actually a separate assumption. Spatiotemporal proximity is

I think we have a challenge in geometry and topology, not philosophy. I'm in the minority of people who see no reason to flirt with abandonment of the notion of reality, nor that of logic. Therefore, as John Bell proved, spatiotemporal locality must go. I have attempted to construct various models of “distance” that do not satisfy the requirements of a metric. I regret my skills are not equal to those of a typical Ph.D. in physics. I offer my idea for better minds to follow if interested.

But I must ask--is this widely known? Is it part of the “oral culture” of the physics community which never gets written down? Or have I made an original contribution? I am a physics grad school dropout, because I never really understood how the physics community operates nor what I had to do to be part of it. Anyway, I have been unable to find any material that seems to follow or refute this reasoning. Please let me know what I have missed. Thank you.

The above shows a violation of the triangle inequality. d(S,A) + d(A,D) is finite, thus less than d(S,D).

Therefore I conclude that spacetime is not metrizable with this metric. However, there seems to be no reason why this metric should be any less valid than many equivalent ones in use.

The idea of spacetime not having a metric is understandably uncomfortable. Obviously, something that acts much like a metric must arise on the large scale of classical objects. We do have a perception of distance. However, it cannot be a true metric. I think this idea is usually rejected on the suspicion that all of physics would collapse if it were true. There are two closely linked ideas here which I consider important to separate.

1. “We must be able to ignore the price of tea in China while measuring the mass of an electron.” The separability of the universe into distinct non-interacting parts is essential to the functioning of physics. We cannot consider everything in the universe, we have to be able to ignore most data in order to perform any effective analysis.

2. “Spacetime locality must not be violated.” This is I suspect justified using #1, but is actually a separate assumption. Spatiotemporal proximity is

*not the only possible means*of separation of the universe into analyzable parts.I think we have a challenge in geometry and topology, not philosophy. I'm in the minority of people who see no reason to flirt with abandonment of the notion of reality, nor that of logic. Therefore, as John Bell proved, spatiotemporal locality must go. I have attempted to construct various models of “distance” that do not satisfy the requirements of a metric. I regret my skills are not equal to those of a typical Ph.D. in physics. I offer my idea for better minds to follow if interested.

But I must ask--is this widely known? Is it part of the “oral culture” of the physics community which never gets written down? Or have I made an original contribution? I am a physics grad school dropout, because I never really understood how the physics community operates nor what I had to do to be part of it. Anyway, I have been unable to find any material that seems to follow or refute this reasoning. Please let me know what I have missed. Thank you.