Escape Velocity, Gravitational Velocity & Time Dilation

[Grav Velocity]

TL;DR Summary

Escape velocity determines gravitational time dilation. This implies that gravitational mass is causing space-time to pass over a location at the speed of free fall. This idea unites special relativity and general relativity, make relativity more intuitive and simplifies some time dilation calculations.

The is a question about gravitational time dilation and escape velocity. As others have pointed out, you may use escape velocity to calculate gravitational time dilation in a gravitational field. (Interestingly, you can't use gravity to calculate gravitational time dilation, which makes "gravitational" time dilation kind of a misnomer doesn't it?)

It has also been pointed out that free-fall velocity is equal to escape velocity.

It seems clear that it is more than a coincidence that "gravitational" time dilation is determined by escape velocity (or free fall velocity), especially when velocity is what determines time dilation for special relativity.

I explain this idea in more detail below, but I also created a short video that I think explains it in more detail and include and how you can use the concept of gravitational velocity to perform actual physics calculations.

https://youtu.be/TB6eQZlDhzQ

My idea is that the gravitational mass is causing space-time to fall over a region or object at the free fall velocity and this free falling space-time is what is causing the time-dilation at that location.

That is, in a gravitational field space-time is falling into the gravity well at the free-fall speed and that is causing the time dilation for a particular location, not the gravity itself.

To illustrate

Special Relativity: object move through space at speed v. Time dilation is calculated as √1-v2/c2 (lorentz transform)
----x---->
-----x--->
------x-->

General Relativity: space moves over object at speed v, where v is the accumulated free fall velocity for that location.
| | |
| x |
| | |
V V V

You can also. calculate time dilation as √1-v2/c2 in this case.

Has anyone ever made this observation or argument? It is actually similar to the application of flux in Maxwell's equations.

To rephrase, it is gravitational mass induced spacetime velocity (gravitational velocity) that is causing the time dilation. This is in contrast to gravity itself, which is measured in m/s^2 (acceleration) and which cannot be used to calculate gravitational time dilation on its own. If you just know the gravity at a location you cannot calculate gravitational acceleration.

Again, "gravitational" time dilation is a complete misnomer.

If nothing else, this idea makes the concept of time dilation in general relativity & special relativity much easier to understand and does a better job of connecting the two. It is just a function of the space-time velocity being experienced at either reference frame and you can use the same equation to calculate time dilation.

To me, something must be accumulating (integrating) speed over the length of the gravity well since that is the only way to explain how two locations that experience the same amount of gravity could have different amounts of time dilation. Saying that space-time gathers that velocity makes the most sense. I know this is also called gravitational potential, but that term does not provide much intuitive understanding as to what is happening or how special relative and general relativity are related in this case does it?

Alternatively, you could argue that movement through space-time causes a curvature of that space time that is the equivalent to the curvature caused by a particular mass at a particular distance. This seems even less useful and intuitive, however.

I think the position that there are two different mechanisms for space time dilation (one for special relativity and one for general relativity) is rather dubious as well, particularly when there is are easily calculated velocity values so readily available.

I would be curious as to what are the arguments against this way of viewing "gravitational" time dilation.

 

[Ibix]

I would be curious as to what are the arguments against this way of viewing "gravitational" time dilation.

It's called the "waterfall model", from a quick skim of what you've written. It doesn't generalise well to more complex spacetimes than Schwarzschild, as far as I know.

 

[Dale]

I would be curious as to what are the arguments against this way of viewing "gravitational" time dilation

It doesn’t work in general, only in very simple spacetimes.

 

[Janus]

(Interestingly, you can't use gravity to calculate gravitational time dilation, which makes "gravitational" time dilation kind of a misnomer doesn't it?)

Sure you can use gravity to calculate gravitational time dilation. It is related to the difference in gravitational potential. Essentially, for our far off observer, the time dilation is related to the difference between the gravitational potential of his position and that where the clock is. Escape velocity is also related to the same difference in gravitational potential. You use the difference in gravitational potential to find both.

 

[Grav Velocity]

Thanks for that reference to the waterfall model. I will do some more research on that.

So, basically you are saying there are situations where time dilation is not equal to the velocity of space-time. Interesting, I would be curious to see an example of that. I will try to track one down.

I also see that the idea of spacetime flow also has similarities with the "river model" of black holes.

