Find Distance on a 2-Sphere Using Metric Tensor

[homology]

Okay, say you're given two points on a sphere, using the metric tensor how do you find the distance between the two points? (along the geodesic connecting them)

By the way, I know how to do it with just plain old vectors in R^3, but I'd like try doing it with the metric tensor.

Thanks,

Kevin

 

[pervect]

Well, if you have the equation of the geodesic, it's fairly easy. Let's suppose the coordinates are x¹ and x², probably lattitude and longitude. Suppose our geodesic is a curve that has coordinates $$\mbox{x^1(\lambda)}$$ , $$\mbox{x^2(\lambda)}$$ i.e
these two functions are the paramaterized geodesic. You can think of $$\mbox{\lambda}$$ as being a time parameter if you envision an object actually moving along the geodesic. Then to get the distance, you just have to integrate


$$\int_{lambda} d \lambda \sqrt{\Sigma_{ij} (g_{ij} {(\frac{d x^i}{d \lambda})(\frac{d x^j}{d \lambda}) )}$$


where $$\mbox{\lambda}$$ varies from the starting point of the geodesic to the ending point

Expanding out the sum over i,j in the square root for expositional purposes, we write
$$\int_{lambda} d \lambda \sqrt{g_{11} (\frac{dx^1}{d\lambda})^2 + 2 g_{12} \frac{dx^1}{d\lambda} \frac{dx^2} {d\lambda} + g_{22} (\frac{dx^2}{d\lambda})^2}$$

If you don't have the equation for the geodesic, you have to solve the differential equations for geodesic motion

$$\frac{d^2 x^u}{d \lambda^2} + \Gamma^u{}_{ab}(\frac{dx^a}{d \lambda})(\frac{dx^b}{d \lambda})$$

 

[homology]

well the geodesic is going to be part of a great circle so I should probably just try to parameterize it...Thanks, I'll try it out.