A simple way to measure the shape of warped spacetime

Question

Background

I've wondered for years about ways to visualize how mass (and hence gravity) affects the curvature of spacetime. The 'bowling ball on a rubber sheet' while being useful for the simplest of explanations is simultaneously not helpful in quantifying the actual metric involved. While this other question about gravity and the volume of space is very related, it is not a duplicate because I plan a followup question with a slightly different take on the actual measurement of the change of volume. This question is about a simple way to measure the effect in a more direct or applied way. Also, in case anyone wonders, no this is not a homework question. I'm an enthusiast who follows cosmology but wished he had taken more math in school.

Setup

Using an empty universe to start with, (Minkowsky space?) we can use geometry to calculate the enclosed space, it's simply pi * r^2 so for this question we want to use a radius of 1 AU (149,597,870,700 meters) to get 7.0307345e+22 square meters.

If we then add a body with the mass of 1 solar mass (our Sun) at the center of the space, this creates a curvature in the space around the body. One question I haven't approached is how the curvature affects the distance measurement, the the current assumption is the radius of interest is still 1 AU since it is perpendicular to the angle of curvature.

Now, we want to place four flat perfectly reflective mirrors in orbit around the body such that if we shine a laser at one of the mirrors, the beam of light reflects off of it and then each of the other 3 in turn to come back along the same path and create a closed loop of light. Of course, for the purposes of this question, we posit that their position and orientation are fixed in such a way as to maintain the closed loop of light. The question is not about astrodynamics and orbits but the curvature of spacetime.

Assumption: The distance between the mass and the mirror is identical between the two scenarios. I realize that there might be some GR/SR weirdness that could affect this assumption so please mention it if this assumption should be discarded. Notice that I did not stipulate a specific angle or orientation for the mirrors as the goal is to discover the desired angle that closes the circuit of light.

Question(s)

A) Is there only one distance from the body where this is possible?
B) What is the angle of reflection of the beam in a single mirror?
C) What is the enclosed volume of the area enclosed by the beam if the distance from the center of mass to the mirror was 1 light year?
D) Does the difference in the measurement between flat space versus warped space imply that it is possible for space itself to have a density? That is, for the same enclosed space (1 AU^2 * pi), does one scenario contain a larger/smaller amount of space for the same volume?

Answer 1

We know for sure it can work for two mirrors: place them on opposite sides of the body where gravitational lensing focuses light from each mirror onto the other mirror. So, it seems very plausible that it can work for three, four, or any number of mirrors.

Specifically, the effect of curvature is that geodesics curve toward the body. There should be a distance from the center, where the light reflects off each mirror at an angle that, in flat space, would circumnavigate a pentagonal arrangement of mirrors -- but where the curvature causes the light beam to circumnavigate a square arrangement of mirrors.

The math is a bit beyond me, but perhaps someone who can do the math can figure out if it works for any arbitrary number of mirrors greater than two. IF it works, then as you guessed, it would provide a way to measure curvature.

Of course, we already can measure curvature via gravitational lensing. Two mirrors would do the trick if there isn't a light source on the opposite side of the body, and no mirrors are needed if there is a suitable light source and the body and light source are at known distances away from us.