All I can say is this model has brought a lot of clarity to me regarding general relativity and the meaning of many of the terms that are thrown around (in a often misleading fashion in my opinion). But is typical of science/industry specific jargon.

Nonetheless, the idea that time dilation for special and general relativity has not been unified is very odd to me, given these similarities.

 

[PeterDonis]

free-fall velocity is equal to escape velocity

Only for the idealized case of an object free-falling in from rest at infinity. (This is just the time reverse of the fact that an object launched outward at exactly escape velocity will end up just coming to rest at infinity.)

More precisely, free-fall speed is equal to escape speed, but the directions are opposite, so the velocities are opposite.

 

[Dale]

time dilation for special and general relativity has not been unified is very odd to me,

They are unified, just not this way. Time dilation always is the ratio of coordinate time and proper time. That applies for all coordinate systems with a time coordinate in all spacetimes in both general and special relativity.

Interesting, I would be curious to see an example of that. I will try to track one down.

Any spacetime except Schwarzschild and Kerr should be an example. So a charged black hole, FLRW, Milne, any of the pp wave spacetimes, a two body spacetime. I am not sure about non-vacuum spherically symmetric spacetimes

 

[Grav Velocity]

Sure you can use gravity to calculate gravitational time dilation. It is related to the difference in gravitational potential. Essentially, for our far off observer, the time dilation is related to the difference between the gravitational potential of his position and that where the clock is. Escape velocity is also related to the same difference in gravitational potential. You use the difference in gravitational potential to find both.

Ah, I disagree with this.

You need to know more than just the gravity to determine time dilation.

For example, if I told you I have two planets that both have 10 m/s2 gravitational acceleration at the surface, which is the amount of gravity, you could not tell me the time dilation at the surface of those planets without more information. In particular you would need the mass and radius or some number that is dependent thereon such as escape velocity.

Or alternatively, two planets with the same surface gravity could have different time dilations at their respective surfaces. You could compare the Earth with a planet that has 10 times the mass and 3.3 times the radius, for example. The later would have more time dilation at the surface.

 

[PeterDonis]

the gravitational mass is causing space-time to fall over a region or object at the free fall velocity and this free falling space-time is what is causing the time-dilation at that location

As applied to a black hole, this model is called the "river model" and is described here:

https://arxiv.org/abs/gr-qc/0411060

The simple version you describe applies to a non-rotating Schwarzschild black hole; there is also a more complicated version that applies to a rotating Kerr black hole. But as has already been mentioned, these models do not generalize to cases that lack the very precise symmetries of these special solutions.

 

[PeterDonis]

time dilation for special and general relativity has not been unified

Because they're not the same. SR time dilation is relative: two observers moving relative to each other in flat spacetime each calculate that the other's clock is running slow. But GR time dilation in the case you describe is absolute: two observers "hovering" at different altitudes in a gravitational field will both agree that the one who is at the lower altitude has the slower clock.

 

[PeterDonis]

Sure you can use gravity to calculate gravitational time dilation.

Not quite. The OP is correct that, if all you know is the "acceleration due to gravity", that's not enough to calculate time dilation. Whereas, if you know escape velocity, that is enough to calculate time dilation (assuming Schwarzschild spacetime in both cases).

The latter claim is easy to see: escape velocity is $v_e = \sqrt{2M / r}$, and time dilation is $\sqrt{1 - 2M/r} = \sqrt{1 - v_e^2}$.

The former claim can be seen by looking at the formula for "acceleration due to gravity":

$$
a = \frac{M}{r^2 \sqrt{1 - 2M / r}}
$$

If we write $D = \sqrt{1 - 2M / r}$ for the time dilation factor, we can invert this formula to obtain:

$$
D = \frac{M}{r^2 a}
$$

This shows that knowing $a$ alone is not sufficient to determine $D$; you also have to know $M / r^2$, which is not determined by $a$.

 

[Grav Velocity]

As applied to a black hole, this model is called the "river model" and is described here:

https://arxiv.org/abs/gr-qc/0411060

The simple version you describe applies to a non-rotating Schwarzschild black hole; there is also a more complicated version that applies to a rotating Kerr black hole. But as has already been mentioned, these models do not generalize to cases that lack the very precise symmetries of these special solutions.

Yes, I saw that and I am reading this document.

I would be curious to have an example where the symmetries are lacking, so to speak. What is the situation that breaks this waterfall model?

