# Distance function in Riemannian normal coordinates

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1. Sep 23, 2015

### shooride

Hi,
I read somewhere the geodesic distance between an arbitrary point $x$ and the base point $x_0$ in normal coordinates is just the Euclidean distance. Why?! That's the part I don't understand. I know that one can write
$g_{\mu \nu} = \delta_{\mu \nu} - \frac{1}{6} (R_{\mu \rho \nu \sigma} + R_{\mu \sigma \nu \rho} ) (x^\rho-x_0^\rho) (x^\sigma-x_0^\sigma) + \dots$
I've tried to figure out the square of the distance (which seems more simple than the distance) in normal coordinates. The calculations wasn't clear, what I get is
$d(x,x_0)^2=g_{\mu\nu} (x^\mu-x_0^\mu)(x^\nu-x_0^\nu) +1/3 R_{\mu\nu\rho\sigma}x^\rho x^\sigma(x^\mu-x_0^\mu)(x^\mu- x_0^\nu) + \dots$
Where am I doing anything wrong?

2. Sep 23, 2015

### bcrowell

Staff Emeritus
This certainly can't be exactly true for points arbitrarily far away. You might want to find a reference that discusses this point and see what it actually says.

3. Sep 23, 2015

### shooride

You are right! AFAIK, one can consider normal coordinates for points which are near to each others, right?! However, it's a little cryptic to me, again.. Under what conditions can one consider the geodesic distance equals to Euclidean distance? The only thing I can say is $d(x,x_0)^2=(x^\mu-x_0^\mu)(x_\mu-{x_0}_\mu)+O(x^4)$ Is this true? Unfortunately, I couldn't find a proper reference..Could you please introduce me a proper reference where I can find distance function in normal coordinates?

4. Sep 23, 2015

### bcrowell

Staff Emeritus
There is more than one thing that people refer to as normal coordinates: https://www.physicsforums.com/threads/gaussian-normal-coordinates.149978/ IIRC there are definitions and theorems about existence in the neighborhood of a point or in the neighborhood of a geodesic. I don't think it matters too much whether the metric is Riemannian or semi-Riemannian. MTW (p. 1055) defines normal coordinates in the neighborhood of a point as obeying the criterion $g_{\mu\nu}=\eta_{\mu\nu}$ and $\partial_\lambda g_{\mu\nu}=0$ at that point. If that's the definition that's appropriate for you, then maybe you can prove that the error is $O(x^4)$. This seems a little tricky to me because presumably $d(x,x_0)$ should be defined along the *exact* geodesic from $x$ to $x_0$.

5. Sep 26, 2015

### shooride

Yeah... I think everything is starting to be a bit clearer now!