i am reading Lillian R. Lieber's book on the einstein theory of relativity and i am a bit confused on page 65. she wants to take the equations:(adsbygoogle = window.adsbygoogle || []).push({});

x=x'cosθ - y'sinθ

y=x'sinθ + y'cosθ

and compare them to:

x'=β(x-vt)

t'=β(t-vx/c^{2})

she takes c as one so:

x'=β(x-vt)

t'=β(t-vx)

she solves for x and t and gets:

x=β(x'+vt')

t=β(t'+vx')

then she replaces t with iτ and t' with iτ' and she gets:

x=β(x'+vt')

iτ = iβτ' + βvx'

x=β(x'+vt')

τ=βτ' + iβvx'

next is the part i am confused about. she sets β = cosθ and -iβv = sinθ. this nicely turns the Lorentz equations into:

x=x'cosθ - τ'sinθ

τ=x'sinθ + τ'cosθ

what i don't understand is how did she choose -iβv = sinθ? it works out all nicely in the end but how did she know that sinθ had to equal -iβv? was it arbitrary as a result of trial and error or did she use β = cosθ in order to figure out that -iβv = sinθ?

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# Question on lorentz transformation equations

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