# Equivalence principle and time dilation

## Main Question or Discussion Point

I'm currently reading https://www.amazon.com/dp/0199236224/?tag=pfamazon01-20&tag=pfamazon01-20. The book claims that, according to the equivalence principle, acceleration and gravity have the same effects. So if gravity slows down time, shouldn't acceleration also slow down time? The book seems to state this as a matter of fact.

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PeterDonis
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The book claims that, according to the equivalence principle, acceleration and gravity have the same effects.
This looks like a pop science book, not a scientific textbook or paper. Unfortunately, you can't learn science from pop science books, even if they're written by scientists. This statement is too vague to really tell you what the science says, and can easily be misunderstood to be saying something false.

Here is a more precise statement of the equivalence principle. Consider two people: one is standing in his spaceship, which is far out in empty space, well removed from all other objects, and is accelerating at 1 g. The other is standing in a room on the surface of the Earth. The equivalence principle says that, locally, these two situations are equivalent. This means that, if each of the people makes identical measurements over a short enough interval of space and time, they will get identical results; no local experiment will be able to distinguish the two, or to allow either one to tell for sure that he is in the spaceship or in the room on Earth, rather than vice versa.

Note carefully that the force the person standing in the room on Earth feels is not "gravity". It is the force of the Earth pushing up on his feet. Gravity is not a force in relativity. This is one reason why the statement in the book is too vague; it doesn't tell you what it means by "gravity". And if you give "gravity" its obvious meaning to a lay person, namely "whatever it is that makes rocks fall and keeps the Moon in its orbit around the Earth", then the statement in the book is simply wrong; that "gravity" is not equivalent to acceleration, as should be evident from the above.

if gravity slows down time, shouldn't acceleration also slow down time?
"Gravity slows down time" is another of those statements that is too vague to really tell you anything, and which can easily be misunderstood to be saying something false. Again, here is a more precise statement.

Consider two people: one is standing on the surface of the Earth, the other is standing at the top of a tall tower directly above the first. If each of them have sufficiently accurate clocks, and they exchange light signals, they will be able to confirm that the clock of the one at the bottom of the tower is running slower; for example, they can verify that there are fewer ticks of the bottom clock between two successive round-trip light signals than there are ticks of the top clock. This is what is actually meant by the common pop science statement that "gravity slows down time".

And in this sense of the statement, it is true that "acceleration also slows down time". For example, consider two more people, inside a spaceship far out in space that is accelerating at 1 g. One is at the rear of the ship; the other is at the front. By similar measurements to those the first pair made (exchaging round trip light signals and counting ticks of each one's clock between successive signals), they can verify that the rear clock is running slower than the front clock, and if the length of the ship is the same as the length of the tower, the difference in clock rates will be the same in both cases. (Strictly speaking, the tower has to be short enough compared to the size of the Earth for this to be true to a good approximation.)

However, note carefully that the difference in clock rates in both these cases is not due to any difference in acceleration; it is purely due to a difference in position. In the above examples, the acceleration of both clocks is the same. The only difference between them is position. This is why we say that acceleration does not affect clock rates, and why the statement in the book is misleading because it invites you to think otherwise. The correct statement is that position in a gravitational field, or in an accelerating spaceship, affects clock rates.

Jimster41
Dale
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The book claims that, according to the equivalence principle, acceleration and gravity have the same effects. So if gravity slows down time, shouldn't acceleration also slow down time?
Gravitational acceleration doesn't slow time down. Simplistically, gravitational potential is what slows time, and the same thing happens in accelerating reference frames.

See Peters answer above for a much more complete explanation.

PAllen
2019 Award
You can also treat both these cases as (essentially) pure relative velocity Doppler if each is analyzed from a momentarily co-moving inertial frame. In the case of the tall building in gravity, the inertial frame is equivalent to special relativity in standard coordinates only up to tidal affects - which are completely negligible in his scenario (unless you building is miles high).

This is why most modern writers recast the equivalence principle to an equivalent form: a local free fall frame near massive bodies is indistinguishable (up to tidal effects) from an inertial frame far away from everything.

