Books say that special relativity is indistinguishable from Newtonian mechanics when the speed of the primed frame ($v$) is small compared to the speed of light ($c$). This is what I mean by the "Newtonian limit" property of special relativity. (I don't know the correct name for it). But the transformation of the time coordinate

$$t' = \gamma \left(t + \frac{vx}{c^2}\right)$$

involves the spatial quantity $x$ which, if large enough, could balance out the smallness of $v/c^2$. So why doesn't this observation mess up the "Newtonian limit" property?


Nice question.

Suppose that you're sitting in your lab on Earth, and you want to plan out an experiment in which you will see a large effect from the $vx/c^2$ term in the Lorentz transformation. Your lab is small, so you'll need to travel some large distance $x$ in order to make this term large. If you travel this distance at velocity $v$, then the travel time required for your experiment will be $t=x/v$. If we want the effect to be large, we will need $vx/c^2$ to be on the same order of magnitude as the time scale $t$ of our experiment. But if $vx/c^2\sim t$ and $t=x/v$, then $v\sim c$, which means that your experiment is not actually being carried out under nonrelativistic conditions.


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