Poisson, relativists toolkit, problem 1.3: Timelike geodesic parametrised by a non-affine parameter

[Gleeson]

Homework Statement

(a) Let $x^a(\lambda)$ describe a timelike geodesic parametrised by a non-affine parameter $\lambda$, and let $t^a = \frac{dx^a}{d \lambda}$ be the geodesic's tangent vector. Calculate how $\epsilon := -t_at^a$ changes as a function of $\lambda$.

(b) Let $\xi^a$ be a killing vector. Calculate how $p := \xi_at^a$ changes as a function of lambda on that same geodesic.


(c) Let $v^a$ be such that in a spacetime with metric $g_{ab}$, $Lie_vg_{ab} = 2cg_{ab}$, where c is a constant. (Such a vector is called homothetic.) Let $x^a(\tau)$ describe a timelike geodesic parametrised by proper time $\tau$, and let $u^a = \frac{d x^a}{d \tau}$ be the four-velocity. Calculate how $q = v_a u^a$ changes with $\tau$.

Relevant Equations

As above

For (a) and (b), since the geodesic is not affinely parametrised, we have that $t^a\nabla_a t^b = f(\lambda) t^b$, for some function f.

As a results, for (a) I get that $t^a \nabla_a \epsilon = 2 f(\lambda) \epsilon$. And for (b) I get that $t^a \nabla_a p = f(\lambda) p$. (I can write out why I got those answers if needed.)

My suspicion is that I am doing something wrong, since I think it is strange to need to give the answer in terms of some unknown function that I introduced.

I'd appreciate some assistance please.