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When the boost velocity ${\displaystyle {\boldsymbol {v}}}$ is in an arbitrary vector direction with the boost vector ${\displaystyle {\boldsymbol {\beta }}={\boldsymbol {v}}/c}$, then the transformation from an unprimed spacetime coordinate system to a primed coordinate system is given by [1]

$${\displaystyle {\begin{bmatrix}ct'\\\\\\\\x'\\\\\\\\y'\\\\\\z'\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{x}&-\gamma \beta _{y}&-\gamma \beta _{z}\\-\gamma \beta _{x}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}\\-\gamma \beta _{y}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{y}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}\\-\gamma \beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{x}\beta _{z}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{y}\beta _{z}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{z}^{2}\\\end{bmatrix}}{\begin{bmatrix}ct\\\\\\x\\\\\\\\y\\\\\\\\z\end{bmatrix}},}$$

where the Lorentz factor is ${\displaystyle \gamma =1/{\sqrt {1-{\boldsymbol {\beta }}^{2}}}}$. The determinant of the transformation matrix is +1 and its trace is ${\displaystyle 2(1+\gamma )}$. The inverse of the transformation is got by reversing the sign of ${\displaystyle {\boldsymbol {\beta }}}$.

I have added the expression for the general boost direction. Xxanthippe (talk) 06:20, 8 September 2024 (UTC). determinant

$${\displaystyle {\begin{bmatrix}ct'{\vphantom {-\gamma \beta _{\text{x}}}}\\x'{\vphantom {1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}^{2}}}\\y'{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}}}\\z'{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma \beta _{\text{x}}&-\gamma \beta _{\text{y}}&-\gamma \beta _{\text{z}}\\-\gamma \beta _{\text{x}}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{z}}\\-\gamma \beta _{\text{y}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}^{2}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}\\-\gamma \beta _{\text{z}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{z}}&{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}&1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{z}}^{2}\\\end{bmatrix}}{\begin{bmatrix}ct{\vphantom {-\gamma \beta _{\text{x}}}}\\x{\vphantom {1+{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}^{2}}}\\y{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{x}}\beta _{\text{y}}}}\\z{\vphantom {{\frac {\gamma ^{2}}{1+\gamma }}\beta _{\text{y}}\beta _{\text{z}}}}\end{bmatrix}},}$$ where the Lorentz factor is ${\displaystyle \gamma =1/{\sqrt {1-{\boldsymbol {\beta }}^{2}}}}$.

The quantity ${\displaystyle c^{2}t^{2}-x^{2}-y^{2}-z^{2}}$ is invariant under the transformation.

The eigenstates of the [ct,x] transformation are eigenvector [1, 1] with eigenvalue ${\displaystyle {\sqrt {(1-\beta )/(1+\beta )}}}$, and eigenvector [1, −1] with eigenvalue ${\displaystyle {\sqrt {(1+\beta )/(1-\beta )}}}$.

In the case of three spatial dimensions [ct,x,y,z], where the boost ${\displaystyle \beta }$ is in the x direction, the eigenstates of the transformation are [1,1,0,0] with eigenvalue ${\displaystyle {\sqrt {(1-\beta )/(1+\beta )}}}$, [1, −1,0,0] with eigenvalue ${\displaystyle {\sqrt {(1+\beta )/(1-\beta )}}}$, and [0,0,1,0] and [0,0,0,1], the latter two with eigenvalue 1.