# How does the clock postulate apply in non-inertial frames?

I've read countless answers and other sources on the question of whether time dilation is caused both by velocity and acceleration or only by velocity, but they all look at things only from an inertial frame and/or use the popular version of the clock postulate, so they don't seem to answer the questions I pose here. Many of them also talk about it in terms of relative lengths of different paths through spacetime, but I think that sidesteps the question of whether acceleration itself can cause those lengths to change in accelerating frames (e.g., by causing a different metric to apply).

It's stated in numerous sources (e.g., the article "Does a clock's acceleration affect its timing rate?" by Don Koks, posted by John Baez, and this section in Wikipedia) and in answers by reputable users of this site (e.g., here and here) that the clock postulate of special relativity says that time dilation (among other relativistic effects) is caused only by relative velocity and that acceleration itself has no direct effect on it.

But this seems problematic to me because it only ever seems to be true in inertial reference frames. For example, in the (non-inertial) frame of the traveling twin in the twin paradox, the earthbound twin's time slows down during each inertial leg of the trip, but during the turnaround, it speeds up to an extent that indicates that the traveler's acceleration towards the earth is equivalent to a gravitational field that he resists while it pulls the earth towards him thus causing him to experience something equivalent to gravitational time dilation relative to the earth. How can it be said that this kind of time dilation is caused by velocity rather than acceleration?

Edit: To be more explicit about what the clock postulate claims according to some: In a comment, Dale says that "In a non-inertial frame, time dilation depends on velocity and position, not acceleration." The answers given here so far, however, don't seem to support this statement. How can it be supported? Edit: Dale has addressed this in his answer.

Now, there seems to be an alternative formulation of the postulate. According to this paper:

The clock hypothesis of relativity theory equates the proper time experienced by a point particle along a timelike curve with the length of that curve as determined by the metric.

Perhaps someone here will show me that this formulation is logically equivalent to the other one (in all frames, including non-inertial ones), but from what I can tell, it actually allows acceleration to cause time dilation. For example, in the traveling twin's frame where you have to use a metric like the Rindler metric during the turnaround (per the paper, "the restriction to Minkowski spacetime and inertial motion has been dropped"), you find that his curve is shorter than the earthbound twin's during that acceleration and thus that his own time dilates relative to the earth's, apparently due to his acceleration towards it. Therefore, unlike the other version of the postulate, this one works in non-inertial frames and seems to be consistent with acceleration directly causing time dilation. Is this true? If so, is this version of the postulate more correct?

• Minor comment to the post (v5): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. Apr 21, 2022 at 4:20
• Special relativity theory applies only to inertial reference frames. If you are searching for gravity/acceleration effects on time dilation and similar effects,- you must take general relativity path,- that's why Einstein have created it. Apr 21, 2022 at 7:39
• @AgniusVasiliauskas GR applies to curved spacetime while SR applies to flat spacetime, and acceleration can happen in the latter. See the last paragraph of the "The spacetime metric, or interval" section of the first link I shared as well as my comment on that. Apr 21, 2022 at 8:36
• @GumbyTheGreen Acceleration is equivalent to curved spacetime, that's why Einstein has invented an equivalence principle. Apr 21, 2022 at 8:41
• @AgniusVasiliauskas That's only true to a point—the equivalence principle has limitations. If you can find a good modern source that specifically says that SR doesn't include non-inertial frames or the Rindler metric, please share the reference. Every one that I've found says otherwise. Apr 21, 2022 at 8:57

The clock hypothesis of relativity theory equates the proper time experienced by a point particle along a timelike curve with the length of that curve as determined by the metric.

This is indeed the correct general formulation of the clock hypothesis. This formulation applies for all reference frames, inertial or not, and for all spacetimes, flat or curved. The only restriction is that the coordinate basis must have one timelike basis, $$dt$$.

