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In the system of units where c=1, the Lorentz transformations are as follows:
## x'=\gamma (x-vt) \\ t'=\gamma (t-vx) ##
In the limit ## v \ll 1 ##, we have ## \gamma \approx 1+\frac 1 2 v^2 ##, so we have, in this limit:
## x' \approx (1+\frac 1 2 v^2)(x-vt)=x-vt+\frac 1 2 v^2 x-\frac 1 2 v^3 t ##
## t' \approx (1+\frac 1 2 v^2)(t-vx)=t-vx+\frac 1 2 v^2 t-\frac 1 2 v^3 x ##
Now if we just keep terms first order in v, we'll get:
## x' \approx x-vt \\ t'\approx t-vx ##
But this is a problem because we are promised to get Galilean transformations because of the correspondence principle!
What's wrong here?
Thanks
## x'=\gamma (x-vt) \\ t'=\gamma (t-vx) ##
In the limit ## v \ll 1 ##, we have ## \gamma \approx 1+\frac 1 2 v^2 ##, so we have, in this limit:
## x' \approx (1+\frac 1 2 v^2)(x-vt)=x-vt+\frac 1 2 v^2 x-\frac 1 2 v^3 t ##
## t' \approx (1+\frac 1 2 v^2)(t-vx)=t-vx+\frac 1 2 v^2 t-\frac 1 2 v^3 x ##
Now if we just keep terms first order in v, we'll get:
## x' \approx x-vt \\ t'\approx t-vx ##
But this is a problem because we are promised to get Galilean transformations because of the correspondence principle!
What's wrong here?
Thanks