Escape Velocity, Gravitational Velocity & Time Dilation

[Dale]

Although not really the focus of this thread I would use the term “gravity” to refer to the whole range of phenomena, not one specific aspect. Then you add qualifiers to refer to specific aspects. So gravitational acceleration and gravitational potential are both specific parts of the overall category of gravity. But I don’t think there is an authoritative stance.

 

[Ibix]

Bottom line: "gravity" typically refers to the attraction between to massive objects, and that bares a closer relation to gravitational acceleration than to gravitational potential.

As noted in passing earlier, what Newton would describe as "gravitational acceleration" corresponds to what Einstein would call "the proper acceleration needed to remain at constant distance". In the case of a black hole, this rises to infinity as you approach the event horizon and cannot be defined inside it. So by your definition of the word "gravity", the interior of a black hole has nothing to do with gravity. That would seem rather odd.

Gravity is a word that acquired a bit of flexibility when general relativity was discovered and it turned out Newton only covered one corner of a much broader topic. But gravitational time dilation is a fairly specific technical term describing a particular phenomenon in general relativity. As others have noted, physics is done with maths and not language because of exactly this kind of issue. Don't try to reason about the words - they're a mish-mash of historical accident and field-specific jargon. Learn the maths.

 

[A.T.]

The OP is correct that, if all you know is the "acceleration due to gravity", that's not enough to calculate time dilation. .

It's not enough to know "acceleration due to gravity" at one point, but if you know it along the path, you can compute both: the time dilation and the escape velocity.

I don't think "gravitational time dilation" is a misnomer. A more explicit term would be "gravitational potential time dilation", but that it's even longer.

 

[PeterDonis]

It's not enough to know "acceleration due to gravity" at one point, but if you know it along the path, you can compute both: the time dilation and the escape velocity.

Only if the path (by which I assume you mean some timelike worldline) is a known function of global invariants. If you don't know where the path is in the global geometry, then knowing the "acceleration due to gravity" along the path doesn't help you.

 

[Grav Velocity]

So, thanks to everyone for the informative replies. I have looked through this forum and other sources for more information about the points you have made.

I just want to be sure I understand the what non-static spacetime is and its implications wrt gravitational potential (which I am calling gravitational velocity) and associated time dilation.

"Static" space time involves only one massive body (gravitational source), but can include any number of test objects that may be moving. The movement of these test objects can include free-fall movement or ‘powered’ movement like a rocket. The test objects are small relative to the massive object and therefore their effect on space time can be ignored.

The waterfall model seems to work ok in static space time, but that is not very interesting.

Non-static space time involves at least two source (massive) objects that are gravitationally interacting with one another and therefore moving. This movement changes the shape of space time as time passes. The most obvious example of this would be two objects orbiting one another and the resulting space time distortions or gravity waves…. These gravity waves would create a fluctuating time dilation (and length) and any given point in space time.

If I have this correct, then the argument is:

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.

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

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), so it would be interesting to understand the cause of the discrepancy between the time dilation and gravitational potential (velocity) over time, assuming I understand this correctly.

The sources I can think of include:

the energy of the sources objects due to their motion...
frame dragging...
what else?

 

[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.