- #1
demonelite123
- 219
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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:
x=x'cosθ - y'sinθ
y=x'sinθ + y'cosθ
and compare them to:
x'=β(x-vt)
t'=β(t-vx/c2)
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?
x=x'cosθ - y'sinθ
y=x'sinθ + y'cosθ
and compare them to:
x'=β(x-vt)
t'=β(t-vx/c2)
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?