I do not know how much of this is true but when people want to go into space. It is interesting to watch.
That first video uses a kind of annoying form of the Lorentz transformation
x' = γ(x - vt)
t' = γ(t - vx/c^2)
with γ=1/(1-v^2/c^2)^(1/2)
It's more clear to take β=v/c and write
ct' = γ(c t - β x)
x' = γ(x - β ct)
to get everything in units of distance. Or even better just make time measured in meters so that the speed of light is
c = (1 meter)/(1 meter [of light travel time]) = 1
and it becomes
t' = γ(t - β x)
x' = γ(x - β t)
with γ=1/(1 - β^2)^(1/2)
and you really see the symmetry. Although the easiest way to use the Lorentz transformations is to write the speed β = tanh θ for the primed frame relative to the unprimed frame so that we get
t' = t cosh θ - x sinh θ
x' = x cosh θ - t sinh θ
which really makes clear the hyperbolic geometry of spacetime while also giving an easier way to do calculations most of the time. Makes it look almost like the hyperbolic geometry version of a rotation in Euclidean geometry and encodes information about the spacetime interval (the spacetime analog to distance in Euclidean geometry) in it.
Anyways, that first video isn't a paradox any more than Zeno's Paradox, which essentially argues that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = ∞ when really 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1 and that Lorentz transformation above makes all the crazy stuff in special relativity make sense.
What's really cool is you can use special relativity to mathematically prove that a wire carrying a current absolutely must induce a magnetic field without having to know any theory whatsoever about magnetism. You can mathematically derive it just using the laws of electrostatics (eg electric fields from charged particles at rest) and the Lorentz transformation. Purcell's awesome Electricity & Magnetism book shows it and it's one of the most incredible things I have ever read.
Einstein's original paper "On the Electrodynamics of Moving Bodies" is a pretty cool read and even derives the doppler effect for light. Then his next paper is like two or three pages long and he derives the rest energy E=mc^2 from the doppler effect.
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