Special Relativity: Solving for τ

[Holystromboli]

I'm on my first pass through special relativity and I can't remember the math that would take me from:

(∂τ/∂x') + (ν/(c²-ν²))(∂τ/∂t) = 0

To

τ = φ(ν)(t - (ν/(c² - ν²))x')

Any help would be appreciated.

Also, sorry for the terrible format, but I haven't taken the time to figure out how to do equations the right way. Any tips in that direction that would apply to an iPhone would be much appreciated as well... :)

 

[PeterDonis]

Any help would be appreciated.

Where are you getting these equations from?

 

[Holystromboli]

Here:
http://fourmilab.ch/etexts/einstein/specrel/www/
Sorry about that. It's an adaptation of the 1905 Einstein paper On the Electrodynamics of Moving Bodies. I followed the math through section 3 up until the derivation of the first equation in my post, but I can't remember why an assumption of linearity would allow me to transform the first equation in my post into the second.

 

[PeterDonis]

I can't remember why an assumption of linearity would allow me to transform the first equation in my post into the second.

Linearity means that $\partial \tau / \partial x'$ and $\partial \tau / \partial t$ must be constants--i.e., they cannot be functions of $x'$ or $t$. (They can still depend on $v$, because $v$ is not a function of any of the coordinates.) So we must have $\tau = k_1 t + k_2 x'$, where $k_1$ and $k_2$ are constants. The first equation in your OP then let's you find the values of $k_1$ and $k_2$, up to an unknown function of $v$ (the $a$ in the second equation).

 

[Holystromboli]

Perfect. Thanks!