Because of this, time dilation can be written as $$\frac{1}{\gamma}=\frac{d\tau}{dt}$$ where $$\gamma$$ is the time dilation factor and $$d\tau$$ is the proper time, which is related to the metric by $$ds^2=-c^2 d\tau^2=g_{\mu\nu}dx^\mu dx^\nu$$

So, for an inertial frame we have $$ds^2= -c^2 dt^2+ dx^2 + dy^2 + dz^2$$ $$\frac{d\tau^2}{dt^2}=1-\frac{dx^2}{c^2 dt^2}-\frac{dy^2}{c^2 dt^2}-\frac{dz^2}{c^2 dt^2}$$ $$\frac{1}{\gamma}=\sqrt{1-\frac{v^2}{c^2}}$$ which is the usual familiar time dilation formula.

Note that it has the property that it depends only on the velocity and not the acceleration. When calculating $$d\tau/dt$$ we get terms like $$dx^2/dt^2$$, which is the square of a component of velocity, not an acceleration like $$d^2x/dt^2$$ would be.

Therefore, unlike the other version of the postulate, this one works in non-inertial frames and seems to be consistent with acceleration directly causing time dilation

Although the general formula does work in non-inertial frames, it still does not give a situation where acceleration directly causes time dilation. Let’s work it out for the Rindler metric you mentioned. For convenience I will use units where $$c=1$$: $$ds^2= -(gx)^2 dt^2+ dx^2+dy^2+dz^2$$ $$\frac{d\tau^2}{dt^2}=(gx)^2-\frac{dx^2}{dt^2}-\frac{dy^2}{dt^2}-\frac{dz^2}{dt^2}$$ $$\frac{1}{\gamma}=\sqrt{(gx)^2-v^2}$$

Now, you might be tempted to say “look, it has $$g$$ which is a pseudo gravitational acceleration, so the Rindler time dilation is directly related to acceleration”. However, closer inspection shows that it is not just $$g$$, but $$gx$$ which is the form of a gravitational potential, not gravitational acceleration.

Furthermore although $$g$$ has units of acceleration, it is a property of the particular coordinates chosen, not the acceleration of any worldline. The time dilation of a specific object does not depend on that object’s acceleration, only its position and velocity with respect to the chosen coordinates. If an object is momentarily at rest ($$v=0$$) at some $$x=x_0$$ then it’s time dilation is $$\gamma=1/(gx_0)$$, regardless of what that object’s acceleration is. So this time dilation term is a function of position, not acceleration.

It turns out that this is a general fact. In the metric the terms $$g_{\mu\nu}$$ are functions of position only, not velocity nor acceleration. And so when you divide by $$dt^2$$ you only get terms that are functions of position times velocities, like $$g_{xy}\frac{dx}{dt}\frac{dy}{dt}$$ You simply cannot get anything else because of the form of the metric.

• Thanks for the detailed, direct replies to my questions and points. I'm following but I can't shake the feeling that it's a matter of interpretation, wording, and the way that the math of relativity has been framed by us humans. If acceleration determines the metric, and the metric—along with position—determines time dilation, can't we say that acceleration determines time dilation as much as position does? Apr 21, 2022 at 10:00
• When the traveling twin is a given distance from Earth, there's a direct causal link between his thrust and his accounting of the speed of Earth's time. Isn't that enough to say that from a physical standpoint, acceleration causes time dilation, at least in part? Finally, you said that "although $g$ has units of acceleration, it is a property of the spacetime", but would it be more correct to say that it's a property of the coordinate system since everything is technically still in flat Minkowski spacetime? If so, does that change anything? Apr 21, 2022 at 10:00
• @GumbyTheGreen yes, you are right on that last point, I will update the wording. On the rest, no, for the reasons already given. First, it is the “gravitational” potential not the acceleration, second it is a function of position, and third the g is a property of the coordinates not the objects in the coordinate system regardless of an object’s proper or coordinate acceleration they have a time dilation that depends only on the object’s position and velocity in the coordinates. There is no causal link as you say. We choose the coordinates for convenience, and could make a different choice
– Dale
Apr 21, 2022 at 11:22
• @GumbyTheGreen thanks for the edit, good catch!
– Dale
Apr 21, 2022 at 22:45
• Sure thing! Hmmm, I'll need to stew on all that. Since acceleration is what causes such a coordinate system to be needed, it seems clear to me that there's a chain of causality between it and time dilation. And this "gravitational" time dilation happens in flat spacetime if and only if there's acceleration. So saying that there's no causal link between them seems like a nonphysical math/accounting/semantic trick to me. But I think you've probably made the case as well as it can be made. Apr 22, 2022 at 2:05

