No, no, no. Three problems, one in the analogy and two in the probabilities.
First, an individual particle can briefly exceeed the speed of light; the group velocity cannot. Go read up on Cerenkov radiation: It’s the blue glow created by (IIRC) neutrons briefly breaking through c, then slowing down. The decrease in energy registers as emitted blue light.
Second: conditional probabilities are not necessarily given by a ratio of densities. You’re conditioning on (or working with) events of measure-zero. These puzzlers are why measure theory exists—to step around the seeming ‘inconsistencies’.
Third: The probability of a probability is superfluous. Probabilities are (thanks to Kolmogorov) just the expectation of indicator variables. Thus P(P()=1) = E(I(E(I())=1)) = 0 or 1; the randomness is all eliminated by the inside expectation.
Leave the musings on probabilities to the statisticians; they’ve already thought about these supposed paradoxes.
Cerenkov radiation: It’s the blue glow created by (IIRC) neutrons briefly breaking through c
I thought it was due to neutrons exceeding the phase velocity of light in a medium, which is invariably slower than c. The neutron is still going slower than c:
Wikipedia
While electrodynamics holds that the speed of light in a vacuum is a universal constant (c), the speed at which light propagates in a material may be significantly less than c. For example, the speed of the propagation of light in water is only 0.75c. Matter can be accelerated beyond this speed (although still to less than c) during nuclear reactions and in particle accelerators. Cherenkov radiation results when a charged particle, most commonly an electron, travels through a dielectric (electrically polarizable) medium with a speed greater than that at which light propagates in the same medium.
No, no, no. Three problems, one in the analogy and two in the probabilities.
First, an individual particle can briefly exceeed the speed of light; the group velocity cannot. Go read up on Cerenkov radiation: It’s the blue glow created by (IIRC) neutrons briefly breaking through c, then slowing down. The decrease in energy registers as emitted blue light.
Second: conditional probabilities are not necessarily given by a ratio of densities. You’re conditioning on (or working with) events of measure-zero. These puzzlers are why measure theory exists—to step around the seeming ‘inconsistencies’.
Third: The probability of a probability is superfluous. Probabilities are (thanks to Kolmogorov) just the expectation of indicator variables. Thus P(P()=1) = E(I(E(I())=1)) = 0 or 1; the randomness is all eliminated by the inside expectation.
Leave the musings on probabilities to the statisticians; they’ve already thought about these supposed paradoxes.
I thought it was due to neutrons exceeding the phase velocity of light in a medium, which is invariably slower than c. The neutron is still going slower than c:
Wikipedia