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spaghetti3451
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Homework Statement
Derive the transformations ##x \rightarrow \frac{x+vt}{\sqrt{1-v^{2}}}## and ##t \rightarrow \frac{t+vx}{\sqrt{1-v^{2}}}## in perturbation theory. Start with the Galilean transformation ##x \rightarrow x+vt##. Add a transformation ##t \rightarrow t + \delta t## and solve for ##\delta t## assuming it is linear in ##x## and ##t## and preserves ##t^{2}-x^{2}## to ##\mathcal{O}(v^{2})##.Repeat for ##\delta t## and ##\delta x## to second order in ##v## and show that the result agrees with the second-order expansion of the full transformations.
Homework Equations
The Attempt at a Solution
We need to find the third term from ##x' = \frac{x+vt}{\sqrt{1-v^{2}}} = (x+vt)(1+\frac{v^{2}}{2}+\cdots)=x+vt+\frac{xv^{2}}{2}+\cdots##, and
we need to find the second and third terms from ##t' = \frac{t+vx}{\sqrt{1-v^{2}}} = (t+vx)(1+\frac{v^{2}}{2}+\cdots)=t+vx+\frac{tv^{2}}{2}+\cdots##.
Using ##x'=x+vt## and ##t'=t+\delta t, \delta t = rx+st+p##,
we have ##t'^{2}-x'^{2}=(t+rx+st+p)^{2}-(x+vt)^{2}##
##t^{2}-x^{2}=((s+1)t+rx+p)^{2}-(x+vt)^{2}##
##t^{2}-x^{2}=(s+1)^{2}t^{2}+2(s+1)(t)(rx+p)+(rx+p)^{2}-x^{2}-2xvt-(vt)^{2}##
##0=(s^{2}+2s-v^{2})t^{2}+r^{2}x^{2}+(2r(s+1)-2v)tx+2p(s+1)t+2rpx+p^{2}##
For the constant term, ##p=0##, so the terms in ##x## and ##t##.
For the term in ##x^2##, we have ##r=0##, so we get a non-zero term in ##xt##.
Where have I made the mistake?
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