I think one way to resolve your conceptual issue is to consider that motion causes the effect of time dilation, and acceleration causes motion, so that it is true to say that acceleration causes time dilation. However, there are some other points to bear in mind, as follows...

The first is that the effect of time dilation is entirely symmetrical between two inertial reference frames, so if you are time dilated by 80% in my frame, I will be time dilated by 80% in yours. That is because inertial motion itself is an entirely relative effect between two inertial frames.

Acceleration, by contrast, is absolute and thus can lead to asymmetries, as is the case in the twin paradox, in which time dilation is reciprocal during the parts of the travelling twin's journey in which she is coasting inertially but not during the acceleration. Note that it is possible to reframe the thought experiment to eliminate acceleration by having the outbound twin hand-over the baton to an third person travelling inertially back to Earth, in which case all of the asymmetry is associated with the switch of reference frame at the hand-over event.

You can model the time dilation effects of more complicated relative movements between two bodies by summing the time differences over a series of inertial legs with a switch of inertial reference frame between each leg. If you consider that you can increase the accuracy of your model by breaking down the motion into progressively shorter and shorter legs, you will see that in the limit you will eventually replace your summation by an integral.

In summary, the form of motion for which time dilation effects can be calculated in the simplest way is inertial motion, in which case only the relative velocity needs to be taken into account using the normal time dilation formula. For more complicated motions, you must perform some form of summation or integration along the timelines of the objects being considered. During any stages in which the motion is inertial, the time dilation effects are due to velocity and are reciprocal- acceleration introduces asymmetry.

• Thanks but this is repeating the things I've read. 1) I was careful to say that the question is whether "acceleration itself has no direct effect on" time dilation. 2) You seem to agree that the time dilation of inertial motion is fundamentally different from that of acceleration, but this disagrees with the common version of the clock postulate as I understand it. 3) The relay race scenario doesn't accurately model the non-inertial twin's frame since it creates a discontinuity in his accounting of earth time at the turnaround. 4) Inertial frames have no proper acceleration and thus [...] Apr 20, 2022 at 19:14
• [...] can't be integrated into a true accelerating frame as Don Koks notes in the first link I gave: "we cannot simply maintain that an acceleration can be treated as a sequence of constant velocities that each exist only for an infinitesimal time interval, for the simple reason that an accelerating body (away from gravity) feels a force, while a constant-velocity body does not." P.S. If you thought the question was worth answering, please consider upvoting it so more people look at it. :) Apr 20, 2022 at 19:15
• I don't understand what distinction you are trying to make. I don't like Don Koks' explanation. Motion doesn't cause clocks to 'run slow'. It is a consequence of the geometry of spacetime, which causes the duration between two events to depend on the path taken between them. If a moving clock records 4 seconds while passing between two stationary clocks which show a time difference of 5 seconds, that is not because the moving clock's time-keeping has been impaired causing it to under-report time- it is because the interval was only 4 seconds and the moving clock faithfully recorded it as such. Apr 20, 2022 at 21:04
• To take a spatial analogy, the distance between two points depends on the route taken between them, and on nothing else. However, acceleration determines which route an object follows, so you can consider acceleration to be a cause of varying path lengths in that sense. Apr 20, 2022 at 21:07
• It sounds like you prefer the second version of the clock postulate, which is about relative lengths of worldlines per the metric. But the question remains: What causes that length to shorten for a particle that travels in a straight line through space (aside from geometry)? The sources I cited say that only the particle's velocity does. But it seems to me that that's only true in an inertial frame and that in a non-inertial frame, we should say that acceleration also causes it (e.g., at the turnaround for the traveling twin). Your 3rd paragraph seems to agree but correct me if I'm wrong. Apr 21, 2022 at 1:34

