Distance function in Riemannian normal coordinates

[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?

 

[bcrowell]

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.

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.

 

[shooride]

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.

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?

 

[bcrowell]

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 $x_0$.

 

[shooride]

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 $x_0$.

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