In school we are taught that Lorentz transformations relate coordinates of observers in uniform motion. Later we are taught that nonlinear coordinate transformations are associated with acceleration. But what about linear non-Lorentz transformations? For example, consider the Galilean transformation of the Minkowski metric

$${\rm d}s^2 = -{\rm d}t^2 + {\rm d}x^2$$ In terms of Galilean-boosted coordinates,

$x’ = x - vt$

$t’ = t$

we obtain $$ {\rm d}s^2 = -{\rm d}t’^2 + ({\rm d}x’ + v {\rm d}t’)^2 $$ which is a perfectly respectable coordinate transformation of Minkowski provided that $|v|<1$. Indeed, in terms of ADM variables it corresponds to choosing a lapse of $1$ and a shift of $v$. On the other hand an object tied to the origin $x’=0$ is clearly not accelerating.

What then is the physical interpretation of such coordinate transformations?



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