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The train will get over the gap. It might get a little bump against the leading wheels. The whole example originates from a very bad idea from the book by Taylor and Wheeler. Shows that even famous physicists mess up at times. One very strong criticism against the scenario is that it attempts to use SR IN THE PRESENCE of GRAVITATIONAL FORCE. This is embarrassing.

This is a miserable video that works ok on the explanation of relativity of simultaneity (the first half). The second half tries to introduce the notion of Born rigid motion and is a disaster. Imagine that the train had many wheels, all powered. Then , instead of being "pushed" in one point (in the back), it will move like a rigid rod, with no "compression". The explanations from "Adam's point of view" , fail miserable now , because the train no longer behaves like an accordion. The first case (the train hitting the front door of the tunnel while compressing against the "infinitely rigid" front door) is not only self-contradicting (there are no "infinitely rigid" objects in relativity) but also unphysical: the train will get deformed in Adam's frame while it goues through the tunnel unscathed in Sarah's frame. The "professor" screwed up royally.

Yes, it is. Problem is, it is also (very) destructive to your rubber.

UV is the ultimate "destructive agent" , so the answer is "no"

The solution is [latex]x(t)=A e^{-t \sqrt{a}}+B e^{t \sqrt{a}}[/latex] where [latex]a=\frac{k}{m}[/latex] You can get A and B from setting the initial conditions. You have them in the problem.

You needed to form the differential equation: [latex]m \frac{d^2x}{dt^2}=kx[/latex] Do you know how to set the initial conditions? Do you know how to solve differential equations?

It is explained well here: https://en.wikipedia.org/wiki/Big_Bang#Inflation_and_baryogenesis Read the next paragraph as well.

Sure, here is the earliest one: https://en.wikipedia.org/wiki/History_of_Lorentz_transformations#Ignatowski_.281910.29 There are many more: L ́evy-Leblond, J.M. (1976). One more derivation of the Lorentz transformation. American Journal of Physics, 44, 271-277 Mermin, N. D. (1984). Relativity without light.American Journal of Physics,52, 119-124 Mitvalsky, V. (1966). Special relativity without the postulate of constancy of light,American Journal of Physics, 34, 825 Pal, Palash B, (2003). Nothing but relativity.European Journal of Physics, 24,315-319 Pauli, W. (1958).Theory of Relativity, (Pergamon, Oxford) Schwartz, H. M. (1984). Deduction of the general Lorentz transformations from a set of necessary assumptions.American Journal of Physics, 52(4), 346-350 Schwartz, M. (1987). Principles of electrodynamics. Dover, New York Sela, O., Tamir, B., Dolev, S. and Elitzur, A.C. (2009). Can special relativity be derived from Galilean mechanics alone?Foundations of Physics, 39,499-509

I answered your question exactly. The answer is not "no", it is "yes". Not only that the speed light is invariant and equal to "c" in vacuum in inertial frames, it can ALSO be PROVEN that it is invariant and equal to "c" in uniformly accelerated frames and in uniformly rotating frames. These are very advanced subjects in relativity.

There are derivations of SR that do not use the axiom of light speed constancy, these papers actually derive this from the principle of relativity PLUS the outcome of the Michelson-Morley experiment.

Not clear what you are asking, can you try again?

Thunder occurs a few fractions of a second after lightning but arrives a few seconds later due to the difference in speed propagation between light and sound. See: https://en.wikipedia.org/wiki/Thunder#Calculating_distance

Yes, if the observer is rotating, he will feel the acceleration. If the electron is rotating, then the observer will feel no acceleration. This is why rotation is absolute (it can be detected with an accelerometer )

you are welcome

Rotation is absolute. As such, observer rotating around an electron is NOT equivalent with electron rotating around the observer. The premise of your OP is false.

This is not correct, I already explained what needs to be done.

You need to know the current law, see here https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws

[latex]\frac{f(x_0+\Delta x)-f(x_0-\Delta x)}{2 \Delta x}=\frac{f(x_0+\Delta x)-f(x_0)+f(x_0)-f(x_0-\Delta x)}{2 \Delta x}[/latex] "mathematic" beat me to it

These are the electrons transitioning between the energy levels WITHIN the atoms of phosphorus coating the CRT screen. NOT the electrons coming from the electron gun as "Sensei" incorrectly posted.

There are several such papers, see here: https://en.wikipedia.org/wiki/Tachyon

The ions are accelerated by a very strong electrostatic potential (of the order of 50000V)

Electrons do not emit photons, the atoms of phosphorus coating the CRT do.

This is false, atomic (cesium or rubidium) clocks do not need any reference, they ARE the reference.

It is correct the way you have it. Let me explain: 1. Light coming directly from the ion is redshifted because the source (the ion) is moving AWAY from the receiver 2. Light going into the mirror has the same frequency as the light reflected from the mirror (due to the energy conservation). So, light reflected by the mirror into the receiver is blueshifted because the source (the ion) is moving TOWARD the mirror. The IS experiment is a very clever one.

You have it right