See, for example:

http://relativity.livingreviews.org/Articles/lrr-2014-4/articlese2.html#x5-20002 [Broken]

for a discussion of modern formulations of the equivalence principle.

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Probably I now found the"bug" of several earlier discussions, and the following reformulation will also be useful for the OP:
[..] And in this sense of the statement, it is true that "acceleration also slows down time". For example, consider two more people, inside a spaceship far out in space that is accelerating at 1 g. One is at the rear of the ship; the other is at the front. By similar measurements to those the first pair made (exchaging round trip light signals and counting ticks of each one's clock between successive signals), they can verify that the rear clock is running slower than the front clock [..]. The only difference between them is position. This is why we say that acceleration does not affect clock rates [..]The correct statement is that position in a gravitational field, or in an accelerating spaceship, affects clock rates.
SR is correct when gravitational fields don't play a role according to GR. That is the case in this rocket example. According to SR, acceleration does not slow down time as function of position but it may appear so according to an observer who does not think to be accelerating. Acceleration involves changes in velocity relative to any Galilean reference system and according to SR it includes the Doppler effect as discussed by PAllen in post #4.

PS compare also this post by pervect in a similar thread: #4 . If the observer ascribes the observations to gravitation instead of acceleration, then his/her explanation will be that the clock rates are different.

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PeterDonis
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According to SR, acceleration does not slow down time as function of position but it may appear so according to an observer who does not think to be accelerating.
Please bear in mind the key distinction between coordinate time and proper time. What you say is correct if by "time" you mean coordinate time. But it is not correct if you mean proper time. The fact that two Rindler observers in flat spacetime can exchange repeated light signals and confirm that (a) they are at rest relative to each other, since the round-trip light travel time does not change; and (b) the rear observer's clock runs slower than the front observer's clock, as shown by their respective proper times elapsed between two successive signals, is an invariant fact, independent of coordinates. This invariant fact is what is referred to by the statement that "position in a gravitational field or in an accelerating spaceship affects clock rates".

Also, the same distinction needs to be observed in the case of curved spacetime. The fact that two observers at rest in a static gravitational field can exchange light signals and make similar observations to the Rindler observers in flat spacetime is, again, an invariant fact, independent of all coordinates, which is usually expressed as the clock at the lower altitude "running slower". But one can choose coordinates for which the clock at the lower altitude does not run slower with respect to coordinate time.

If one fails to keep in mind this key distinction, it is easy for discussions of this sort to degenerate into people simply talking past each other, using the same words to refer to different concepts and then wondering why there is never any resolution. There have been plenty of previous threads here which have exhibited exactly this phenomenon.

If the observer ascribes the observations to gravitation instead of acceleration, then his/her explanation will be that the clock rates are different.
But, as I have shown above, the observer also has the option of not "ascribing" the observations to anything, but simply defining "the clock rates are different" purely in terms of invariant quantities that are directly observable, without making any commitment whatsoever about what the observations are "ascribed" to. Again, this is a key distinction that needs to be kept in mind to avoid repeated talking past each other in discussions of this sort.

pervect
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I'm currently reading Relativity: A Very Short Introduction. The book claims that, according to the equivalence principle, acceleration and gravity have the same effects. So if gravity slows down time, shouldn't acceleration also slow down time? The book seems to state this as a matter of fact.
To repeat what a few others have said but a little less abstractly, suppose we have an accelerating elevator, typically called "Einstein's elevator". A person riding in the elevator will feel an inertial force from the acceleration that is locally indistinguishable from gravity.

A clock on the elevator will keep the same time as a non-accelerating clock with the same velocity, called a co-moving clock, presumed to be at the same location.

So acceleration is not directly causing time dilation. The accelerating clock and the inertial clock keep the same time. This is sometimes called the "clock postulate".

What is true, and what your textbook is probablly getting at, is that two clocks at DIFFERENT HEIGHTS on the elevator will tick at different rates when you use the accelerating coordinate system. Two clocks at the same height will still tick at the same rate. Two clocks at the same height will tick at the same rate evein if one of them is accelerating an the other is not accelerating. So the point is that it's not acceleration that causes the clock to tick slowly, it's the product of acceleration * height. This is what is meant when some posters mention that "gravitational potential" (which is acceleartion * height) and not just acceleration which "causes" time dilation.

