Drunk. Probably still are. Definitely had too much, but flatmate finally got his M.Eng after years of bullshit, so at least it seemed like a good occasion. I have a math major and loved courses like real analysis, so obviously started drinking and deriving from Lorentz transform. For kb's case I got: t' = (c/g) asinh(gt/c) x - distance from stationary frame's perspective. Mea maxima culpa if I fucked it up along the way, but it looks reasonable to me.but if you really hate yourself, do d(𝛾*t' = t), with 𝛾(t), and then do algebra for days.
t = sqrt(x²/c² + 2x/g)
Looks good to me. I might come back with pictures of my chickenscratch, but yeah, I can see how some geometric substitution in the integral leads to a sinh function. Perhaps somewhat ironic is that the asinh function is sometimes pronounced "a cinch", a.k.a. American slang for something considered easy. But some would say "arc shine". As for "inverse hyperbolic tangent", it's just that no one wants to take the time to say an 8 syllable thing. The fact that pronunciation runs the gamut is proof that we've needed to standardize communication since science got off the ground. It's just dumb luck that I'm born into the English-speaking contingent of the world. But meanwhile, you know at least two languages thoroughly, and your brain is all the more ductile for it. Long term benefit! But yeah, I do genuinely feel kinda guilty for just having been born in the U.S., for what seems like an ever-increasing number of reasons, when I consider it. For now, at the very least, enjoy that booze. Kill off the weak brain cells. You'll make more, no worries.