Finally, here's a generalization of equality (5/7):

\(\newcommand{\eaf}{\varepsilon_{(\Phi \, A)}}\newcommand{\eap}{\varepsilon_{(A \, \Psi)}}\newcommand{\efp}{\varepsilon_{(\Phi \, \Psi)}}
a_A:=c\,\text{lim}\left[\sqrt{\frac{\begin{vmatrix}0&1&1&1\\1&0&\ell^2[\eaf,\eap]&\ell^2[\eaf,\efp]\\1&\ell^2[\eaf,\eap]&0&\ell^2[\efp,\eap]\\1&\ell^2[\eaf,efp]&\ell^2[\efp,\eap]&0\end{vmatrix}}{\ell^2[\eaf,\eap]\,\ell^2[\eaf,\efp]\,\ell^2[\efp,\eap]}\right]\)

(7/7) #Relativity #TeachRelativity #SpaceTime

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

\( \forall t_i\ne t\ne t_f:\)
\(\newcommand{\Sti}{c^2\,(t-t_i)^2-(c^2/a)^2\left(\sqrt{1+(a/c\, t)^2}-\sqrt{1+(a/c t_i)^2}\right)^2}\newcommand{\Stf}{c^2\,(t-t_f)^2-(c^2/a)^2\left(\sqrt{1+(a/c\, t_f)^2}-\sqrt{1+(a/c t)^2}\right)^2}
\newcommand{\Sif}{c^2\,(t_f-t_i)^2-(c^2/a)^2\left(\sqrt{1+(a/c\, t_f)^2}-\sqrt{1+(a/c\, t_i)^2}\right)^2}
a :=c^2\,\sqrt{\frac{\begin{vmatrix}0&1&1&1\\1&0&\Sti&\Stf\\1&\Sti&0&\Sif\\1&\Stf&\Sif&0\end{vmatrix}}
{\Sti\,\Stf\,\Sif}.\)

(5/7) #Relativity #TeachRelativity

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

@ben_crowell_fullerton
> <em> Why can't you go faster than light? </em>

[Cherenkov rad.]() aside:

Why can't you arrive sooner than the #SignalFront emitted + propagating from any specific event in which you took part ? -- Simple enough:

Any receiver you met after having taking part in a specific event
can learn about this event either (at the latest) from you, as you meet;
or else even before having met you (having rec.d the corresponding signal front already).

#Relativity #TeachRelativity

@pustam_egr

From the perspective of (experimental) physics, #Spacetime is a set of events, individuated through [coincidence determinations](http://einsteinpapers.press.princeton.edu/vol6-trans/165?highlightText=coincidences), pertaining to identifiable (and distinct) so called [material points](http://einsteinpapers.press.princeton.edu/vol6-trans/165?highlightText=material).

Problems to consider:

- How to assign one specific topology to a given Spacetime ?

- How to assign one specific "differentiable structure" to a given Spacetime (of specific topology) ?
.

#Relativity #TeachRelativity

When you teach relativity, do you introduce the notions of "straightness", "planarity", "flatness" (as well as "acceleration") through vanishing [Cayley-Menger determinants](https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant), of corresponding order, in terms of values of [Synge's world function \(\sigma\)](https://en.wikipedia.org/wiki/Synge%27s_world_function), or more generally, in terms of (the squares of) values of Lorentzian distance \(\ell\) ?

Viva Cayley-Menger determinants!
3 cheers for Lorentzian distance!
...

#Relativity #TeachRelativity

@heafnerj
Joe Heafner wrote:
> <em> I’m (re)reading Ch. 2 of Laurent’s Intro. to #Spacetime [...] </em>

But have you read Ch. 1 ?

Should students swallow this defeatism:
<blockquote> For us it suffices in principle as a definition of [#Duration as measure of] time to point at Big Ben and say: ‘That's a clock; it measures time’. </blockquote>
?? ...

No!: it can + should be taught what motivates e.g. the #MarzkeWheelerClock

Moreover: Why + How to improve on that!
#Relativity #TeachRelativity