There is a subtle but important issue here. That issue is how we determine whether clocks "tick at the same rate" can depend on the observer and the exact procedure we use to compare the clocks. However, I won't get into much detail on this subtle point, I'll just mention that the propagation time delay when you send a signal from the lower clock to the upper clock (or back) appears to be constant, and this allows an easy and natural way of comparing their rates - so natural that one doesn't even think much about alternatives.

Nugatory
PAllen
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At risk (certainty?) of completely losing the OP, I'll state my take on the unifying principle behind my post #4 and Peter's #6.

The core principle, stated as simply as possible, is that SR holds everywhere up to second order (tidal) effects. Then to abstract the difference between #4 and #6 from coordinates, you introduce congruences. You get a 'Doppler interpretation' if you introduce a (space-time locally static) timelike geodesic congruence, and you find that clocks of the congruence show no position dependence of rate (as always, up to tidal effects), and other clock rates depend only on velocity relative to this congruence. Alternatively, you can always find a (locally static) congruence such that each world line of the congruence has constant proper acceleration (but it differs for different world lines). Then, you find that the rate of clocks of the congruence depends on position. The universality of SR guarantees you can always locally introduce such congruences.

Then, a key difference between flat spacetime and curved spacetime is that in the former, both of these congruences can be globally* exact. In curved spacetime, the timeilke geodesic congruence can never have the indicated properties exactly or globally; while in specialized situations, the constant acceleration (on each world line) congruence can be globally exact. We call such situations stationary solutions in GR.

*strictly, by globally I mean exact for a finite region. The Rindler congruence cannot fill all of Minkowski space.

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PeterDonis
PAllen
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At risk (certainty?) of completely losing the OP, I'll state my take on the unifying principle behind my post #4 and Peter's #6.

The core principle, stated as simply as possible, is that SR holds everywhere up to second order (tidal) effects. Then to abstract the difference between #4 and #6 from coordinates, you introduce congruences. You get a 'Doppler interpretation' if you introduce a (space-time locally static) timelike geodesic congruence, and you find that clocks of the congruence show no position dependence of rate (as always, up to tidal effects), and other clock rates depend only on velocity relative to this congruence. Alternatively, you can always find a (locally static) congruence such that each world line of the congruence has constant proper acceleration (but it differs for different world lines). Then, you find that the rate of clocks of the congruence depends on position. The universality of SR guarantees you can always locally introduce such congruences.

Then, a key difference between flat spacetime and curved spacetime is that in the former, both of these congruences can be globally* exact. In curved spacetime, the timeilke geodesic congruence can never have the indicated properties exactly or globally; while in specialized situations, the constant acceleration (on each world line) congruence can be globally exact. We call such situations stationary solutions in GR.

*strictly, by globally I mean exact for a finite region. The Rindler congruence cannot fill all of Minkowski space.
I thought of a funny aside to to this: Einstein's goal of a 'general principle of relativity' by virtue of admitting non-inertial frames + (gravity = connection coefficients, as Einstein preferred), is achieved much more successfully in SR rather than GR (per modern definitions)!! That is, the equivalence is exact and global in flat spacetime, while only local and approximate in curved spacetime. In curved spacetime, there are generally fewer (or no(!)) killing vector fields, while flat spacetime has a maximal set of killing vector fields.

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Dale
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SR is correct when gravitational fields don't play a role according to GR.
In addition to the "key distinction" mentioned by Peter Donis, there is another important distinction. That is what is meant by "gravitational fields". This statement is correct if by "gravitational fields" you mean "curvature tensor". It is not correct if by " gravitational field" you mean "Christoffel symbols". Not all authors use the term the same way.

This thread has caught me in a misconception. I thought the clock at the top of the tower ran faster precisely because of tidal effects present in curved spacetime and that clocks at the top and bottom of an accelerating elevator in flat (Minkowski) spacetime would both tick at the same rate because there are no tidal effects associated with linear acceleration.

