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Write down the transformation from a frame S to a frame S' moving at +0.5 c in the x direction and then to another frame S'' moving at +0.5 c in the x direction relative to S'. What is the complete transformation from S to S''? What relative speed between frames S and S'' does your answer imply?

Answer?

Well, the Lorentz transformation matrix is just

[tex]

\hat{L}=

\begin{pmatrix}

\gamma & -\gamma\beta & 0 & 0\\

-\gamma\beta & \gamma & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1\\

\end{\pmatrix}

[/tex]

Now, it happens that Lorentz matrices are closed under multiplication (ie [tex]\hat{L} \cdot \hat{L} = \hat{L_1}[/tex]). If we let

[tex]

\hat{L_1} =

\begin{pmatrix}

G & -GB & 0 & 0\\

-GB & G & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1\\

\end{pmatrix}

[/tex]

we find that [tex]B = 2\beta/(1+\beta^2)[/tex], [tex]G = (1+\beta^2)\gamma^2[/tex]. Now, since [tex]\beta = 0.5[/tex], we have that [tex]B = 0.8[/tex], so the relative speed between S and S'' would be [tex]0.8 c[/tex]. I'm a bit confused here... why wouldn't the answer just be [tex]0.5c + 0.5c = c[/tex]?

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# Homework Help: Basic Lorentz Transformation

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