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Yes.
Great. I see that most of the time the formula for gravitational potential is noted as -GmM / r . Based on what I said, this formula shows the energy that is needed to bring an object from a radius of infinity down to a radius of r (for example the radius of Earth). Correct?

That being said, now that I understand that just a difference in potential energy causes time dilation even if g is constant, does this mean that in case of a varying g with height, time dilation should change even stronger over a Δh compared to the same Δh in a constant g? I'm thinking this because the difference between the velocities between emitter and receiver would be larger in case of a varying g with height.

jbriggs444
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difference between the velocities between emitter and receiver would be larger in case of a varying g with height.
It is the difference in potential, not the difference in velocity. Do you think that a varying g means an increasing g?

It is the difference in potential, not the difference in velocity. Do you think that a varying g means an increasing g?
No, decreasing with distance. The thing is, that according to the Equivalence Principle, the difference in velocities between emitting and receiving is the cause for time dilation, even if the acceleration of the emitter and receiver are the same. If it's the difference in gravity potential that causes this instead, then explaining gravitational time dilation by using the Equivalence Principle would be obsolete since it's explaining it by the difference in velocities.

PeterDonis
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Based on what I said, this formula shows the energy that is needed to bring an object from a radius of infinity down to a radius of r (for example the radius of Earth). Correct?
Yes. With the convention for the "zero point" of potential energy that that formula implies, an object at infinity has zero potential energy, and an object at a finite radius has negative potential energy, which means the process of moving the object from infinity to a finite radius can be made to produce energy.

does this mean that in case of a varying g with height, time dilation should change even stronger over a Δh compared to the same Δh in a constant g? I'm thinking this because the difference between the velocities between emitter and receiver would be larger in case of a varying g with height.
You're mixing up two scenarios. The scenario with constant g is a local approximation, in a small patch of spacetime; that is the scenario in which the equivalence principle applies. The scenario with g varying with height is not local--if you can see any variation of g with height, then you are dealing with a region of spacetime that is too large to apply the equivalence principle. So you can't make the comparison you are trying to make here; there is no such thing as a constant g with height over a large range of height.

according to the Equivalence Principle, the difference in velocities between emitting and receiving is the cause for time dilation
Only locally. You can't use the equivalence principle to understand time dilation over a large range of heights. See above.

If it's the difference in gravity potential that causes this instead, then explaining gravitational time dilation by using the Equivalence Principle would be obsolete since it's explaining it by the difference in velocities.
No, this is not correct. The equivalence principle and potential energy are not two different competing explanations for time dilation. They are two different aspects of the same explanation.

No, this is not correct. The equivalence principle and potential energy are not two different competing explanations for time dilation. They are two different aspects of the same explanation.
Ah, I see. The problem is, I can understand time dilation if it's explained mathematically by the difference in velocity, as shown in de graph in my OP. But how can time dilation be explained by the difference in gravity potential? Especially when the equivalence principle can't be used for time dilation over a large range of heights?

vanhees71
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In the non-relativistic limit the pseudo-metric reads
$$g_{00}=1-\frac{2MG}{r}, \quad g_{0j}=g_{j0}=0, \quad g_{ij}=-\delta_{ij}.$$
Here, latin indices run from 1 to 3, and I set ##c=1##. From this you get
$$\mathrm{d} \tau^2 = \left (1-\frac{2MG}{r} \right) \mathrm{d} t^2-\mathrm{d} \vec{x}^2, \quad r=|\vec{x}|$$
A clock's proper time is thus
$$\mathrm{d} \tau = \mathrm{d} t \sqrt{1-\frac{2MG}{r}} \simeq \mathrm{d} t \left (1-\frac{MG}{r} \right).$$
The coordinate time ##t## is the time a clock very far from the mass ##M## (which is located around the origin of the spatial coordinate system).

In the non-relativistic limit the pseudo-metric reads
$$g_{00}=1-\frac{2MG}{r}, \quad g_{0j}=g_{j0}=0, \quad g_{ij}=-\delta_{ij}.$$
Here, latin indices run from 1 to 3, and I set ##c=1##. From this you get
$$\mathrm{d} \tau^2 = \left (1-\frac{2MG}{r} \right) \mathrm{d} t^2-\mathrm{d} \vec{x}^2, \quad r=|\vec{x}|$$
A clock's proper time is thus
$$\mathrm{d} \tau = \mathrm{d} t \sqrt{1-\frac{2MG}{r}} \simeq \mathrm{d} t \left (1-\frac{MG}{r} \right).$$
The coordinate time ##t## is the time a clock very far from the mass ##M## (which is located around the origin of the spatial coordinate system).
Thanks. I've still yet to delve into pseudo-metrics and its formula derivations.

