The discussion centers on deriving the geodesic equation to find the shortest path between two points on Earth, modeled as a perfect sphere. Participants emphasize that the solution involves solving two coupled differential equations for the angles θ and φ, which represent the geodesic on a two-sphere. The conversation highlights the importance of explicitly defining the metric tensor components and utilizing the Einstein summation convention to proceed with the calculations. Additionally, the constant 'k' related to curvature is clarified as a non-factor in the context of the two-sphere.
PREREQUISITESStudents and researchers in physics, particularly those focusing on general relativity, differential geometry, and geodesic calculations on spherical surfaces.
https://www.physicsforums.com/file:///C:\Users\acer\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gifFightfish said:It would be helpful if you could supply more details about the part(s) where you got stuck and any relevant equations.
If you had treated the Earth as a two-sphere, you should be able to solve the geodesic equations to find that the shortest path between two points lies on the arc of a great circle.
NihalRi said:So I've gone through the process of deriving the geodesic equation, I thought I understood it. I hoped that once the equation was obtained I'd be able to do simple replacements and find the shortest path between two locations on earth. I'm really stuck right now though so does anyone know how I'd go about doing this ?
Tensorslavinia said:How are you representing the metric on the sphere?
That's not the most illuminating answer; its a metric tensor after all. I'm pretty sure what lavinia's referring to is the explicit form of the metric components that you have.NihalRi said:Tensors
Right. That is what I meant.Fightfish said:That's not the most illuminating answer; its a metric tensor after all. I'm pretty sure what lavinia's referring to is the explicit form of the metric components that you have.
Could you try to fix the image or typeset the relevant equations? If not, we will find it quite tricky to guide you through.
Yes. I'll do that and get back to you shortly thank you:)Fightfish said:Since you managed to arrive there, I will assume that you are familiar with the Einstein summation convention then? To proceed from that equation, you need to insert the explicit expressions for your metric tensor g_{ij} and your coordinates x^{i} into the expression. Since the two-sphere is a two-dimensional manifold, it is parameterised by the two angles (\theta,\phi).
Could you please try doing that and see where that gets you (and try to show us your work - so we can advise accordingly)?
(P.S. if any staff happens to see this, I think this would be better placed in the homework section)
So I'm working on expaning the summation and this, "k," is confusing me. It looks like something that should be summed over but besides it being a representation of spsce curvature I have no idea what it represents (is it a constant?)Fightfish said:Since you managed to arrive there, I will assume that you are familiar with the Einstein summation convention then? To proceed from that equation, you need to insert the explicit expressions for your metric tensor gijg_{ij} and your coordinates xix^{i} into the expression.
Well I'm still working on that as k also showed up in the geodesic equation(the form i sent the link of) I found out that it is guassian curvature, and I think and it's one over radius squared for a sphere. It seems different to the k in the robertson walker metric but there are a lot of similarities(both k's can tell us if a universe is open or closed.)Fightfish said:I presume the k you're referring to is the one in the Robertson-Walker metric? It's just a constant. In fact, only the sign of k matters, because the scale factor can always be redefined such that |k| = 1 (except when k = 0, of course).
On a side note - have you been successful in your original problem here of deriving the geodesics on a two-sphere?