But this seems problematic to me because it only ever seems to be true in inertial reference frames. For example, in the (non-inertial) frame of the traveling twin in the twin paradox, the earthbound twin's time slows down during each inertial leg of the trip, but during the turnaround, it speeds up to an extent that indicates that the traveler's acceleration towards the earth is equivalent to a gravitational field that he resists while it pulls the earth towards him thus causing him to experience something equivalent to gravitational time dilation relative to the earth. How can it be said that this kind of time dilation is caused by velocity rather than acceleration?

The postulate is meant for inertial frames only. As you point out in the example, in non-inertial frames, it is not valid. A clock in artificial gravity well (in accelerating rocket) runs slower than a clock that is higher in that well. This is true also in real gravity well, even without any apparent acceleration.

We can ask the same question about two clocks that have no motion relative to each other at all but one is higher than the other in a rocket that accelerates "upward" (or in a gravitational well if the postulate also applies to general relativity). In this case, the lower clock's time is dilated. How is this caused by velocity rather than acceleration? In other words, how is it consistent with this formulation of the clock postulate?

In case of accelerating rocket, the difference in speed of two clocks can be explained in any inertial frame as a result of the standard time dilation; the front of the accelerating rocket is a little slower than the back (due to increasing Lorentz contraction), and this difference makes for the clock speed difference.

Now there seems to be an alternative formulation of the postulate. ... The clock hypothesis of relativity theory equates the proper time experienced by a point particle along a timelike curve with the length of that curve as determined by the metric. Perhaps someone here will show me that this formulation is logically equivalent to the other one (in all frames, including non-inertial ones), but from what I can tell, it actually allows acceleration to cause time dilation.

Only in the sense that different acceleration will result in different velocity and different path in spacetime. The statement on metric is a result of the postulates of special relativity, which do not allow time dilation to depend on acceleration; time dilation in SR is determined by speed in inertial frame.

In reality, hypothetically, clock speed can be influenced by their acceleration, and when this is experimentally found to be a universal effect, not due to some failure of the clock design, then special relativity will be disproven.

For example, in the traveling twin's frame where you have to use a metric like the Rindler metric during the turnaround (per the paper, "the restriction to Minkowski spacetime and inertial motion has been dropped"), you find that his curve is shorter than the earthbound twin's during that acceleration and thus that his own time dilates relative to the earth's, apparently due to his acceleration towards it. Therefore, unlike the other version of the postulate, this one works in non-inertial frames and seems to be consistent with acceleration directly causing time dilation. Is this true? If so, is this version of the postulate more correct?

Again, this is from a non-inertial frame. Postulates of special relativity are not meant to be valid in non-inertial frames.

• Thanks for the answer. "The postulate is meant for inertial frames only." But all discussion of it that I've read is worded in an absolute way with no caveats about which frames it applies to. If you're saying that it only applies to scenarios that only have inertial frames that never change, then the postulate is unnecessary since all time dilation is then fully symmetrical/reciprocal. Are you agreeing that acceleration does cause time dilation in the accelerating frame (even when there's no gravity)? Apr 20, 2022 at 19:58
• Note that I've removed the paragraph about two clocks in an accelerating rocket to shorten the question (and because including it was a brainfart—I'm familiar with Bell's spaceship paradox, etc). P.S. If you thought the question was worth answering, please consider upvoting it so more people look at it. :) Apr 20, 2022 at 19:59
• In accelerating frames clock speed depends not directly on acceleration, but on acceleration times distance along the acceleration direction, or on effective gravity potential. So it depends on position in the frame too, acceleration by itself is not determining factor. Apr 20, 2022 at 20:23
• > But all discussion of it that I've read is worded in an absolute way with no caveats about which frames it applies to. -- This is probably because most sources will discuss clock speed from the point of view of inertial observer, implicitly. One cornerstone of general relativity is understanding how things are in inertial frames (SR), and then transforming this understanding to non-inertial ones. Apr 20, 2022 at 20:31
• True, I'm aware of the dependence on distance in conjunction with acceleration. But by saying that acceleration can have any direct effect on clock speed in any frame, I think you and I are contradicting the common version of the clock postulate. Now, you seem to be implying that SR only handles inertial frames while non-inertial frames are in the domain of GR but from everything I've read, SR includes everything that happens in flat Minkowski space, i.e, in the absence of gravity. E.g., the Rindler metric is a coordinate system of an accelerating frame in flat space and is part of SR. Apr 21, 2022 at 1:50

