# How to express the magnitude of proper acceleration through spacetime intervals

Given the trajectory of participant $P$ in a flat region $\mathcal S$ of spacetime through the set of events $\mathcal E_P \subset \mathcal S$ in which $P$ had taken part,
and given the values of spacetime intervals $s^2 : \mathcal E_P \times \mathcal E_P \rightarrow \mathbb R$ for all pairs of events in set $\mathcal E_P$,
is it possible to express the magnitude of proper acceleration $\mathbf a^P_Q$ of $P$, at event $\varepsilon_{PQ} \in \mathcal E_P$ in terms of the values $s^2$ ?

In order to find some suitable expression of the magnitude of the proper acceleration of participant $P$, at event $\varepsilon_{PQ}$, consider the momentarily co-moving inertial reference system; consisting specificly of participant $Q$, say, along with suitable additional participants $U$, $W$, ... (whom $P$ met and passed, in this order, subsequently to having met and passed $Q$; whereby the events $\varepsilon_{PU}, \varepsilon_{PW}, ... \in \mathcal E_P$), who remained at rest wrt. each other.

Let $\tau Q[ \, \_P, \circledS U \circ P \, ]$ denote $Q$'s duration from having indicated $P$'s passage until its indication simultaneous to $U$'s indication of $P$'s passage.

The interval between events $\varepsilon_{PQ}$ and $\varepsilon_{PU}$ can accordingly be expressed (formally) as

$$s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] := c^2 \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 - (\mathbf x[ \, Q, U \, ] \cdot \mathbf x[ \, Q, U \, ]),$$

where $\sqrt{ (\mathbf x[ \, Q, U \, ] \cdot \mathbf x[ \, Q, U \, ]) }$ is the distance of $Q$ and $U$ wrt. each other. Likewise

$$s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] := c^2 \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^2 - (\mathbf x[ \, Q, W \, ] \cdot \mathbf x[ \, Q, W \, ]),$$

and

$$s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] := c^2 \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^2 - (\mathbf x[ \, U, W \, ] \cdot \mathbf x[ \, U, W \, ]).$$

Expressing vector $\mathbf x[ \, Q, U \, ]$ as Taylor series wrt. duration $\tau Q[ \, \_P, \circledS U \circ P \, ]$, explicitly up to third degree, we have

$$\mathbf x[ \, Q, U \, ] \approx \mathbf x[ \, Q, Q \, ] + \mathbf v_{(Q)}[ \, P \, ] \, \tau Q[ \, \_P, \circledS U \circ P \, ] + \frac{1}{2} \, \mathbf a_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 + \\ \frac{1}{6} \, \mathbf j_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^3 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^4 \, ].$$

Obviously, $\mathbf x[ \, Q, Q \, ]$ is null; and $\mathbf v_{(Q)}[ \, P \, ]$ vanishes as well, according to the requirement of $Q$ being momentarily co-moving wrt. $P$. Therefore also, the sought magnitude of proper acceleration of $P$ equals

$$\| \mathbf a^P_Q \| = \sqrt{ (\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) }.$$

Consequently

$$\mathbf x[ \, Q, U \, ] \approx \frac{1}{2} \, \mathbf a_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 + \\ \frac{1}{6} \, \mathbf j_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^3 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^4 \, ].$$

Likewise

$$\mathbf x[ \, Q, W \, ] \approx \frac{1}{2} \, \mathbf a_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^2 + \\ \frac{1}{6} \, \mathbf j_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^3 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^4 \, ].$$

On the other hand, again formally

$$\mathbf x[ \, U, W \, ] \approx \mathbf x[ \, U, U \, ] + \mathbf v_{(U)}[ \, P \, ] \, \tau U[ \, \_P, \circledS U \circ P \, ] + \frac{1}{2} \, \mathbf a_{(U)}[ \, P \, ] \, (\tau U[ \, \_P, \circledS W \circ P \, ])^2 + \\ \frac{1}{6} \, \mathbf j_{(U)}[ \, P \, ] \, (\tau U[ \, \_P, \circledS W \circ P \, ])^3 + \mathcal O[ \, (\tau U[ \, \_P, \circledS W \circ P \, ])^4 \, ].$$

As above, $\mathbf x[ \, U, U \, ]$ is obviously null; but $\mathbf v_{(U)}[ \, P \, ]$ does not necessarily vanish exactly, since at event $\varepsilon_{PU} \not\equiv \varepsilon_{PQ}$ participants $P$ and $U$ are not necessarily co-moving wrt. each other. Instead, velocity vector $\mathbf v_{(U)}[ \, P \, ]$ as well as acceleration vector $\mathbf a_{(U)}[ \, P \, ]$ may in turn be expressed as Taylor series wrt. duration $\tau Q[ \, \_P, \circledS U \circ P \, ] = \tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ]$ to corresponding relevant order

$$\mathbf v_{(U)}[ \, P \, ] \approx \mathbf v_{(Q)}[ \, P \, ] + \mathbf a_{(Q)}[ \, P \, ] \, \tau Q[ \, \_P , \circledS U \circ P \, ] + \frac{1}{2} \, \mathbf j_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P , \circledS U \circ P \, ])^2 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^3 \, ] = \mathbf a_{(Q)}[ \, P \, ] \, \tau Q[ \, \_P, \circledS U \circ P \, ] + \frac{1}{2} \, \mathbf j_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^3 \, ],$$

and

$$\mathbf a_{(U)}[ \, P \, ] \approx \mathbf a_{(Q)}[ \, P \, ] + \mathbf j_{(Q)}[ \, P \, ] \, \tau Q[ \, \_P , \circledS U \circ P \, ] + \mathcal O[ \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 \, ].$$

