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If the velocity of the train is in the $x$-direction, and $y$ is vertical, with the laser in the train frame shooting straight up. A vertically propagating light wave is described by: $$ E(x', y', t') …

Focusing on the 1st question (which is a great place to start if you're trying to wrap your head around SR): You can't add velocities, but you can add "rapidities", which is (with $c=1$): $ w = \tanh …

In the fluid frame, the light is moving at $u' = c/n$. If the lab is moving colinearly at $\pm v$, then the velocity of the light in the lab frame is: $$ u = \frac{u'\pm v}{1+\frac{vu'}{c^2}} = \frac{ …

It possible that you can't define the 2nd postulate, since according to https://math.stackexchange.com/questions/650718/does-infinite-equal-infinite $$ \infty \ne \infty $$ Regarding the 1st postulate …

Aligning the slots is indeed a problem. How do you spin the thing up? If you start from one side, then the slots are aligned if torque propagates down the axle FTL (c.f. Born Rigidity). OK, so you can …

This is really a "Do the Lorentz Transform" situation, since it is once of the easiest. In the trolly's frame ($S$), there are two events (labeled by $(ct, x)\rightarrow (t, x)$, with $c=1$): Transmis …

This question really shouldn't receive down votes (punctuation notwithstanding) since it is entirely logical for Galilean relativity. You fire a laser pulse with wavenumber $\vec k$, and then just boo …

If you are 1 light year from Earth, and the Earth time is 1/1/2020 12:00:01 AM, then you are seeing Earth's light that was emitted on 1/1/2019. If you are stationary with respect to Earth, then that l …

In the reference frame of the sail boat, no one is moving. The laser pointer at the top of the mast is pointed straight down, and to no one's surprise, the laser takes 1 second to traverse the mask an …

No. How would time dilation and space contraction work? If you are at rest in your car and you emit a flash, you see it propagating out at $c$ in a spherical shell. What if you are moving at $v=0.99 …

When once speaks of the color of light, there is a implicit reference frame. A plane wave (in the $z$-direction) takes the form: $$ E(x,y,z,t) = E_0e^{i(kz-\omega t)}=E_0e^{i\phi(z, t)}$$ where $$\phi …

So the spacetime velocity refers to the 4-velocity. Most of our favorite 3D vectors have a 4-vector counter-part, and it is this 4-vector counter part Lorentz transforms between various interial frame …

The index of refraction for gases is commonly discussed using: $$ \eta \equiv (n-1)\times 10^6 $$ The general value for air is $\eta=273$; however, if you need to consider dispersion, the accepted f …

There an many ways to look at this problem. I am going to add, light refracts because its path satisfies Fermat's Principle : "light travels between two points along the path that requires the least t …

It doesn't take any energy to counter gravity on Earth. Just stand on the ground, which applies a for $f=mg$. Integrating that force over the distance it is applied gives the work done: $$ W = \int_{z …