(SR) Lorentz low speed approximations In Special Relativity, the standard Lorentz transformations are:
$t' = \gamma (t - \frac{vx}{c^2}) \\
x' = \gamma (x - vt) \\
y' = y \\
z' = z$
However, if we make a low speed approximation where $v \ll c$, then $\gamma \approx 1$ and we get:
$t' = t - \frac{vx}{c^2} \\
x' = x - vt \\
y' = y \\
z' = z$
These are the Galilean transformations almost, except for that remaining $vx/c^2$ which we can't ignore for large $x$. What does this term mean and what is it's physical significance?
 A: In Newtonian mechanics (using Galilean relativity) we should take the limit $c\to\infty$ so the term in the time transformation vanishes.
A little bit more formal:
Galilean spacetime (whis is treated in Newtonian mechanics) is a 4-manifold $M$ endowed with a smooth application $t: M\rightarrow\mathbb{R}$ s.t. $\textrm{grad}(t)\not=0$, uniquely defined up to affine transformations (i.e: there is an absolute time which can be measured by any observer with the same result up to an election of $t=0$ and a choice of units).
In that spacetime, two events $p,q$ are simultaneous if and only if $t(p)=t(q)$, and since $t$ is a scalar, in every frame of reference. As in SR, the set of points simultaneous to $p$ is an spatial section which divides $M$ in two regions: past and future of $p$. So two points can be connected causally if and only if $t(p)<t(q)$ or vice versa in every frame of reference. This condition requires that interactions at a distance must be simultaneous, therefore the speed of light must be infinite.
Conclusion: Newtonian mechanics is incompatible with a finite  speed of light due the existence of this application $t$, so this term must be disregarded.
On the other hand, and may be clearer: let's assume that Newtonian mechanics is, in certain approximation, compatible with finiteness of speed of light. Let be $\delta t$ the minimum amount of time which a measure can achieve, at any distance $x>c~\delta t$ the approximation fails, since two points in this spacetime cannot be simultaneous. So the "physical" interpretation of this term is that the approximation is not good enough.
A: At flippiefanus's suggestion, I am converting this from a comment to an answer:
The physical significance of the term is exactly what it looks like --- when $v$ is small and $x$ is comparably large, $t\neq t'$ (even approximately). Thus when you and I pass each other even at very slow speeds, we will still disagree substantially about the times we assign to far-distant events.
