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Assume a four-vector in the most general case in frame $\Sigma$: $$ \begin{bmatrix} t \\ x(t,y,z) \\ y(t,x,z) \\ z(t,x,y) \end{bmatrix} $$ This can be a four-velocity, four-acceleration or other things. (given that the derivatives of $x,y,z$ with respect to time or anything else would result in a function of those again this can be interpreted as any general four-vector). Now if the frame $\Sigma'$ is moving with velocity $\vec v$ in three dimensional space with respect to the initial frame, what is the lorentz transformation resulting $t',x',y',z'$?

NOTE: I don't want to rotate the initial system so that its x-axis aligns with the direction of the second frames velocity. I would like to have it a general way.

Can such transformation exist? And to be more general what are the steps of finding a lorentz transformation for a given four-vector?

I came across a general boost function but that seems to be only for four-velocity.

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  • $\begingroup$ It exists, but is very ugly and does not teach you any useful physics for all that work. Why are you trying to find it? You should be able to find it online somewhere if you really look for it. $\endgroup$
    – naturallyInconsistent
    May 7 at 11:13
  • $\begingroup$ I am trying to look for how do we derive such transformation, any source I found would just consider it along the x axis and find the simple transformations @naturallyInconsistent $\endgroup$
    – Kid A
    May 7 at 11:15
  • $\begingroup$ Usually it is z axis. There are those that directly boost in arbitrary axis, and there are those that apply Euler rotations with one simple boost. The solution simply exists out there. $\endgroup$
    – naturallyInconsistent
    May 7 at 11:19

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In units where $c=1$ a Lorentz boost in a general direction is: $$\left( \begin{array}{cccc} \gamma & -\gamma v_x & -\gamma v_y & -\gamma v_z \\ -\gamma v_x & \frac{(\gamma -1) v_x^2}{v^2}+1 & \frac{(\gamma -1) v_x v_y}{v^2} & \frac{(\gamma -1) v_x v_z}{v^2} \\ -\gamma v_y & \frac{(\gamma -1) v_x v_y}{v^2} & \frac{(\gamma -1) v_y^2}{v^2}+1 & \frac{(\gamma -1) v_y v_z}{v^2} \\ -\gamma v_z & \frac{(\gamma -1) v_x v_z}{v^2} & \frac{(\gamma -1) v_y v_z}{v^2} & \frac{(\gamma -1) v_z^2}{v^2}+1 \\ \end{array} \right)$$

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  • $\begingroup$ Thank you for the response, is this transformation true for any four-vector? $\endgroup$
    – Kid A
    May 11 at 8:17
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    $\begingroup$ @KidA yes, all four vectors transform the same $\endgroup$
    – Dale
    May 11 at 10:57

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