If three events are (pairwise) spacelike related to each other then their interval ratios do not necessarily satisfy the triangle inequality. (For example, consider the case that two of the three event pairs are "almost lightlike".)

But what about three events which are (pairwise) spacelike related and which are moreover constrained "on the same lightcone" -- are their interval ratios guaranteed to satisfy the triangle inequality?

To be concrete, consider the

  • three spacelike separated events $$\varepsilon_{A B}, \qquad \varepsilon_{J K}, \qquad \varepsilon_{P Q}$$ (where there is no one participant in any two of these events),

  • one signal event $\varepsilon_{S U}$ (with signal source $S$ participating) and

  • one ping event $\varepsilon_{S W}$ (with $S$ participating, too, whereby the events $\varepsilon_{S U}$ and $\varepsilon_{S W}$ are timelike related to each other),

with interval ratios assigned such that

$$\frac{s^2[~\varepsilon_{A B}, \varepsilon_{J K}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} \lt 0, \quad \frac{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} \lt 0, \quad \frac{s^2[~\varepsilon_{J K}, \varepsilon_{P Q}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} \lt 0,$$

and such that all other pairs among these events are lightlike related to each other, therefore with interval ratios

$$\begin{array}[rcl] ~ \frac{s^2[~\varepsilon_{A B}, \varepsilon_{S U}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} = \frac{s^2[~\varepsilon_{A B}, \varepsilon_{S W}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} & = & ~ \cr \frac{s^2[~\varepsilon_{J K}, \varepsilon_{S U}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} = \frac{s^2[~\varepsilon_{J K}, \varepsilon_{S W}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} & = & ~ \cr \frac{s^2[~\varepsilon_{P Q}, \varepsilon_{S U}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} = \frac{s^2[~\varepsilon_{P Q}, \varepsilon_{S W}~]}{s^2[~\varepsilon_{S W}, \varepsilon_{S U}~]} & = & 0. \end{array}$$

Then, do the interval ratios $\frac{s^2[~\varepsilon_{A B}, \varepsilon_{J K}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]}$ and $\frac{s^2[~\varepsilon_{J K}, \varepsilon_{P Q}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]}$ satisfy the triangle inequality (in Heron's form):

$$\begin{array}[lr] ~ 2~\frac{s^2[~\varepsilon_{A B}, \varepsilon_{J K}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]} + 2~\frac{s^2[~\varepsilon_{J K}, \varepsilon_{P Q}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]} + 2~\frac{s^2[~\varepsilon_{A B}, \varepsilon_{J K}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]}~\frac{s^2[~\varepsilon_{J K}, \varepsilon_{P Q}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]} & \ge \cr 1 + \left( \frac{s^2[~\varepsilon_{A B}, \varepsilon_{J K}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]} \right)^2 + \left( \frac{s^2[~\varepsilon_{J K}, \varepsilon_{P Q}~]}{s^2[~\varepsilon_{A B}, \varepsilon_{P Q}~]} \right)^2 \end{array}$$



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