Einstein's Solution to the PDE for Tau in 1905 SR Paper

[DoobleD]

This is maybe more a maths question.

In part 3 of his 1905 SR paper, how does Einstein solves the following PDE :

to get :

?

 

[Ibix]

He can vary x' and t' independently. If he keeps t' constant then $\partial\tau/\partial t'$ does not change - but he's still free to vary x'. That tells him that $\partial\tau/\partial x'$ is constant and independent of t' - in other words $\tau=Ax'+f (t')$. He can make the same argument the other way around to get the dependence on t'. So the only possible solution is of the form $\tau=Ax'+Bt'$. He then just substitutes that solution into get the ratio of A to B, and chooses that he wants the constant a to be dimensionless.

 

[DoobleD]

If he keeps t' constant then ∂τ/∂t′∂τ/∂t′\partial\tau/\partial t' does not change - but he's still free to vary x'.

Hmm, if t is kept constant, shouldn't ∂τ/∂t be 0 ?

 

[martinbn]

To solve an equation of the form
$$
\frac{\partial\tau}{\partial x'}+b\frac{\partial\tau}{\partial t} = 0
$$
where $b$ is a constant, you need the change the variables to $\xi=x'+bt$ and $\eta=x'-bt$. Then the equation becomes very simple and you can solve it. The general solutions is
$$
\tau = F(x'-bt)
$$
where $F$ is any differentialble function. If it is linear, then you get the result in the paper.

 

[DoobleD]

Got it (also with the help of a video on solving the transport equation), thank you !