Spacetime line elements and proper time (specific problem)

Question

Sometimes, when we have a certain line element and we are given a worldline parametrised by a path parameter that is not necessarily proper time I don't completely understand some of the standard manipulations done. A specific example:

Godel spacetime has the line element $ ds^2=a^2(dt^2-dr^2 + f(r)d\phi^2 + 2g(r)d\phi dt- dz^2)$.

I came across the following question:

A spaceship travels n a closed timelike worldline given by $$t(\lambda)=0,\, r(\lambda)=R, \, \phi(\lambda)=w\lambda, \, > z(\lambda)=0,$$ where $w>0$ and $\lambda$ is a suitable path parameter. Calculate the proper time to go around the loop once.

The normal thing to do is to use $ds^2=c^2\ d\tau^2$. But then I don't understand if when we say that $d\tau=1/c\cdot \sqrt(f(R))d\phi$ we mean that we take $\lambda=\tau$ (are we allowed to do this?) or if we simply divide both sides by the infinitesimal $d\lambda$ then use the expressions for the worldline given and then "cancel out" the infinitesimal again. Both approaches seem a bit suspicious for me.

Answer 1

Let's see. Along a path of constant $r=R$, $t=0$, $z = 0$ the line element becomes (in $c = 1$ units) $$d\tau^2 = ds^2 = a^2 f(R) d\phi^2$$ Because $\phi = w \lambda$ we have $$d\tau^2 = ds^2 = a^2 f(R) w^2 d\lambda^2$$ so we have a relation here between $d\tau$ and $d\lambda$ which can be easily integrated, and which shows clearly that $\lambda \neq \tau$ in general.