# [SR] How does Einstein solve the PDE for Tau?

1. Jan 16, 2016

### 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 :

?

2. Jan 16, 2016

### 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 in to get the ratio of A to B, and chooses that he wants the constant a to be dimensionless.

3. Jan 17, 2016

### DoobleD

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

4. Jan 17, 2016

### 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.

5. Jan 18, 2016

### DoobleD

Last edited: Jan 18, 2016