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In school, or even in university, we are only taught Lorentz transformation in one direction. It is quite easy: $$\begin{align} t' &= \gamma \left( t - \frac{vx}{c^2} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{align}$$

However, when it comes to any direction:

$$\begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \gamma&-\gamma\,\beta_x&-\gamma\,\beta_y&-\gamma\,\beta_z\\ -\gamma\,\beta_x&1+(\gamma-1)\dfrac{\beta_x^2}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_y}{\beta^2}&(\gamma-1)\dfrac{\beta_x \beta_z}{\beta^2}\\ -\gamma\,\beta_y&(\gamma-1)\dfrac{\beta_y \beta_x}{\beta^2}&1+(\gamma-1)\dfrac{\beta_y^2}{\beta^2}&(\gamma-1)\dfrac{\beta_y \beta_z}{\beta^2}\\ -\gamma\,\beta_z&(\gamma-1)\dfrac{\beta_z \beta_x}{\beta^2}&(\gamma-1)\dfrac{\beta_z \beta_y}{\beta^2}&1+(\gamma-1)\dfrac{\beta_z^2}{\beta^2}\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\,.$$

I give up.

A friend of mine ask what happen if we travel around Earth. Of course we can't due to the massive we have, but let's say we are light, what would happen? Is there anyone who has solved this problem? Searching with some keywords like light travel around Earth gives me unrelated results, however I find one interesting result proving that it is possible for light can bend into an arc (I haven't read all the article thought).

Here I list some thoughts:

  • Earth will always be an ellipsoid shape, the long axis always parallel with your direction.
  • Time always be freeze, as usual. However, you can always check your time with the clock in Greenwich. This will lead to a paradox. If you travel, say 10000 times, won't you see the Greenwich clock move? If it moves, doesn't it prove that your time isn't freeze?
  • If you are massive, then of course you can only travel just below light speed. By then, Earth will be extremely massive since your angular acceleration will be identify as gravitational force.
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  • $\begingroup$ before undertaking the problem of what happens when traveling around the Earth, I suggest you to go step by step, i.e. first calculate the value of $\gamma$ for the case $v \to c$. $\endgroup$ – Sofia Feb 8 '15 at 15:38
  • $\begingroup$ the Lorentz factor is still the same as the one direction problem, right? Therefore it's $\gamma\rightarrow\infty$ $\endgroup$ – Ooker Feb 8 '15 at 16:00
  • $\begingroup$ You might be interested in my answer to Can a ultracentrifuge be used to test general relativity? because I treat the case of relativistic motion in a circle. $\endgroup$ – John Rennie Feb 8 '15 at 16:00
  • $\begingroup$ The problems both for special and general relativity have been solved for the GPS system , which are satellites going around the earth. astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html $\endgroup$ – anna v Feb 8 '15 at 16:16

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