Escape Velocity, Gravitational Velocity & Time Dilation

[PeroK]

@Grav Velocity just a question from me. In general, how is "gravitational time dilation" defined?

 

[PeterDonis]

"Static" space time involves only one massive body

More precisely, one non-rotating massive body. If the massive body is rotating, the spacetime will be stationary, but not static.

Non-static space time involves at least two source (massive) objects that are gravitationally interacting with one another and therefore moving.

The spacetime in this case will be non-stationary, not just non-static.

This movement changes the shape of space time as time passes.

No, it's more than that: the concept of "space as time passes" has no well-defined meaning in a non-stationary spacetime, unless that spacetime has some other symmetry that picks out a particular set of spacelike hypersurfaces. In the case of FRW spacetimes, which are used in cosmology to model the universe, there is a particular set of spacelike hypersurfaces which are all homogeneous and isotropic, and the spacetime as a whole can be viewed as an infinite series of such hypersurfaces, each one labeled by a different value of a coordinate that we can call "time". But in these spacetimes, the shape of each spacelike hypersurface is the same; all that changes with time is the scale factor, i.e., the "size".

The most obvious example of this would be two objects orbiting one another and the resulting space time distortions or gravity waves….

Note that there can be cases of two objects orbiting one another and not emitting gravitational waves (not "gravity waves", btw, that term refers to something completely different that has nothing to do with general relativity), but these are highly idealized cases in which the two objects have exactly the same mass and are in exactly circular orbits about their common center of mass. In any real case of two objects in mutual orbits, there will be gravitational waves emitted, yes.

These gravity waves would create a fluctuating time dilation (and length) and any given point in space time.

No, because "gravitational time dilation" is not well-defined in such spacetimes, since they are not stationary.

Also, a point in spacetime can't have changing anything; it is a particular point in a particular 4-dimensional geometry, and no geometric properties can "change" at a single point in a geometry; that wouldn't even make sense, since the geometric properties at each point are what define the geometry.

The space time distortions created by the orbiting planets in non-static space time causes a time dilation at a particular location that does not match the corresponding gravitational potential (velocity) at that same point.

Wrong. I have already explained why several times.

this difference between gravitational potential and time dilation breaks the waterfall model.

Wrong. The "waterfall" model does not work because the spacetime is not stationary.

I can say the math for gravitational potential and time dilation for two massive object affecting a single point in space time seems to work out without any problems for various positions of two objects orbiting one another (just some simple vector additions)

First, as has already been said several times, "gravitational potential" and "time dilation" aren't even meaningful in non-stationary spacetimes, so you can't possibly have correct math for them.

Second, "simple vector additions" do not work in GR because the Einstein Field Equation is nonlinear; adding two solutions together does not give you another solution.

 

[Nugatory]

The space time distortions created by the orbiting planets in non-static space time causes a time dilation...

No, that is not right. Some of the underlying problem here is that you still don't understand what time dilation is, and that is confusing you. In post #7 of this thread @Dale described time dilation as a ratio between coordinate time and proper time, and I have suggested that you understand the role of relativity of simultaneity in velocity-based time dilation. Until you've done these things or similar exercises, you will find it very difficult to construct a workable mental model.

And if you are really interested in learning this stuff, you might give "Spacetime Physics" by Taylor and Wheeler a try. It's a fairly sophisticated treatment of relativity, but is within the grasp of a committed high school senior.

at a particular location that does not match the corresponding gravitational potential (velocity) at that same point

It's not that it "does not match", the issue is that the concept of gravitational potential is not meaningful here. Talking about the potential at a point in one of these spacetimes is like talking about the color and weight of an emotion - a grammatically correct but meaningless string of words.