 

[Ibix]

time dilation is not equal to the velocity of space-time

This concept doesn't really make sense in general. Frankly, I think it barely makes sense even in Schwarzschild spacetime, and raises an awful lot of questions (what's falling in, and what's the source of whatever it is?) that you can't really answer. Some people seem to like it as an introductory model, but it's always seemed silly to me.

By the way, I see I misremembered the name - it's the "river model" not the "waterfall model". Apologies for the confusion. Either way, it's modelling the gravitational field as some unspecified thing flowing along.

Nonetheless, the idea that time dilation for special and general relativity has not been unified is very odd to me, given these similarities.

They both boil down to different clocks following paths of different "lengths" between surfaces of simultaneity. In that sense, the two are the same. The reason the paths are different lengths is rather different though.

if I told you I have two planets that both have 10 m/s2 gravitational acceleration at the surface, which is the amount of gravity

I'm not sure that "the amount of gravity" has an agreed meaning. You are defining it as the proper acceleration required to hover, and you are correct that you cannot determine the time dilation directly from that - but so what? It's time dilation that arises as a consequence of the presence of a gravitational field.

 

[Ibix]

Yes, I saw that and I am reading this document.

I would be curious to have an example where the symmetries are lacking, so to speak. What is the situation that breaks this waterfall model?

Pretty much anything which doesn't have time-translation symmetry, I would think. So two black holes on a collision course - this immediately prompts the question of what the source of whatever it is that's flowing in, particularly on the direct line between the two holes. Cosmological spacetimes like FLRW have properties that are neatly described as curvature, but I've no idea how you'd imagine a "flow" there.

 

[Grav Velocity]

I'm not sure that "the amount of gravity" has an agreed meaning. You are defining it as the proper acceleration required to hover, and you are correct that you cannot determine the time dilation directly from that - but so what? It's time dilation that arises as a consequence of the presence of a gravitational field.

I am going to call this sloppy wording, which was part of my point that "gravitational velocity" is a misnomer. But you are not alone in this regard.

Gravity is something specific and it is measured in m/s2. When you ask what is the gravity on this or that planet, you are going to be given a number in m/s2.

Gravitational potential is something else, and you can not go directly from gravitational potential to gravity without additional information.

And what is interesting is that the units of gravitational potential are velocity units. So I think it makes more sense, and provides more clarity, if you just call gravitational potential gravitational velocity.

What makes this interesting is that it is velocity that causes time dilation in special relativity. Hence the idea of gravitational velocity (or space-time velocity or space-time flux) is something that unifies the two concepts conceptually.

And just as an aside, I have been told that gravity does not cause time dilation. I have been told it is gradient of the time dilation that causes gravity (or gives the illusion of gravity), no?

 

[PeterDonis]

What is the situation that breaks this waterfall model?

Any spacetime other than Schwarzschild or Kerr.

 

[Dale]

I would be curious to have an example where the symmetries are lacking, so to speak. What is the situation that breaks this waterfall model?

I listed a bunch in post 7.

 

[PeterDonis]

Gravity is something specific and it is measured in m/s2.

If you're going to criticize other people for sloppy use of language, you need to not be sloppy yourself.

What you describe is acceleration due to gravity. It is not "gravity" without qualification; the term "gravity" can have several different meanings.

the units of gravitational potential are velocity units

More precisely, they are energy (potential energy) per unit mass (because objects of different masses follow the same trajectories if gravity alone is acting on them) units.

I think it makes more sense, and provides more clarity, if you just call gravitational potential gravitational velocity

No, because the fact that two things happen to have the same units does not mean they are the same thing. Potential energy per unit mass is not the same as velocity.

What makes this interesting is that it is velocity that causes time dilation in special relativity. Hence the idea of gravitational velocity (or space-time velocity or space-time flux) is something that unifies the two concepts conceptually.

No, it doesn't. See my previous post #10.

I have been told that gravity does not cause time dilation. I have been told it is gradient of the time dilation that causes gravity (or gives the illusion of gravity), no?

In situations where the concepts of "gravitational potential", "gravitational time dilation", and "acceleration due to gravity" make sense, the acceleration due to gravity is minus the gradient of the gravitational potential, and gravitational time dilation depends on the gravitational potentail. But those concepts only make sense in stationary spacetimes.

 

[Ibix]

I am going to call this sloppy wording, which was part of my point that "gravitational velocity" is a misnomer. But you are not alone in this regard.