PAllen
2019 Award
This thread has caught me in a misconception. I thought the clock at the top of the tower ran faster precisely because of tidal effects present in curved spacetime and that clocks at the top and bottom of an accelerating elevator in flat (Minkowski) spacetime would both tick at the same rate because there are no tidal effects associated with linear acceleration.
Yep, that is a BIG misconception. The tidal effect over a tall building remains too small to detect. Pound-Rebka was, alternatively, either a test of pure Doppler (viewed from a free fall frame), or position dependence of clock rate for a non-inertial frame. The principle of equivalence was tested in that measurement for a building on earth matched the pure SR prediction for a tall rocket in deep space.

Note, consistent with this, in the link I gave earlier on tests of EP, Pound-Rebka is listed as a test of 'local position invariance' : equivalent local SR scenario behaves the same near a massive body as in deep space.

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Fielder
Please bear in mind the key distinction between coordinate time and proper time. What you say is correct if by "time" you mean coordinate time. But it is not correct if you mean proper time.[..].
The "bug" that I identified is much more basic. With "accelerating rocket" we mean by definition that d(v-rocket)/dt ≠ 0. All textbooks that I know define acceleration as such, and most probably that is also how the OP understands the meaning of that word.

Dale
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The "bug" that I identified is much more basic. With "accelerating rocket" we mean by definition that d(v-rocket)/dt ≠ 0.
Yes, that is indeed another ambiguity. What you write there is the coordinate acceleration. By "accelerating rocket" you could also mean ##\frac{DU^{\mu}}{d\tau}\ne 0## which is known as proper acceleration.

Thanks everyone for your replies. I went over the relevant section in the book, and it specifies spaceship acceleration rather than gravitational acceleration.

All of this brings me to the following understanding:

Suppose I go to the top of the Leaning Tower of Pisa, and drop an atomic clock. The moment before the clock hits the ground, (1) the clock will be ticking more slowly than when released because its speed is higher; (2) it will be ticking even more slowly because it's closer to the Earth's surface; (3) the acceleration due to gravity does not slow down the clock, although the speed produced by the gravitational acceleration does slow down the clock.

PAllen
2019 Award
Thanks everyone for your replies. I went over the relevant section in the book, and it specifies spaceship acceleration rather than gravitational acceleration.

View attachment 82696
All of this brings me to the following understanding:

Suppose I go to the top of the Leaning Tower of Pisa, and drop an atomic clock. The moment before the clock hits the ground, (1) the clock will be ticking more slowly than when released because its speed is higher; (2) it will be ticking even more slowly because it's closer to the Earth's surface; (3) the acceleration due to gravity does not slow down the clock, although the speed produced by the gravitational acceleration does slow down the clock.
Just to complete the description, instead of unqualified tick rate, you should state: compared to a clock remaining at the top of the tower via e.g. signal exchange, with light travel time factored out (otherwise, you would be including direction dependent classical Doppler). If you want to arrive at a factor of (elevation red shift facgtor)*(speed relative to ground time dilation), you need to specify removal light signal travel times. If you want to describe the raw observation, you would instead have:

(elevation red shift factor) * (SR Doppler relative to ground clock)

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PeterDonis
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With "accelerating rocket" we mean by definition that d(v-rocket)/dt ≠ 0.
Maybe you mean that, but "we" don't necessarily mean that. Again, you are confusing coordinate acceleration with proper acceleration. Proper acceleration can be defined without reference to coordinates at all; and if we define "accelerating rocket" in terms of proper acceleration, we don't have to talk about coordinates or ##dv / dt## at all.

All textbooks that I know define acceleration as such
Really? You've never read a relativity textbook that talks about proper acceleration? Have you read Misner, Thorne, and Wheeler?
most probably that is also how the OP understands the meaning of that word
If so, he is welcome to clarify that point. Then I will be happy to clarify the key distinction between coordinate acceleration and proper acceleration and how it helps to resolve his confusion.

