Curvature of Light Paths Near a Mass

[Sonderval]

If I understand everything correctly, space near (but outside) a mass is curved negatively, so that if I create a triangle with, for example, rigid rods and the mass in its center, the angles would sum up to less than 180°. (If I am mistaken, please correct me.)

On the other hand, the typical light-bending pictures look like this

I do understand that the space projection of the iight geodesic does not coincide with the shortest path in space as gotten by laying out rods (this is obvious because there is a dt² in the formula for the space-time distance).

But how does the projection of the light geodesics look like, exactly? If I had three stars situated on a triangle with the mass in the middle, would the angular sum of the space projections of the geodesics be smaller or larger than 180°? From the fact that images are shifted away from the central mass, I would think that the sum has to be larger, but I'm not sure.

 

[ChrisVer]

I don't understand what you mean. The shortest path in spacetime is given by the geodesics (they are straight lines).
Now whether the sum of angles is less or bigger than 180 depends on the geometry of your spacetime (universe). An hyperbolic geometry (or negative curvature) would lead in less than 180 degrees, whereas in a geometry with positive curvature, it's greater than 180...

 

[Sonderval]

@ChrisVer
Imagine a second star in the lower left corner of the picture above and imagine light-rays from each star to the Earth and between the stars. Does the space projection of the light rays form a triangle with asum of more or less than 180°?

 

[ChrisVer]

again I say, it depends on the curvature of the spacetime. you can't see that on such a diagram...

 

[A.T.]

If I understand everything correctly, space near (but outside) a mass is curved negatively, so that if I create a triangle with, for example, rigid rods and the mass in its center, the angles would sum up to less than 180°. (If I am mistaken, please correct me.)

You are mistaken. If you include the mass in the triangle, the net spatial curvature will be positive, so a triangle made of spatial geodesics (straight rigid rods) will have an inner angle sum of more than 180°. If you keep the mass outside of the triangle, it will be lees than 180°.

I do understand that the space projection of the iight geodesic does not coincide with the shortest path in space as gotten by laying out rods (this is obvious because there is a dt² in the formula for the space-time distance).

That is important to keep in mind. The light paths in space are not geodesics. The light worldlines in space-time are geodesics. But in this case the bending direction of a spatial geodesic and light paths in space is the same.

If I had three stars situated on a triangle with the mass in the middle, would the angular sum of the space projections of the geodesics be smaller or larger than 180°? From the fact that images are shifted away from the central mass, I would think that the sum has to be larger, but I'm not sure.

This is correct, the angle sum for a light triangle around a mass will be greater than 180°.

 

[Sonderval]

@A.T.
Thanks a lot for clearing up my initial mistake, that was very helpful. Now it all fits together.