Frame of simultaneity and of cospatiality

We have seen that for a space-type pair there exists a frame in which the two events are simultaneous, and for a time-type pair a frame in which they are cospatial (same point). What remains is a quantitative question: how fast does that frame move relative to the laboratory? The answer is geometrically transparent in the Minkowski plane.

Speed of the frame of simultaneity (space-type events)

If $\{E_1,E_2\}$ is space-type, there exists a frame in which $\Delta t'=0$. In the lab’s $(t,s)$ plane, events simultaneous in $\text{SR}'$ lie along lines parallel to the spatial axis of $\text{SR}'$. By the Lorentz transformations, the $s'$ axis (the set of events at $t'=0$ in $\text{SR}'$) has, in the lab plane, equation $t = v\,s$ (in units $c=1$), i.e. a line of slope $1/v$ relative to the $t$ axis. Imposing that this line passes through the two events immediately gives the required speed:

$\ev{v_\text{simult} = \frac{\Delta t}{\Delta s}}$

valid for space-type events, for which $|\Delta t/\Delta s|<1$. The ratio $\Delta t/\Delta s$ is precisely the inverse of the “apparent propagation speed” between the two events; and $|v_\text{simult}|<1$ always holds, as is necessary for such a frame to genuinely exist.

Speed of the frame of cospatiality (time-type events)

If $\{E_1,E_2\}$ is time-type, there exists a frame in which $\Delta s'=0$: the two events happen at the same point. It is found immediately, because it must be the frame of an observer moving with uniform motion precisely from $E_1$ to $E_2$:

$\ev{v_\text{cosp} = \frac{\Delta s}{\Delta t}}$

valid for time-type events, for which $|\Delta s/\Delta t|<1$. Geometrically, the time axis of $\text{SR}'$ is precisely the line joining $E_1$ and $E_2$ in the lab plane: a tilted line passing through the two events.

The two formulae are each other’s inverse, and this is no coincidence: it reflects the geometric duality between the spatial axis and the time axis when passing from one inertial frame to another. For a space-type pair it makes sense to ask where to make it simultaneous; for a time-type pair it makes sense to ask in what motion to make it cospatial. Trying to swap the two cases would lead to a speed greater than $c$, i.e. to a non-existent frame — the algebraic way in which the theory reminds itself which question is legitimate.

Links

Topics: Special relativity Concepts: Relativity of simultaneity · Spacetime invariant Skills: Changing reference frame Methods: Minkowski diagram

Related exercises: Space-type or time-type · Relativistic western duel · No signals faster than light