Genady

@mistermack, There are two time dilations involved. One is a gravitational time dilation. You could call it "actual" as it depends only on a change in gravitational field between events. The other is a relativistic time dilation, due to the observers' relative motion. This one differs "from the perspective of" different observers. (BTW, don't "reunite" the clocks, as the motion will cause the relativistic time dilation and the test will be ruined.) In your scenario, the two effects add in different ways: A and B don't move relative to each other, so there is only the "actual" gravitational time dilation, and it makes A to run slower than B. A and C are in the same place at the same time, so there is no gravitational time dilation between them. But they move relative to each other, which causes a relativistic time dilation. Thus, A is slower than C from the C's perspective, and C is slower than A from the A's perspective. B and C are in different positions in the gravitational field. This causes the gravitational time dilation which makes C to run slower than B. But they also move relative to each other causing the relativistic time dilation. For B, it makes the C to run even slower. However, for C, it makes the B to run slower, compensating for the effect of the gravitational time dilation. In case of B being in infinity and C free falling straight from a rest in infinity, this compensation will be exact, and thus for C, they will be running in sync.

In addition, for the observer C, A will be slower. And (assuming that B is extremely far, and that C is in free radial fall from extremely far) for C, B will be running at the same speed. So, I think that is correct, but only from the C's perspective.

No light is coming from the black letters. Did you see my last post above? What do you think about that solution of the puzzle?

And, of course, "Galilean" (Newtonian) gravity acts instantaneously at any distance.

For an observer with clock A, C will be slower. For an observer with B, C will be slower, too.

I think I got the solution. We don't see light coming from a letter, the letter is black. We see the light from background (white), the letter is where there is no light. Because of the object being out of focus, there is progressively less light coming through as we get closer to the object's edge. So, near the object's edge more light coming through from a bit farther from the object and it gets darker as we get closer. The letter appears where it is dark. The dark area moves progressively closer to the edge. Thus, the lines of the letter appear to move closer to the edge.

This is the curvature I'm talking about. It applies to any "smooth" manifold with any number of dimensions: Riemann curvature tensor - Wikipedia @geordief Let me add an example of the three possibilities for the space of a homogeneous isotropic universe. Take any three points in space, far from each other, for which the distances between them can be measured. They make a triangle. Knowing the distances, calculate the sum of angles of the triangle. If the sum is 1800 then the space is flat. Otherwise, it is curved. If it is less than 1800, the space is "open" or "hyperbolic". If more, it is "closed" or "spherical".

You've replied before my last addition in the post above. Here it is: When the finger is a bit farther away, the letter "i" is definitely vertical:

I think a short answer to the "gravity in flat space" question is that the gravity is a curvature of spacetime, a 4D object, rather than that of space, which is 3D.

I took a snapshot. The line of the "i" appears bent, i.e., not parallel to other "vertical" lines, without a movement. Here is a "before" picture: To put the optical illusion possibility finally to rest, I've rotated the image to make "vertical" lines vertical and have added the vertical grid. "i" is obviously not vertical:

In this lecture Alan Guth explains problems with applying Newton's law of gravity, including the shell theorem, to infinite space. Basically, it boils down to a conditionally convergent integral, which doesn't have a unique answer. Lecture 6: The Dynamics of Homogeneous Expansion, Part II | The Early Universe | Physics | MIT OpenCourseWare Unless, that world contains ether...

To clarify, in GR based cosmology, the flat space (together with open and closed ones) is a possible solution for a homogenous isotropic universe.

In a homogenous isotropic universe flat space with gravity is one of the possibilities, which btw looks like what we in fact got.

Yes, of course.

I guess it would look like Newtonian gravity.

Perhaps I am a bad experimentalist. I could not reproduce the effect at all. Anyway, this is what I got. The first pic has F6.3 and the second has F16. The lines look very parallel to me all the time.

Or, by doing the same test in different ambient light intensity.

I'd expect this to progressively dim parts of the image on the retina. But why would they move? (If they in fact do.)

The following test might distinguish between an optical effect and an optical illusion. Instead of the finger or pencil, try an object that is of the same color as a background, i.e., something white, maybe just a piece of paper. If it is an optical illusion, the phenomenon will disappear. If it rather is an optical effect, it will be the same.

Well, congratulations.

Optical illusion, akin this?

I suggest moving this thread to the Speculations forum.

I also had once my hair in a solid crust, unrelated to ice. Back in Baku, after swimming in the Caspian Sea. It was a blob of mazut. Evidently, I swam through it. Caspian Sea, especially around Baku, was heavily polluted back then, as the main source of oil in the USSR was there.

The clocks that measure time without measuring a movement which have been considered so far: 1. A mean lifetime of decay of an unstable elementary particle. 2. A mean time of flipping of electron spin in a magnetic field. Here I propose another: 3. Change over time in a wavelength of photons of the CMB.

It looks like being out of focus is a requirement for this phenomenon. Perhaps, there is a key for its explanation.