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I consider the geometry around a massive homogeneous static spherical object (for example, a neutron star)

This system is static, so I am not interested in the space-time distance between two events.

I am interested in the spatial distance (without the influence of time) and the shortest path between two points?

Questions:

1. If a light source was turned on at point A, did the ray of light that first reaches point B has traveled along the shortest path? ( Fermat's principle )

2. Is the (null geodesic) path of the ray of light the (locally) shortest path between a two points?

The null geodesic is defined as ##{g_{\mu\nu}} \frac {dx^{\mu}} {ds} \frac {dx^{\nu}} {ds}=0 ##.

Can I remove time ##\mu =0, \nu =0## in my static (pure space-like) geometry in order to get non-zero distance?

$$ d(A,B)=L=\int_P \sqrt{g_{\mu\nu} dx^{\mu} dx^{\nu}} \,ds$$

3. Is the length of that path the spatial distance between those two points?

This system is static, so I am not interested in the space-time distance between two events.

I am interested in the spatial distance (without the influence of time) and the shortest path between two points?

Questions:

1. If a light source was turned on at point A, did the ray of light that first reaches point B has traveled along the shortest path? ( Fermat's principle )

2. Is the (null geodesic) path of the ray of light the (locally) shortest path between a two points?

The null geodesic is defined as ##{g_{\mu\nu}} \frac {dx^{\mu}} {ds} \frac {dx^{\nu}} {ds}=0 ##.

Can I remove time ##\mu =0, \nu =0## in my static (pure space-like) geometry in order to get non-zero distance?

$$ d(A,B)=L=\int_P \sqrt{g_{\mu\nu} dx^{\mu} dx^{\nu}} \,ds$$

3. Is the length of that path the spatial distance between those two points?

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