# Question re spacial curvature K(r) w/r/t the Shwarzchild metric

• I
Gold Member

## Summary:

I understand that K(infinity) = zero, and K(Schwarzchild radius) = infinity, but what is K(r) between these limits?

## Main Question or Discussion Point

I understand that
K(∞) = 0,​
and
K(rs) = ∞​
where
rs = 2GM/c2.​
What is an equation for K(r) when
rs < r < ∞?​
I have tried the best I can to search the Internet to find the answer, but I came up empty. I would very much appreciate the answer, or a reference that discusses the desired answer. I have seen some tensor equations, but my math skills cannot deal with those.

In
in the section
Flamm's Paraboloid​
the following equation is derived:
w(r) = 2rs√((r/rs)-1).​
The coordinate system is described as
The Euclidean metric in the cylindrical coordinates (r, φ, w) is written:​
ds2 = dw2 + dr2 + r22.​
I am guessing (and may certainly be mistaken) that w(r) is related to the spatial curvature. In particular, I am guessing that w(r) is the radius of curvature, and the curvature is
k(r) = 1/w(r).​

I am hoping that some reader can confirm that my guesses are correct or incorrect, and if incorrect, what the curvature might be.

I am also wondering if this thread might be better placed in the Special and General Relativity forum. I also apologize for mispelling "spatial" in the title.

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