As I understand it, the value of a 4-vector [itex]x[/itex] in another reference frame ([itex]x'[/itex]) with the same orientation can be derived using the Lorentz boost matrix [itex]\bf{\lambda}[/itex] by [itex]x'=\lambda x[/itex]. More explicitly,(adsbygoogle = window.adsbygoogle || []).push({});

$$\begin{bmatrix}

x'_0\\

x'_1\\

x'_2\\

x'_3\\

\end{bmatrix}

=

\begin{bmatrix}

\lambda_{00}&\lambda_{01}&\lambda_{02}&\lambda_{03}\\

\lambda_{10}&\lambda_{11}&\lambda_{12}&\lambda_{13}\\

\lambda_{20}&\lambda_{21}&\lambda_{22}&\lambda_{23}\\

\lambda_{30}&\lambda_{31}&\lambda_{32}&\lambda_{33}\\

\end{bmatrix}

\begin{bmatrix}

x_0\\

x_1\\

x_2\\

x_3\\

\end{bmatrix}

$$

I have seen examples of these components written in terms of [itex]\beta[/itex] and [itex]\gamma[/itex], which are defined as

$$\beta=\frac{v}{c}$$

$$\gamma=\frac{1}{\sqrt{1-\beta\cdot\beta}}$$

where [itex]v[/itex] is the 3-velocity and [itex]c[/itex] is the speed of light. My question is this: How can the components of [itex]\lambda[/itex] be written in terms of the 4-velocity [itex]U[/itex] alone?

I know that [itex]U_0=\gamma c[/itex] and [itex]U_i=\gamma v_i=\gamma c\beta_i[/itex] for [itex]i\in\{1,2,3\}[/itex], but I'm having trouble deriving the components for [itex]\lambda[/itex] using the matrices based on [itex]\beta[/itex] and [itex]\gamma[/itex]. An example of one of these matrices can be found at Wikipedia. How can I rewrite this matrix in terms of [itex]U[/itex] alone?

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# Lorentz boost matrix in terms of four-velocity

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