# Clarifying Equivalence Principle

Frames under identical acceleration, regardless of whether due to gravity or accelerated motion, are locally identical but time runs slower in a gravity well. For example, at the center of a large mass (like the one pictured, below) time runs more slowly than for an inertial observer far from any large masses. jfizzix
Gold Member
Time runs slower in any accelerating reference frame, not just in gravity wells.
The equivalence principle really makes no distinction between a constant uniform gravity field and an accelerating reference frame.
The only real qualitative difference in practice between being near a gravity well, and accelerating in deep space is that there are no minute tidal forces in the accelerating reference frame (because the gravity field would not be completely uniform).

A.T.
Frames under identical acceleration, regardless of whether due to gravity or accelerated motion, are locally identical but time runs slower in a gravity well.
In an accelerating rocket you will also find a frequency shift between back and front.

Jonathan Scott
Gold Member
Frames under identical acceleration, regardless of whether due to gravity or accelerated motion, are locally identical but time runs slower in a gravity well. For example, at the center of a large mass (like the one pictured, below) time runs more slowly than for an inertial observer far from any large masses.
Time dilation is not directly related to acceleration, so I don't understand your point here.

A static gravitational field does not affect the energy of anything in free-fall, so any fixed observer would see a falling object or light beam to have constant energy or frequency. Redshift in this situation depends on the potential at the location of the observer. Any fixed observer sees a falling or rising light-beam to have a constant frequency as observed at all heights (and sees a falling object to have constant total energy). However, an observer at a higher potential will see the frequency of a beam emitted at a lower potential to be red-shifted (slowed) relative to the value as seen by an observer at the potential at which the beam was emitted.

stevendaryl
Staff Emeritus
Frames under identical acceleration, regardless of whether due to gravity or accelerated motion, are locally identical but time runs slower in a gravity well.

It's not clear from the above statement whether you are using "time runs slower in a gravity well" as an example of how gravity is different from acceleration. If you are, then you may not realize that the way that Einstein predicted gravitational time dilation was by first proving that inside an accelerating rocket (or elevator, to be accurate) clocks that are higher up run faster than clocks that are lower. Then he applied the equivalence principle to predict that the same would be true for clocks in a gravitational field.

So gravitational time dilation is a prediction that follows from Special Relativity and the Equivalence Principle.

• PeterDonis
My point is that, though an observer at the center of the sphere experiences zero acceleration, time runs slower for them than for an inertial observer far from any large masses. Both experience zero acceleration but not the same time rate.

PeterDonis
Mentor
So gravitational time dilation is a prediction that follows from Special Relativity and the Equivalence Principle.

"Gravitational time dilation" can be used to mean at least two different things, and only the first is a local phenomenon that follows from SR and the EP. That is the fact that, as you say, a clock at the top of an accelerating rocket will run faster than a clock at the bottom. But the argument for this based on SR and the EP is limited to a single local inertial frame (and requires the two clocks to be accelerating in that frame). It cannot cover the second meaning of "gravitational time dilation", an example of which is described in post #6. That example involves two inertial observers, which sharpens the point, but the general point applies to any pair of observers which cannot be contained in a single local inertial frame. The difference between the "rate of time flow" of any such pair of observers (assuming we are in a spacetime where that concept makes sense, i.e., a stationary spacetime) arises from the global geometry of spacetime and cannot be explained by the SR/EP argument.

stevendaryl
Staff Emeritus
"Gravitational time dilation" can be used to mean at least two different things, and only the first is a local phenomenon that follows from SR and the EP. That is the fact that, as you say, a clock at the top of an accelerating rocket will run faster than a clock at the bottom. But the argument for this based on SR and the EP is limited to a single local inertial frame (and requires the two clocks to be accelerating in that frame). It cannot cover the second meaning of "gravitational time dilation", an example of which is described in post #6. That example involves two inertial observers, which sharpens the point, but the general point applies to any pair of observers which cannot be contained in a single local inertial frame. The difference between the "rate of time flow" of any such pair of observers (assuming we are in a spacetime where that concept makes sense, i.e., a stationary spacetime) arises from the global geometry of spacetime and cannot be explained by the SR/EP argument.

Yes, but if I remember correctly, Einstein came up with his prediction of gravitational time dilation a few years before he developed General Relativity. The more general case you are talking about requires full GR.

My point is that, though an observer at the center of the sphere experiences zero acceleration, time runs slower for them than for an inertial observer far from any large masses. Both experience zero acceleration but not the same time rate.

That is because gravitational time dilation is not directly the result of locally measured acceleration - as Jonathan Scott has already pointed out -, but is due to the geometry of spacetime "in-between" the two frames you are comparing. To understand what I mean by this, the analogy (!) of the table mountain may be useful - the top of the mountain is perfectly flat, just like the land the mountain itself stands on, but it sits at a different altitude than does the surrounding terrain. The same in your example - both observers are stationary with respect to one another ( one at infinity, the other one at the centre of the mass ), and measure no proper acceleration in their rest frames, but they sit at different places in a globally curved spacetime. Hence, the relationship between them is non-trivial, and time dilation is one manifestation of this.

Jonathan Scott
Gold Member
One way to see the close relationship between time dilation due to velocity and due to gravity is to consider the example of the rotating space station, rotating with angular velocity ##\omega##.

At distance ##r## from the axis, the speed is ##v = r\omega## and from special relativity the time rate relative to an external observer is then ##\sqrt{1-v^2/c^2}##, which is approximately ##1 - (1/2) (v^2/c^2)## for non-relativistic speeds, so the fractional difference in time rate at two different distances is equal to the difference in ##(1/2) (v^2/c^2)##.

