Is there no curvature due to mass when travelling along a sphere outside a black hole?

Question

I'm trying to do actual Schwarzschild Metric calculations. In looking at this video Schwarzschild Proper Distance at 1:20 he shows the calculation for moving directly outward on a radius from the black hole. This equation has the $M$ figure in it so it has significant curvature the closer we are to the mass of the black hole. But then at 3:00 he shows the calculation for moving along the sphere (with constant $r$) and there is no use of $M$ in the equation. This would indicate to me that there is no curvature due to mass when moving along the sphere, regardless of how close I am to the black hole. Is this a correct interpretation? It's very hard to envision the first curvature without the second.

Answer 1

It's due to the way the radial coordinate $r$ is defined. It's natural for beginners to assume it's a radial distance, but it isn't. It is defined as the circumference of a circle centred on the black hole divided by $2\pi$. So if you are in a circular orbit round any black hole the circumference of your orbit is always $2\pi r$ regardless of the mass of the black hole.

The actual radial distance has to be calculated by integrating $ds$ along a line of constant $t$, $\theta$ and $\phi$. For more on this see Exact meaning of radial coordinate of the Schwarzschild metric.