Why Are My Geodesic Calculations Dependent on an Unknown Function?

Thread starter Gleeson
Start date Dec 19, 2023
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General relativity

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SUMMARY

The discussion centers on the dependency of geodesic calculations on an unknown function, denoted as f(λ), in the context of non-affinely parametrized geodesics. The user derives equations for the geodesic equations, specifically t^a∇_aε = 2f(λ)ε and t^a∇_ap = f(λ)p. The user expresses concern over the necessity of introducing an unknown function, indicating a potential misunderstanding in their calculations. The need for clarification on this dependency is emphasized.

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Understanding of differential geometry concepts, particularly geodesics
Familiarity with the notation of covariant derivatives, such as ∇_a
Knowledge of parametrization methods in the context of geodesics
Basic grasp of functions and their dependencies in mathematical equations

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Dec 19, 2023

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.