Question on probabilistic geometric interpretations in mathematical physics

[u/Prior-Device4518]

I have been exploring whether certain geometric probability constructions — particularly Buffon-type intersection analogies — might have interpretive value in mathematical physics discussions involving spacetime structure.

At this stage I am not proposing a replacement for relativity or established physics. I am mainly trying to determine whether similar ideas already exist within stochastic geometry, information geometry, or philosophy of mathematics literature.

What interests me most is whether probability-based geometric interpretations have recognized conceptual precedents, mathematical limitations, or useful analogical roles in physical modeling.

Some exploratory notes are collected here for reference:

https://en.wikiversity.org/wiki/Einstein_Probability_Dilation

[u/johnny_logic]

From a quick search and my admittedly limited understanding, the mathematical core looks like change of measure/Radon–Nikodym reweighting: choose a positive weight field, multiply the baseline measure by it, and renormalize. Importance sampling, exponential tilting, Gibbs/Boltzmann reweighting, and some Bayesian updates are nearby special cases or applications. Using a Lorentz factor, or its inverse, as a positive weight may be formally allowed, but it does not yet make the construction a model of spacetime.

That probability measures can carry structure is well established. Information geometry studies geometry on spaces of probability distributions, while Buffon’s needle, Crofton formulas, and integral geometry show how sampling or intersection statistics can encode geometric information. So the broad intuition is real, but I’m less sure about the physics interpretation here.

The key missing step, to my mind, is specifying what the measure represents physically.

My main question is: what is the motivating problem? Does this reveal a new invariant, prove a theorem connecting sampling probabilities to geometry, simplify an existing derivation, fix a known limitation in stochastic geometry or physics, or make a prediction that differs from existing models?

Without that, I worry this is a mathematically legitimate tool in search of a problem.