One can also consider the point of view of a person in the space station who feels an acceleration like gravity, equal to ##v^2/r = r\omega^2## in the radial direction, so the difference in effective gravitational potential (in dimensionless units, potential energy per total energy) is ##1/c^2## times the integral of the acceleration between two values of ##r##, which is the change in ##(1/c^2) (r^2/2) \omega^2 = (1/2) (v^2/c^2)##. Both of these methods therefore give the same fractional change in time rate.

(Another method is of course to consider clocks at the front and back of a rigid spaceship undergoing a changing Lorentz contraction, but I find the rotation one more enlightening).

So, would it be sensible to state that ultimately the phenomenon of time dilation between frames - be it of the kinematic or the gravitational kind - is due to the fact that those frames do not share the same notion of simultaneity ?

stevendaryl
Staff Emeritus
So, would it be sensible to state that ultimately the phenomenon of time dilation between frames - be it of the kinematic or the gravitational kind - is due to the fact that those frames do not share the same notion of simultaneity ?

I don't think that's correct. Having a shared notion of simultaneity is a matter of convention, but the effects of gravitational time dilation are independent of convention. I start with two identical clocks, set to the same time. I take one of them to the top of Mount Everest, and leave the other at the foot of the mountain. I wait 10 years (or however long). Then I bring the first clock down from the mountain, and compare the two clocks. The one that was on top of the mountain will be slightly ahead of the one at the bottom. You don't need any notion of simultaneity to see this effect.

stevendaryl
Staff Emeritus
I don't think that's correct. Having a shared notion of simultaneity is a matter of convention, but the effects of gravitational time dilation are independent of convention. I start with two identical clocks, set to the same time. I take one of them to the top of Mount Everest, and leave the other at the foot of the mountain. I wait 10 years (or however long). Then I bring the first clock down from the mountain, and compare the two clocks. The one that was on top of the mountain will be slightly ahead of the one at the bottom. You don't need any notion of simultaneity to see this effect.

Both velocity-dependent time dilation and gravitational time dilation can be described in a way that is independent of simultaneity convention. It really amounts to this: Given two different events (points in spacetime, which can be characterized in a coordinate system by 4 numbers: x,y,z,t), A and B, there can be multiple ways to "travel" from A to B, and the proper time for the trip (time as measured by a comoving clock) depends on which path you take.

The corresponding fact about roads on Earth is unsurprising: Two different ways to go from Paris to Berlin can take different amounts of time. Analogously, two different ways to go from event A to event B can take different amounts of proper time.

• PeterDonis
Nugatory
Mentor
Both velocity-dependent time dilation and gravitational time dilation can be described in a way that is independent of simultaneity convention. It really amounts to this: Given two different events (points in spacetime, which can be characterized in a coordinate system by 4 numbers: x,y,z,t), A and B, there can be multiple ways to "travel" from A to B, and the proper time for the trip (time as measured by a comoving clock) depends on which path you take.
That explanation works well for the twin paradox and variants such as your example in the previous post, in which the clock goes up the mountain and down again. However, the traditional velocity-dependent time dilation examples involve three events, not two: the clocks are colocated and read the same; later one clock is here and reads T; at the same time the other clock is somewhere else and reads T'. The gravitational time dilation examples usually involve four events, as the clocks are not colocated at either the start or the end of their respective intervals, which are assumed to start at the same time.

In these cases, the notion of shared simultaneity is essential - we can't even state the conditions of the problem without using that phrase "at the same time".

My personal preference is to think of differential aging, the "different paths have different lengths" thing illustrated by the twin paradox, as a different (and more physically "real") phenomenon than time dilation. One is independent of simultaneity conventions; the other is caused by them.

• PeterDonis and Markus Hanke
stevendaryl
Staff Emeritus
My personal preference is to think of differential aging, the "different paths have different lengths" thing illustrated by the twin paradox, as a different (and more physically "real") phenomenon than time dilation. One is independent of simultaneity conventions; the other is caused by them.

They are sort-of related, though, in that any consistent simultaneity convention will give a "history" that is consistent with the convention-independent differential aging. The same factor of $\sqrt{1-\frac{v^2}{c^2}}$ comes into play, whether you are computing time dilation or differential aging.

PeterDonis
Mentor
if I remember correctly, Einstein came up with his prediction of gravitational time dilation a few years before he developed General Relativity.

Yes, he did.

The more general case you are talking about requires full GR.

Yes, and that means "gravitational time dilation" in the more general sense, which includes pretty much any scenario described by the phrase "time runs slower in a gravity well" (since AFAICT it's extremely rare for that phrase to be used to describe something that is confined to a single local inertial frame), cannot be explained by SR and the EP. It requires full GR.

The change in gravitational potential energy gives the change in wavelength and that gives the change in spacetime.

PeterDonis
Mentor
The change in gravitational potential energy gives the change in wavelength

For cases where the concept of "gravitational potential energy" is well-defined, yes. (More precisely, it gives the change in wavelength for light traveling radially.)

that gives the change in spacetime.

Spacetime doesn't "change". It just is. That is, the 4-dimensional spacetime geometry already contains all the information about "space" at different "times"; nothing in it has to change.

Also, the gravitational redshift does not give complete information about the spacetime geometry. It only gives one particular aspect of it. (And for it to give any information at all about the spacetime geometry, you have to go beyond a single local inertial frame, which means going beyond the scope of the equivalence principle.)

Also, the gravitational redshift does not give complete information about the spacetime geometry. It only gives one particular aspect of it. (And for it to give any information at all about the spacetime geometry, you have to go beyond a single local inertial frame, which means going beyond the scope of the equivalence principle.)

Exactly - it is crucially important to understand this, or else all manner of misconceptions will arise.