Together (leaving the term with $\mathbf j_{(U)}[ \, P \, ]$ unchanged, and using $\tau U[ \, \_P, \circledS W \circ P \, ] = \tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ]$)

$$\mathbf x[ \, U, W \, ] \approx (\mathbf a_{(Q)}[ \, P \, ] \, \tau Q[ \, \_P, \circledS U \circ P \, ] + \frac{1}{2} \, \mathbf j_{(Q)}[ \, P \, ] \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2) \, \tau Q[ \, \circledS U \circ P , \circledS W \circ P \, ] + \frac{1}{2} \, (\mathbf a_{(Q)}[ \, P \, ] + \mathbf j_{(Q)}[ \, P \, ] \, \tau Q[ \, \_P , \circledS U \circ P \, ]) \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^2 + \frac{1}{6} \, \mathbf j_{(U)}[ \, P \, ] \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^3 + \mathcal O[ \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^4 \, ].$$

Inserting these expansions in the expressions of intervals we obtain (explicitly only up to fourth order in the durations, as is needed in the following)

$$s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] \approx c^2 \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 - \frac{1}{4} \, (\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^4 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^5 \, ],$$

$$s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] \approx c^2 \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^2 - \frac{1}{4} \, (\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^4 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^5 \, ],$$

and

$$s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] \approx c^2 \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^2 - \frac{1}{4} \, (\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) \, (2 \, \tau Q[ \, \_P , \circledS U \circ P \, ] + \tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^2 \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^2 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^5 \, ],$$

where $$\tau Q[ \, \_P, \circledS W \circ P \, ] = \tau Q[ \, \_P, \circledS U \circ P \, ] + \tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ]$$

and the rest terms were all estimated accordingly as $\mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^5 \, ]$.

Now, in turn substituting these expansions of intervals in the following formula in terms of intervals

$$\frac{\begin{vmatrix} 0 & s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] & s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] & 1 \cr s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] & 0 & s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] & 1 \cr s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] & s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix}}{ s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] \, s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] \, s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] }$$

(which formally recalls the formula for the inverse of the radius of a circle circumscribed to a triangle, in terms of the triangle sides, written conveniently in terms of a Cayley-Menger determinant, this fortuitously reduces to a fairly simple fraction

$$\frac{(\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) \, \left( (\tau Q[ \, \_P, \circledS W \circ P \, ])^2 \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^2 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^7 \, ] \right)}{c^4~\left( (\tau Q[ \, \_P, \circledS W \circ P \, ])^2 \, (\tau Q[ \, \_P, \circledS U \circ P \, ])^2 \, (\tau Q[ \, \circledS U \circ P, \circledS W \circ P \, ])^2 + \mathcal O[ \, (\tau Q[ \, \_P, \circledS W \circ P \, ])^8 \, ] \right)}.$$

Consequently, taking the limit of ever smaller durations, or equivalently, the limit of time-like intervals of ever smaller magnitude, the sought magnitude of the proper acceleration of participant $P$, at event $\varepsilon_{PQ}$, is obtained

$\| \mathbf a^P_Q \| = \sqrt{ (\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) } =$ $$\Large \matrix{ c^2~\text{lim}_{\{ s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] \rightarrow 0 \}} \! \! \Big[ \, \Big( & \frac{\begin{vmatrix} 0 & s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] & s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] & 1 \cr s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] & 0 & s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] & 1 \cr s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] & s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix}}{ s^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] \, s^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] \, s^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] } & \Big)^{\large (1/2)} \, \Big] . }$$

It may be interesting to note that the above formula for flat regions in terms of values of (timelike) intervals can be readily adapted to generally curved regions by substituting values of Lorentzian distances $\ell$, namely:

$\| \mathbf a^P_Q \| = \sqrt{ (\mathbf a_{(Q)}[ \, P \, ] \cdot \mathbf a_{(Q)}[ \, P \, ]) } =$ $$\Large \matrix{ c^2~\text{lim}_{\{ \ell[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] \rightarrow 0 \}} \! \! \Big[ \, \Big( & \frac{\begin{vmatrix} 0 & \ell^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] & \ell^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] & 1 \cr \ell^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] & 0 & \ell^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] & 1 \cr \ell^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] & \ell^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] & 0 & 1 \cr 1 & 1 & 1 & 0 \end{vmatrix}}{ \ell^2[ \, \varepsilon_{PQ}, \varepsilon_{PU} \, ] \, \ell^2[ \, \varepsilon_{PQ}, \varepsilon_{PW} \, ] \, \ell^2[ \, \varepsilon_{PU}, \varepsilon_{PW} \, ] } & \Big)^{\large (1/2)} \, \Big] . }$$