Let's say bunch of physicists on a rocket try to figure out this problem.

So far they have noticed that

1. The more they press the gas pedal the more the clock on earth accelerates
2. The more they press the gas pedal the more the clock on earth is time dilated
3. The more they press the gas pedal the more the frame of rocket is non-inertial

Now it might be so that the increase of time dilation is caused by the increase of acceleration of the clock. Or it might be caused by the increase of the non-inertialness of the rocket frame.

As they are good experimental physicists, they conduct an experiment in which the clock does not accelerate more when the gas pedal is pressed more. They bolt a pre-programmed rocket on to the clock. Said rocket produces such thrust that the acceleration of the clock does not change when the gas pedal is pressed more.

They then notice that the time dilation of the clock still changes, although the change of acceleration of the clock is zero.

So their conclusion is that acceleration of the clock does not have an effect on the clock.

The above mentioned "gas pedal" is a pedal that alters the acceleration of the rocket in an inertial frame. In the rocket frame it is a pedal that alters the amount of non-inertialness of the frame, in other words it's a pseudo-gravity adjustment pedal.

• Note that "time dilated" is supposed to mean that a frame's unit of time is dilated (e.g., each of its seconds is longer) and thus that its time is slower, not faster. Anyways, let me see if I understand... "Said rocket produces such thrust that the acceleration of the clock does not change when the gas pedal is pressed more." So there are two rockets now that accelerate together so that they have no acceleration or motion relative to each other? "the time dilation of the clock still changes" Relative to the first rocket's time? I don't think that's true. How are you concluding that? Apr 21, 2022 at 9:35
• There are 2 things in your question: a rocket and a clock. When the rocket blasts, there is an inertial force, which accelerates the clock. The rocket does not accelerate because the thrust of the rocket cancels out the inertial force. That was in the rocket frame. In any inertial frame the rocket accelerates, and the clock does not care about that at all. Apr 21, 2022 at 22:55
• @GumbyTheGreen "Relative to the first rocket's time? I don't think that's true. How are you concluding that?". . . In an inertial frame light from clock has a harder time catching up with the rocket after the pedal has been pressed down more, this causes the physicists to say that the clock runs slower. Apr 22, 2022 at 1:06
• Ok, I'm not clear. So there are two rockets—the one the physicists are on (Rocket 1) and the one the clock is on (Rocket 2)—right? Is Rocket 1 on the earth or away from it? When they press the gas pedal, does it accelerate toward the earth or away from it? After bolting Rocket 2 to the clock, is the clock still on the earth? Does its thrust push it toward Rocket 1 or away from it? Also, note that time dilation can't be observed in real time (what the physicists see is determined only by the relativistic Doppler effect), so observation #2 isn't realistic, but I'll go with it for now. Apr 22, 2022 at 1:33
• @GumbyTheGreen Rocket1 is away from earth, it accelerates away from earth, rocket2 with a clock glued on it is on earth and starts accelerating towards rocket1. . . . . Here's a simpler experiment: clocks are sprinkled all around from a rocket, then the gas pedal is pressed to the floor, which causes a horizon to appear. Clocks near said horizon are seen to move slowly, tick slowly and accelerate slowly. Acceleration small, time dilation large. Apr 22, 2022 at 3:59