You are the only one using this term.

Gravity is something specific and it is measured in m/s2.

Not in general relativity it isn't. The quantity measured in ms^-2 is the proper acceleration required to hover. That isn't even a meaningful concept outside of stationary spacetimes - which is a good part of why your model fails in more general cases.

And what is interesting is that the units of gravitational potential are velocity units. So I think it makes more sense, and provides more clarity, if you just call gravitational potential gravitational velocity.

You are, of course, free to invent whatever terminology you like. Nobody will understand you unless you use the conventional terms, but you can talk a private language if you want. However, note that the units of gravitational potential per unit mass are velocity squared, not velocity.

What makes this interesting is that it is velocity that causes time dilation in special relativity. Hence the idea of gravitational velocity (or space-time velocity or space-time flux) is something that unifies the two concepts conceptually.

No. What unifies the concepts is, as has been noted at least twice already, that they are ratios of proper time to coordinate time. The reasons that those ratios are not unity differ between the two circumstances.

And just as an aside, I have been told that gravity does not cause time dilation. I have been told it is gradient of the time dilation that causes gravity (or gives the illusion of gravity), no?

I don't think "causes" is a helpful term here. You can recover Newtonian gravity in the weak field/low velocity limit of relativity, and it turns out to be the same term in the metric that leads to time dilation that leads to Newtonian gravity.

And "illusion of gravity" makes no sense. There's either a gravitational field or there isn't. There's nothing illusory going on.

 

[Grav Velocity]

Any spacetime except Schwarzschild and Kerr should be an example. So a charged black hole, FLRW, Milne, any of the pp wave spacetimes, a two body spacetime. I am not sure about non-vacuum spherically symmetric spacetimes

Interesting, so when applying the Einstein field equations to these spacetimes you could obtain a gravitational potential that would not match the time dilation for particular location in the field?

Or perhaps you mean the gravitational potential and gravitational velocity would not longer match in these spacetimes...

 

[Ibix]

Interesting, so when applying the Einstein field equations to these spacetimes you could obtain a gravitational potential that would not match the time dilation for particular location in the field?

You can't obtain a gravitational potential at all except in the cases of Kerr and Schwarzschild (and a few others that exhibit time-translation symmetry).

 

[Grav Velocity]

I don't think "causes" is a helpful term here. You can recover Newtonian gravity in the weak field/low velocity limit of relativity, and it turns out to be the same term in the metric that leads to time dilation that leads to Newtonian gravity.

And "illusion of gravity" makes no sense. There's either a gravitational field or there isn't. There's nothing illusory going on.

Well, I have heard it phrased that way.

It it trying to capture the notion that gravity is not a true force, but rather a result of the gradient of the time dilation (space-time dilation, but really it is time dilation that causes the movement).

 

[Grav Velocity]

I listed a bunch in post 7.

I don't see any... Here is what I see in post 7:

They are unified, just not this way. Time dilation always is the ratio of coordinate time and proper time. That applies for all coordinate systems with a time coordinate in all spacetimes in both general and special relativity.

I am looking for an example where the gravitational potential at a location would not match the free-fall velocity of a particle in that same location, or where the time dilation at that location does not match that indicated by the associated gravitational potential (or velocity).

 

[Dale]

Interesting, so when applying the Einstein field equations to these spacetimes you could obtain a gravitational potential that would not match the time dilation for particular location in the field?

Or perhaps you mean the gravitational potential and gravitational velocity would not longer match in these spacetimes...

As @Ibix mentioned, most don’t even have a gravitational potential. Of those that do, the gravitational potential is directly related to time dilation, but it is surprisingly difficult to get a “flowing space” idea to fit.

I don't see any... Here is what I see in post 7:

You already quoted the list in your post 20.

 

[Nugatory]

I don't see any... Here is what I see in post 7:

You may not have scrolled down to to the rest of the post, which says

Any spacetime except Schwarzschild and Kerr should be an example. So a charged black hole, FLRW, Milne, any of the pp wave spacetimes, a two body spacetime. I am not sure about non-vacuum spherically symmetric spacetimes

 

[Grav Velocity]

the term "gravity" can have several different meanings.

Well, that is bad on physics. One of the most important terms in the science has several different meanings?

