- #1
CarlosMarti12
- 8
- 0
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,
$$\begin{bmatrix}
x'_0\\
x'_1\\
x'_2\\
x'_3\\
\end{bmatrix}
=
\begin{bmatrix}
\lambda_{00}&\lambda_{01}&\lambda_{02}&\lambda_{03}\\
\lambda_{10}&\lambda_{11}&\lambda_{12}&\lambda_{13}\\
\lambda_{20}&\lambda_{21}&\lambda_{22}&\lambda_{23}\\
\lambda_{30}&\lambda_{31}&\lambda_{32}&\lambda_{33}\\
\end{bmatrix}
\begin{bmatrix}
x_0\\
x_1\\
x_2\\
x_3\\
\end{bmatrix}
$$
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?
$$\begin{bmatrix}
x'_0\\
x'_1\\
x'_2\\
x'_3\\
\end{bmatrix}
=
\begin{bmatrix}
\lambda_{00}&\lambda_{01}&\lambda_{02}&\lambda_{03}\\
\lambda_{10}&\lambda_{11}&\lambda_{12}&\lambda_{13}\\
\lambda_{20}&\lambda_{21}&\lambda_{22}&\lambda_{23}\\
\lambda_{30}&\lambda_{31}&\lambda_{32}&\lambda_{33}\\
\end{bmatrix}
\begin{bmatrix}
x_0\\
x_1\\
x_2\\
x_3\\
\end{bmatrix}
$$
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?