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Given a flat region of spacetime as set $\mathcal S$ of events together with values of spacetime intervals (up to a common non-zero constant) for each pair of events, $s^2 : \mathcal S \times \mathcal S \rightarrow \mathbb R$,
and considering three (distinct) participants $A$, $B$, and $P$ contained in this region, such that

  • $A$ and $B$ had been and remained at rest to each other,
  • $A$ and $P$ had been passing each other (i.e. as coincidence event $\varepsilon_{AP} \in \mathcal S)$, and
  • $B$ and $P$ had been passing each other (i.e. as coincidence event $\varepsilon_{BP} \in \mathcal S$),

how can $\| \overline{ \mathbf v}_{AB}[~P~] \|$, i.e. the average speed of $P$'s motion with respect to $A$ and $B$, be expressed in terms of the given spacetime interval values ?

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user12262
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1 Answers1

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In reference to the given participants $A$ and $B$ there can be another participant $M$ (uniquely) identified as the "middle between" $A$ and $B$ by the conditions that

  • $M$ and $A$ had been and remained at rest to each other (which can be expressed in terms of spacetime interval values, following the prescription linked in the OP), likewise

  • $M$ and $B$ had been and remained at rest to each other,

  • for each event $\varepsilon_{AP} \in \mathcal E_A$ (i.e. in which $A$ and $P$ took part) there exist events $\varepsilon_{AS},\varepsilon_{AV} \in \mathcal E_A$, $\varepsilon_{MR},\varepsilon_{MU} \in \mathcal E_M$, and $\varepsilon_{BT} \in \mathcal E_B$, such that $$s^2[~\varepsilon_{AP}, \varepsilon_{MR}~] = 0, \qquad s^2[~\varepsilon_{AP}, \varepsilon_{BT}~] = 0, \qquad s^2[~\varepsilon_{MR}, \varepsilon_{AS}~] = 0,$$ $$s^2[~\varepsilon_{AS}, \varepsilon_{MU}~] = 0, \qquad s^2[~\varepsilon_{MU}, \varepsilon_{AV}~] = 0, \qquad s^2[~\varepsilon_{BT}, \varepsilon_{AV}~] = 0,$$ $$s^2[~\varepsilon_{AP}, \varepsilon_{AS}~] = s^2[~\varepsilon_{AS}, \varepsilon_{AV}~],$$

  • likewise, for each event $\varepsilon_{BQ} \in \mathcal E_B$ there exist events $\varepsilon_{BT}, \varepsilon_{BW} \in \mathcal E_B$, $\varepsilon_{MR},\varepsilon_{MU} \in \mathcal E_M$, and $\varepsilon_{AS} \in \mathcal E_A$, such that $$s^2[~\varepsilon_{BQ}, \varepsilon_{MR}~] = 0, \qquad s^2[~\varepsilon_{BQ}, \varepsilon_{AS}~] = 0, \qquad s^2[~\varepsilon_{MR}, \varepsilon_{BT}~] = 0,$$ $$s^2[~\varepsilon_{BT}, \varepsilon_{MU}~] = 0, \qquad s^2[~\varepsilon_{MU}, \varepsilon_{BW}~] = 0, \qquad s^2[~\varepsilon_{AS}, \varepsilon_{BW}~] = 0,$$ $$s^2[~\varepsilon_{BQ}, \varepsilon_{BT}~] = s^2[~\varepsilon_{BT}, \varepsilon_{BW}~],$$

  • and for each event $\varepsilon_{MR}, \in \mathcal E_M$ there exist events $\varepsilon_{MU}, \in \mathcal E_M$, $\varepsilon_{AS}, \in \mathcal E_A$, and $\varepsilon_{BT}, \in \mathcal E_B$ such that $$s^2[~\varepsilon_{MR}, \varepsilon_{AS}~] = 0, \qquad s^2[~\varepsilon_{AS}, \varepsilon_{MU}~] = 0,$$ $$s^2[~\varepsilon_{MR}, \varepsilon_{BT}~] = 0, \qquad s^2[~\varepsilon_{BT}, \varepsilon_{MU}~] = 0.$$

With the given event $\varepsilon_{AP}$, the therefore (according to the conditions sketched above) identified events $\varepsilon_{AS}$ and $\varepsilon_{BT}$, and along with the given event $\varepsilon_{BP}$ the average speed of $P$ wrt. (the system to which belong) $A$ and $B$ can be expressed as

$$\| \overline{ \mathbf v }_{AB}[~P~] \| := $$ $$c~\sqrt{\frac{ s^2[~\varepsilon_{AP}, \varepsilon_{AS}~] }{ s^2[~\varepsilon_{AP}, \varepsilon_{AS}~] + 2~\text{sgn}[~s^2[~\varepsilon_{AP}, \varepsilon_{AS}~]~]~\sqrt{s^2[~\varepsilon_{AP}, \varepsilon_{AS}~] ~ s^2[~\varepsilon_{BT}, \varepsilon_{BP}~] } + s^2[~\varepsilon_{BT}, \varepsilon_{BP}~]} },$$

where the case is understood that $(s^2[~\varepsilon_{BT}, \varepsilon_{BP}~])^2 < (s^2[~\varepsilon_{AP}, \varepsilon_{BP}~])^2$
and of course $\text{sgn}[~s^2[~\varepsilon_{AP}, \varepsilon_{AS}~]~] = \text{sgn}[~s^2[~\varepsilon_{BT}, \varepsilon_{BP}~]~]$.

user12262
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