I've got 2 questions regarding your formula:

1. Is the Equivalence Principle explanation some kind of alternative substitute for the gravitational potential explanation in case of a local scenario?
2. If so, does this mean that if you apply the gravitational potential formula in your post for a local scenario, it would dilate the time by the same factor the Equivalence Principle would? Thus, time dilated by a infinitesimal small change in g (local) would be approximately equivalent to the time dilated by the difference in velocities?

jbriggs444
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2. If so, does this mean that if you apply the gravitational potential formula in your post for a local scenario, it would dilate the time by the same factor the Equivalence Principle would? Thus, time dilated by a infinitesimal small change in g (local) would be approximately equivalent to the time dilated by the difference in velocities?
It's not the change in g over distance. It's the integral of g over distance.

It's not the change in g over distance. It's the integral of g over distance.
Sorry, that's what I meant in the question.

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PeterDonis
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I can understand time dilation if it's explained mathematically by the difference in velocity
Which only works locally, in a small enough patch of spacetime such that there is no difference in g within the patch. So this explanation doesn't help you over a large enough range of height that differences in g are observable.

how can time dilation be explained by the difference in gravity potential?
Because that's how the math works. It's different math from the math you're using in a local inertial frame, but it still works.

To put it another way, the time dilation between observers at different heights in a gravity well is simply part of the geometry of spacetime in this particular scenario. That is the explanation.

Especially when the equivalence principle can't be used for time dilation over a large range of heights?
The equivalence principle can't tell you anything about the geometry of spacetime anyway, because by definition it only works in a small enough patch of spacetime that tidal gravity (which is the physical observation referred to by "the geometry of spacetime") is unobservable.

Basically, you're looking at things backwards. The geometry of spacetime is the fundamental entity in GR; the equivalence principle is just one of many possible approximations. You're trying to make the EP the fundamental entity and derive everything else from it. That won't work.

PeterDonis
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Is the Equivalence Principle explanation some kind of alternative substitute for the gravitational potential explanation in case of a local scenario?
No. It's just a different way of looking at the same explanation (or a local approximation of it). See my post #29.

does this mean that if you apply the gravitational potential formula in your post for a local scenario, it would dilate the time by the same factor the Equivalence Principle would?
The gravitational potential formula is exact and applies over any height change, however small. The local EP formula is an approximation to it that only works for small enough height changes that no change in g is observable. So you are once again looking at it backwards: the question isn't whether the potential formula gives the same answer as the EP formula, the question is whether, and under what condtions, the EP formula (the approximation) gives the same answer (to within the desired accuracy) as the potential formula (the exact formula).

The equivalence principle can't tell you anything about the geometry of spacetime anyway, because by definition it only works in a small enough patch of spacetime that tidal gravity (which is the physical observation referred to by "the geometry of spacetime") is unobservable.

Basically, you're looking at things backwards. The geometry of spacetime is the fundamental entity in GR; the equivalence principle is just one of many possible approximations. You're trying to make the EP the fundamental entity and derive everything else from it. That won't work.

Yes, what you describe here was how I was considering EP previously. But I have already noticed and am aware that it only works in a small enough patch of spacetime which made me ask how gravitational potential mathematically proves time dilation. Vanhees71 replied to that but I still have yet to understand this different math problem.

The gravitational potential formula is exact and applies over any height change, however small. The local EP formula is an approximation to it that only works for small enough height changes that no change in g is observable. So you are once again looking at it backwards: the question isn't whether the potential formula gives the same answer as the EP formula, the question is whether, and under what condtions, the EP formula (the approximation) gives the same answer (to within the desired accuracy) as the potential formula (the exact formula).
Ah, ok. So in case of an infinitesimal small g change, EP would give a sufficient approximation for the potential formula?

I think an analogue for what EP is for the Potential formula is what Newton's formulas are for relativistic ones.

PeterDonis
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in case of an infinitesimal small g change, EP would give a sufficient approximation for the potential formula?
Basically, yes.

I think an analogue for what EP is for the Potential formula is what Newton's formulas are for relativistic ones.
Not really. The Newtonian formulas are approximations to the relativistic ones that hold in a particular kind of spacetime, in which there is an isolated source of gravity surrounded by empty space, and in which the source of gravity is weak and all objects are moving slowly compared to the speed of light.

The EP is valid in a sufficiently small patch of any spacetime whatsoever; there is no limitation that gravity be weak or that objects be moving slowly. The only limitation is that the patch of spacetime be small enough that tidal gravity is not observable.

vanhees71