Bosko said:
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?
Well, the space-time geometry inherently includes time, so to exclude time what you need to do is define a projection operator that projects the 4-d space-time manifold into a 3-d spatial manifold in order to answer your question.
In general, there is no unique projection operator, but in the case of the static geometry, the time-like symmetry (the time-like killing vector field) specifies a projection operator via the requirement that it preserves the symmetry.
It may seem like a fine point, but sloppy thinking about what you are doing can cause confusion.
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 )
I don't have a detailed calculation, but I believe the answer to this is "no".
To sketch the required detailed calculations, specify a static space-time Schwarzschild geometry. There are several choices, I will chose to use isotrorpic Schwarzschild coordinates (t,x,y,z). (If this isn't clear, I could provide more detail, but hopefully it's not necessary). Then the projection operator maps a point (t,x,y,z) in the spacetime manifold to the point (x,y,z) in the spatial manifold, which is a quotient manifold. If you prefer to use other coordinates, feel free. The isotropic coordinates I used are conceptually simple (IMO), but troublesome to work through the mathematics.
You'll find that the path of the light beam is "bent" in these coordinates, not a straight line. This is referred to commonly as the "deflection of light by gravity".
2. Is the (null geodesic) path of the ray of light the (locally) shortest path between a two points?
Locally, yes. Ignoring my point about sloppy thinking, I'll just say that while light is bent by gravity, locally it is so insignificant it doesn't matter.
You might consider a different example. Consider Rindler coordinates on an accelerating elevator, again using (t,x,y,z) coordinates for the space-time manifold and (x,y,z) as coordinates for the spatial submanifold. You will again see that light follows a curved path, but that radar methods give the correct notion of distance for nearby points on the elevator.
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?
It's not exactly the problem you are interested in, but some of the questions you ask are covered in the context of a rotating platform in
https://arxiv.org/abs/gr-qc/0309020, "The Relative Space: Space Measurements on a Rotating Platform". I would say that the paper in question solves a problem that is actually harder than the one you are interested in - but I'm not aware of any other formal reference that touches on your question offhand.
arxiv said:
We can introduce the local spatial geometry of the disk, which defines the
proper spatial line element, on the basis of the local optical geometry. To this
end we can use the radar method[18], [24].
Thus, they do something similar to what you describe - they use light to determine the local spatial geometry.
Another quote.
arxiv said:
That is, the relative space is the ”quotient space” of the world tube of the
disk, with respect to the equivalence relation RE, among points and space plat-
forms belonging to the lines of the congruence Γ.
Instead of using the world tubes of the rotating disk, as in the paper, we use the world tubes of the "static observers" in your example. The important point is that in order to answer the question, we need to define some projection operator to reduce the 4-d space-time manifold to a 3-d manifold. A congruence of worldlines is one way to define the projection operator - every worldline in the 4 dimensonal congruence is mapped to a single point in the "quotient space", which is the 3d projection of the 4d manifold. Thus we map some curve (t,x,y,z) representing a worldline (defined by the set of points with t varying and x,y,z fixed) to a single point in the "quotient space", namely (x,y,z).