- #1

"Don't panic!"

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in which they claim that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames of reference, thus spacetime events are absolute i.e. they exist independently of any reference frame.

Is this simply to do with the fact that if two spacetime events are coincident then their spacetime interval is null, i.e. ##ds^{2}=0##, and as this is a frame independent quantity this must hold in any inertial reference frame, hence they are coincident in all inertial frames? Or does it simply follow from the fact that spacetime is modeled as a 4-D smooth manifold, and the points on a manifold exist independently of how you choose to represent them?

Is there anyway to prove the statement that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames?

One of the reasons for me asking this is to try and understand why Lagrangian densities ##\mathscr{L}## are always expressed in terms of field values (and their derivatives) at a single spacetime point? Is this simply because the Lagrangian At a given instant in time is given by $$\mathcal{L}(t)=\int d^{4}x\;\mathscr{L}(t,\mathbf{x})$$ (I know that there is no explicit dependence on spacetime points, but this is just to highlight my question) and so fields can only interact with objects at a single spatial point (or infinitesimally close to that point), as otherwise they would be separated by a spacelike interval and their interaction would correspond to an instantaneous action at a distance (prohibited by the finite speed of light)?