Yes, that is indeed another ambiguity. What you write there is the coordinate acceleration. By "accelerating rocket" you could also mean ##\frac{DU^{\mu}}{d\tau}\ne 0## which is known as proper acceleration.
I have no specific coordinates in mind... Do you mean that according to you the Wikipedia definitions of "acceleration" and "proper acceleration" are wrong? They look correct to me:
"Acceleration, in physics, is the rate of change of velocity of an object."
"proper acceleration is [...] acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured."

PS. I suppose that this comment in a thread on "proper acceleration" is correct: #4 . In other words, for the analysis of a rocket "in deep space" it is quite irrelevant if d(v-rocket)/dt is supposed to refer to acceleration with v<<c, or to proper acceleration in which v-rocket(t+dt)=dv, which is also <<c; that ambiguity does not really matter here.

Apparently the bug here is really a bug in the physics, which is only somewhat obscured by different ways of phrasing.

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Thanks everyone for your replies. I went over the relevant section in the book, and it specifies spaceship acceleration rather than gravitational acceleration.

View attachment 82696
All of this brings me to the following understanding:

Suppose I go to the top of the Leaning Tower of Pisa, and drop an atomic clock. The moment before the clock hits the ground, (1) the clock will be ticking more slowly than when released because its speed is higher; (2) it will be ticking even more slowly because it's closer to the Earth's surface; (3) the acceleration due to gravity does not slow down the clock, although the speed produced by the gravitational acceleration does slow down the clock.
Albert Einstein would no doubt say that that is all correct, assuming that you use the ECI frame as reference for your analysis (speed is of course frame dependent). The people who actually did such an experiment with a falling atomic clock ("Gravity probe A") used an equation that says the same, also for the ECI frame (a third term was added to correct for the acceleration of the earth station).

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PeterDonis
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proper acceleration is [...] acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured
I would say this is a description of how proper acceleration could be modeled in coordinate terms, but it's not a definition of proper acceleration. The definition of proper acceleration is the derivative of 4-velocity with respect to proper time. Physically, it's what is measured by an accelerometer. You don't need a co-located inertial observer to define or measure it.

Apparently the bug here is a bug in the physics.
I don't see any bug at all in the physics. The only bug I see is that you are confusing different concepts of acceleration.

Mentz114
PeterDonis
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the acceleration due to gravity does not slow down the clock
Note that this "acceleration due to gravity" is coordinate acceleration, not proper acceleration. So the question of whether this kind of acceleration affects clock rates (it doesn't) is different from the question of whether proper acceleration (i.e., acceleration that is actually felt by the clock--the clock dropped from the Leaning Tower of Pisa is in free fall, feeling zero acceleration) affects clock rates (it doesn't as long as the felt acceleration does not disrupt the physical structure of the clock).

Dale
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I have no specific coordinates in mind.
Then both v and t are undefined so your definition of "accelerating rocket" is invalid. Without specific coordinates you are limited to coordinate-invariant quantities like proper acceleration.

Apparently the bug here is really a bug in the physics, which is only somewhat obscured by different ways of phrasing.
I don't think this is a bug in the physics at all. This appears to me to simply be a confusion stemming from an ambiguous use of an ambiguous term.

If you believe that it is a bug in the physics, then please state the bug using math or the corresponding unambiguous terms. I.e. never just "time" but either "proper time" or "coordinate time", similarly with "acceleration", and avoid "gravitational field" altogether in favor of the actual quantity intended ("curvature tensor" or "Christoffel symbols").

Mentz114
Dale
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The definition of proper acceleration is the derivative of 4-velocity with respect to proper time.
I would clarify by saying "covariant derivative".

PeterDonis
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I would clarify by saying "covariant derivative".
Actually, the correct technical term is "absolute derivative". Written out in components, the covariant derivative of the 4-velocity ##u^a## would be ##\nabla_b u^a##; this is a 1-1 tensor, not a 4-vector. The proper acceleration, written in components, is ##u^b \nabla_b u^a##, which is a 4-vector. In other words, the "derivative with respect to proper time" operator ##d / d\tau##, which is called the "absolute derivative", when written out in components, is ##u^a \nabla_a##, i.e., it's the contraction of the 4-velocity with the covariant derivative.

Dale
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Actually, the correct technical term is "absolute derivative". Written out in components...
Excellent, even better.