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?

 

[quantumdude]

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!