Why Are My Geodesic Calculations Dependent on an Unknown Function?

  • #1
Gleeson
30
4
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.
 

FAQ: Why Are My Geodesic Calculations Dependent on an Unknown Function?

Why are my geodesic calculations dependent on an unknown function?

Geodesic calculations often involve solving differential equations that describe the shortest path between points on a curved surface. These equations can include unknown functions, which represent the curvature of the surface or other geometric properties that are not explicitly defined. The dependency on an unknown function arises because the exact geometry of the surface may not be fully specified.

How can I determine the unknown function in my geodesic calculations?

To determine the unknown function, you typically need additional information about the geometry of the surface. This information can come from boundary conditions, symmetry considerations, or specific properties of the surface. In some cases, empirical data or measurements may be required to define the unknown function accurately.

Can I simplify my geodesic calculations to avoid the unknown function?

Simplification is possible in certain cases, especially if the surface has a high degree of symmetry or if approximations are acceptable. For example, on a sphere, the geodesics are great circles, and the calculations can be simplified using spherical coordinates. However, for more complex surfaces, the dependency on the unknown function may be unavoidable.

What role does the unknown function play in the accuracy of my geodesic calculations?

The unknown function is crucial for accurately describing the geometry of the surface. If this function is not correctly identified or approximated, the resulting geodesic calculations may be inaccurate or misleading. Ensuring the correct form and values of the unknown function is essential for precise geodesic determinations.

Are there numerical methods to handle the unknown function in geodesic calculations?

Yes, numerical methods such as finite element analysis, finite difference methods, and iterative solvers can be employed to handle the unknown function in geodesic calculations. These methods allow for the approximation of the unknown function and the solution of the differential equations governing the geodesics, even when analytical solutions are not feasible.

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