For reference, here's an implementation (fitting the even tighter and painful limitations on the maximum number of characters) of the equality (5/7) in [Wolfram Alpha](https://www.wolframalpha.com/input?i=Simplify%5B+ReplaceAll%5BDet%5B%7B%7B0%2C1%2C1%2C1%7D%2C%7B1%2C0%2CJ%2CK%7D%2C%7B1%2CJ%2C0%2CN%7D%2C%7B1%2CK%2CN%2C0%7D%7D%5D%2F%28J+K+N%29%2C%7BJ+-%3Et%5E2-%28Sqrt%5B%28c%2Fa%29%5E2%2Bt%5E2%5D-c%2Fa%29%5E2%2CK+-%3Ef%5E2-%28Sqrt%5B%28c%2Fa%29%5E2%2Bf%5E2%5D-c%2Fa%29%5E2%2CN+-%3E%28f-t%29%5E2-%28Sqrt%5B%28c%2Fa%29%5E2%2Bf%5E2%5D-Sqrt%5B%28c%2Fa%29%5E2%2Bt%5E2%5D%29%5E2%7D%5D+%5D).

(6/7) #Relativity #TeachRelativity #HyperbolicMotion #SpaceTime

Now integrating:

\(\int a\,{\rm d}t=\int\frac{{\rm d}v}{(\sqrt{1-(v/c)^2})^3},\)

\(a\, t=\frac{v}{\sqrt{1-(v/c)^2}},\)

\(\frac{{\rm d}x}{{\rm d}t}:=v=\frac{a\, t}{\sqrt{1+(a/c\, t)^2}}.\)

Integrating (and normalizing) again:

\(\int{\rm d}x =\int{\rm d}t\,\frac{a\, t}{\sqrt{1+(a/c\, t)^2}},\)

\(x=(c^2/a)\, (\sqrt{1+(a/c\, t)^2}-1) \), as shown in (2/7).

The main point follows: expressing constant \(a\) through #Spacetime intervals: ...

(4/7) #Relativity #TeachRelativity #HyperbolicMotion

Digression to show that \(a\) is the magnitude of \(A\)'s proper acceleration (i.e. wrt. the respective momentarily co-moving inertial frame):

\(\rm d v := \)
\(\frac{v + (a \, \rm d \tau)}{1 + (v/c) \, (a \, \rm d \tau)/c} - v = \)
\(\frac{(a \, \rm d \tau) \, (1 - (v/c)^2)}{1 + (v/c) \, (a \, \rm d \tau)/c} \approx \)
\(a \, \rm d \tau) \, (1 - (v/c)^2) = \)
\(a \, \rm d t) \sqrt{1 - (v/c)^2} \, (1 - (v/c)^2).\)

(3/7) #Relativity #TeachRelativity #HyperbolicMotion #SpaceTime

#HyperbolicMotion of participant \(A\) in a "flat #spacetime region", in terms of Minkowski coordinates \((t, x)\) (which are of course adapted to the geometric relations between members of an #InertialFrame \(\mathcal F\)):

\(x_{\mathcal F}[ \, A \, ] = (c^2/a) \, \left(\sqrt{1+(a/c\, t_{\mathcal F}[ \, A \, ])^2}-1\right) \)

which can be re-arranged to "the canonical form" of an [hyperbola equation](https://en.wikipedia.org/wiki/Hyperbola#Equation) (wrt. Cartesian coordinates).

(2/7) #Relativity #TeachRelativity

Elaborating on the utility of hyperbolas (hat tip to Joe Heafner) and the utility of [Cayley-Menger determinants (CMDs)](https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant) for expressing (defining?) "acceleration" (of an identifiable "material point", or participant) as radius of curvature of the corresponding #worldline in "flat" (pseudo-plane) 1+1 #SpaceTime --

for #HyperbolicMotion, magnitude and direction of acceleration are constant.
In terms of Minkowski coordinates: ...

(1/7) #Relativity #TeachRelativity