Physics of coordinate transformations In Special Relativity, coordinate transformations between inertial frames are encoded by the Lorentz transformations. Coordinate transformations are changes in the labelling of events, however the Lorentz transformations encode information about physical effects, such as length contraction and time dilation. How can the re-labelling of events lead to the emergence of new physical phenomena?
 A: If there's a line in the Euclidean plane that is parallel to the $x$ axis, with tick marks on it that are one unit apart, then in a primed coordinate system that is rotated by $θ$ relative to the unprimed coordinates, the $x'$ coordinates of the tick marks are $\cos θ$ units apart. If you express the rotation as a slope, $m = \tan θ$, instead of an angle, then the tick marks are $1/\sqrt{1+m^2}$ units apart.
If there's a line in the Minkowskian plane that is parallel to the $t$ axis, with tick marks on it that are one unit apart, then in a primed coordinate system that is rotated by a rapidity $α$, the $t'$ coordinates of the tick marks are $\cosh α$ units apart. If you express the rotation as a slope, $v = \tanh α$, instead of an angle, then the tick marks are $1/\sqrt{1-v^2}$ units apart.
These two examples are almost exactly the same; the minor differences arise from the flipped sign in the Minkowskian version of the Pythagorean theorem. The Minkowskian case is called time dilation.
Length contraction is similar: if you have a strip parallel to the $x$ axis whose width along the $y$ axis is 1 unit, then its width along the $y'$ axis is $\sec θ = \sqrt{1+m^2}$, and if you have a strip parallel to the $t$ axis whose width along the $x$ axis is 1 unit, then its width along the $x'$ axis is $\mathrm{sech}\,α = \sqrt{1-v^2}$.
The Euclidean version of the twin paradox is the triangle inequality. The Euclidean version of the Ehrenfest paradox is that if you want to cover an infinite cylinder with strips of wallpaper of infinite length and finite width, you need fewer of them if they're at an angle than if they're straight. (In the most extreme case, you can cover the cylinder with just one strip if you wrap it in a tight spiral.)
Are these physical phenomena? I suppose it's a matter of opinion. The Minkowskian ones are exactly as physical as the Euclidean ones.
