Special Relativity | Part IB, 2003

Prove that the two-dimensional Lorentz transformation can be written in the form

x=xcoshϕctsinhϕct=xsinhϕ+ctcoshϕ\begin{aligned} x^{\prime} &=x \cosh \phi-c t \sinh \phi \\ c t^{\prime} &=-x \sinh \phi+c t \cosh \phi \end{aligned}

where tanhϕ=v/c\tanh \phi=v / c. Hence, show that

x+ct=eϕ(x+ct)xct=eϕ(xct)\begin{aligned} &x^{\prime}+c t^{\prime}=e^{-\phi}(x+c t) \\ &x^{\prime}-c t^{\prime}=e^{\phi}(x-c t) \end{aligned}

Given that frame SS^{\prime} has speed vv with respect to SS and SS^{\prime \prime} has speed vv^{\prime} with respect to SS^{\prime}, use this formalism to find the speed vv^{\prime \prime} of SS^{\prime \prime} with respect to SS.

[Hint: rotation through a hyperbolic angle ϕ\phi, followed by rotation through ϕ\phi^{\prime}, is equivalent to rotation through ϕ+ϕ\phi+\phi^{\prime}.]

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