Am I correct in thinking that if two spacetime events are coincident in one frame of reference, then they are coincident in all frames of reference, i.e. coincidence of spacetime events is a Lorentz invariant concept?
If so, is the following the correct way to prove this claim?
Let $x^{\mu}$ and $y^{\mu}$ be the coordinates of two spacetime events in some inertial frame $S$. Suppose that these events are coincident in this frame, i.e. $x^{\mu}=y^{\mu}$. Now, consider another inertial frame $S'$. In this frame the spacetime coordinates of the two events are $x'^{\mu}$ and $y'^{\mu}$ respectively. The coordinates of the events in $S'$ are related to those in $S$ by a Lorentz transformation in the following manner $x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu}$ and $y'^{\mu}=\Lambda^{\mu}_{\;\;\nu}y^{\nu}$. It follows then, that as $x^{\mu}=y^{\mu}$ in $S$, $$x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu}=\Lambda^{\mu}_{\;\;\nu}y^{\nu}=y'^{\mu}$$ and hence if the two events are coincident in $S$, then they are also coincident in $S'$. Furthermore, as these two inertial frames were chosen arbitrarily, it follows that this holds true for any inertial frame.
Is this the reason why we construct Lagrangian densities (in field theory) in terms of fields (and their first-order derivatives) at a single point in spacetime, as this is the only case in which the location of an interaction is Lorentz invariant, thus providing a Lorentz invariant notion of locality in the theory?
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