Line element in spherical coordinates

[broegger]

Hi,

I was just reading up on some astrophysics and I saw the line element (general relativity stuff) written in spherical coordinates as:

ds^2 = dr^2 + r^2(d\theta^2 + \sin\theta\d\phi)

I don't get this. dr is the distance from origo to the given point, so why isn't ds^2 = dr^2 without the other stuff?

[Tom Mattson]

I don't get this. dr is the distance from origo to the given point, so why isn't ds^2 = dr^2 without the other stuff?

Because you aren't after the distance between some point and the origin, you're after the distance between 2 arbitrary points in space. If you want to see how this expression comes about then start from the more intutive expression for the line element in Cartesian coordinates:

ds^2=dx^2+dy^2+dz^2

Then use the following transformation equations:

x=r\sin(\theta)\cos(\phi)
y=r\sin(\theta)\sin(\phi)
z=r\cos(\theta)

Take the differentials dx, dy, and dz and verify that ds^2 \neq dr^2 in general.

[broegger]

Thanks, Tom!