This effectively means that "gravitational time dilation" is a poor term in and of itself, since it is unclear which of the several meanings of gravity is being referred to. Unfortunately, it still receives common usage and I say it is vague at best, misleading at worst.

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.

Again, when some asks you for the "amount of gravity" 99% of the time you are going to give gravitational acceleration. Probably 99.999%

More precisely, they are energy (potential energy) per unit mass (because objects of different masses follow the same trajectories if gravity alone is acting on them) units.

Now this is more interesting. I would say if you are going to make potential energy per unit mass the defining metric for time dilation, than it would be useful to connect that metric to special relativity and velocity in that frame.

Then again, velocity is more intuitive.

No, because the fact that two things happen to have the same units does not mean they are the same thing. Potential energy per unit mass is not the same as velocity.

Hmm, but they both cause time dilation by the exact same amount. What a coinsidence.

In situations where the concepts of "gravitational potential", "gravitational time dilation", and "acceleration due to gravity" make sense, the acceleration due to gravity is minus the gradient of the gravitational potential, and gravitational time dilation depends on the gravitational potentail. But those concepts only make sense in stationary spacetimes.

Stationary spacetime would seem to cover a lot of situations.

Nonetheless, I guess I will have to investigate.

I find it hard to believe that somewhere buried in the field equations there is not a term that will yield a value that is effectively the space-time velocity even in non-stationary spacetimes.

 

[Nugatory]

Well, that is bad on physics. One of the most important terms in the science has several different meanings?

You are forgetting that math and not English is the language of physics. The math is crystal clear and unambiguous; the ambiguity in the natural language description is a problem with the natural language not the way we state the physics mathematically.

At this point, I suggest that you back off for a moment and study two important concepts: how relativity of simultaneity leads to velocity-based time dilation; and how gravitational time dilation is predicted by Einstein’s elevator thought experiment. This will help you understand what’s really going on with time in relativity.

 

[PeterDonis]

when applying the Einstein field equations to these spacetimes

You don't apply the Einstein Field equation to a spacetime. You obtain a spacetime by solving the Einstein Field equation with some particular set of assumptions. For example, if you assume vacuum and spherical symmetry and solve the Einstein Field Equation, you get Schwarzschild spacetime.

you could obtain a gravitational potential that would not match the time dilation for particular location in the field?

No, what you get in spacetimes that are not stationary is that the concepts of "gravitational potential" and "gravitational time dilation" don't even make sense.

 

[PeterDonis]

that is bad on physics

No, it's bad on you for forgetting that, as @Nugatory has pointed out, physics is not done in ordinary language, it's done in math.

This effectively means that "gravitational time dilation" is a poor term in and of itself

No, it doesn't; for the spacetimes in which "gravitational time dilation" makes sense at all, its meaning is unambiguous. But "gravitational time dilation" and "gravity" are not the same term; the former is much more specific.

I say it is vague at best, misleading at worst.

I would strongly advise you to spend considerable time improving your understanding instead of making such pronouncements.

"gravity" typically refers to the attraction between to massive objects

In Newtonian physics, yes, but GR is not Newtonian physics.

I would say if you are going to make potential energy per unit mass the defining metric for time dilation, than it would be useful to connect that metric to special relativity and velocity in that frame.

You are wrong. They are two different things, as I already explained way back in post #10. Did you read it?

they both cause time dilation by the exact same amount

No, they don't; the "time dilation" in the two cases does not work the same way. Go back and read post #10. Your understanding is flawed.

Stationary spacetime would seem to cover a lot of situations.

It covers a lot of commonly analyzed ones, sure. But there are also a lot of commonly analyzed ones that it does not cover. To name just the two most important cases:

Any system with more than one gravitating body: binary star systems, the solar system if you try to take into account the spacetime curvature produced by anybody in addition to the Sun, neutron star mergers, black hole mergers, galaxies, etc.

The universe as a whole, i.e., the entire field of cosmology.

I guess I will have to investigate.

Good idea.

I find it hard to believe that somewhere buried in the field equations there is not a term that will yield a value that is effectively the space-time velocity even in non-stationary spacetimes.

Whether or not you find it hard to believe, it is true.

 

[PeterDonis]

they both cause time dilation by the exact same amount. What a coinsidence.

Your attitude here is getting you close to a warning and having your thread closed. You have a number of knowledgeable people in this thread who are giving you valuable information that you can use as a starting point to improve your understanding. Please take heed.

 

[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?