# Special relativity and accelerated frames

## Main Question or Discussion Point

SR directly addresses only inertial frames. Can SR be applied to analyze accelerated frames?

For example, consider a point on the rim of a rotating disc. I have see an analysis which makes claims by assigning an associated, instantaneous inertial frame comoving with the point at each position. My question is whether this can be used correctly since the frame of the point is always accelerating toward the center of the disc whereas any comoving inertial frame is not accelerating. Can you justify ignoring the acceleration in such an analysis?

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Nugatory
Mentor
SR directly addresses only inertial frames. Can SR be applied to analyze accelerated frames?
Special relativity works in any flat (that is, no gravity) spacetime, whether you use an inertial frame or not; that's just a choice of coordinates and is handled as is any other coordinate transformation. Rindler coordinates are an example; google wand a serach of this forum will find some decent examples and explanations. (In fact, Einstein considers a rotating frame in his 1905 paper). Most introductory treatments don't cover this material because the math is more complex but doesn't provide much more insight into the fundamental principles.

You should work through the Rindler solution before you take on circular motion; the latter is appreciably more complicated.

For example, consider a point on the rim of a rotating disc. I have see an analysis which makes claims by assigning an associated, instantaneous inertial frame comoving with the point at each position. My question is whether this can be used correctly since the frame of the point is always accelerating toward the center of the disc whereas any comoving inertial frame is not accelerating. Can you justify ignoring the acceleration in such an analysis?
The acceleration is just ##dv/dt##, so I'm not sure what you mean by "ignore it"; you calculate it, it is what it is, and it tells you what the instantaneous second derivative of the position is. However, when you're working with instantaneous comoving inertial frames ("ICIF" - the concept is used enough to justify its own acronym) there is one potential pitfall: You cannot blithely transfer values from one ICIF to another, just as you cannot blithely transfer values between the different inertial frames of two observers in constant relative motion. We all know to look for this mistake in constant motion case, but it's easy to get it wrong in the accelerated cases because we don't have different observers in the different frames, we have the same observer at different times.

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WannabeNewton
My question is whether this can be used correctly since the frame of the point is always accelerating toward the center of the disc whereas any comoving inertial frame is not accelerating. Can you
Why focus on SR? We do the same thing in Newtonian mechanics when defining the rest frame of an observer in uniform circular motion around an origin. It's an instantaneously comoving inertial frame so the only thing we care about is that the velocity of the inertial frame matches the velocity of the accelerating observer at the given instant, hence the term "comoving".

The acceleration is a higher order effect, by which I mean it is the second proper time derivative of the trajectory, and in order to analyze its physical effects a single instantaneously comoving inertial frame is not enough. You need a continuous sequence of such frames, one for each instant of time on the clock carried by the accelerating observer. This is how the rest frame of the observer is defined: it is a continuous family of instantaneously comoving inertial frames.

After we have this one-parameter family we can start looking at second order effects like acceleration because we can differentiate velocity given this family. When we only have one instantaneously comoving frame we clearly cannot differentiate velocity and the effects of acceleration cannot be analyzed but that does not mean they are ignored. This is the same in SR and in Newtonian mechanics.

I will look up the "Rindler solution". I think I understand your response.

My question is about a problem I see in attempting to derive properties (such as time and distance) in an accelerated frame (relative to a particular IRF) by appealing to a series of ICRFs where we know how time and distance behave relative to that IRF.

The problem I see is that the ICRFs are unlike the accelerated frame at every instant. Although the ICRFs have infinitesimal motion relative to the accelerated frame, they always have a substantial (non-infinitesimal) difference in acceleration. SR itself doesn't say anything about acceleration.

I'm asking: Can you ignore this difference in acceleration when using ICRFs to analyze properties of the accelerated frame? In other words, is it nevertheless valid to analyze an accelerated frame using incremental ICRFs and integratomg? If so, how do you justify ignoring the difference in acceleration in formulating infinitesimals from ICRFs and integrating?

You mention that you have to be careful in transferring values (e.g. in order to do an integration?).

Why focus on SR? We do the same thing in Newtonian mechanics when defining the rest frame of an observer in uniform circular motion around an origin. It's an instantaneously comoving inertial frame so the only thing we care about is that the velocity of the inertial frame matches the velocity of the accelerating observer at the given instant, hence the term "comoving".
Because in Newtonian mechanics time are space are absolute, that is, independent of motion. This is not true in SR. In Newtonian mechanics, we do not have scratch our heads trying to figure out how much time passes on a moving clock while in arbitrary motion.

The acceleration is a higher order effect, by which I mean it is the second proper time derivative of the trajectory, and in order to analyze its physical effects a single instantaneously comoving inertial frame is not enough. You need a continuous sequence of such frames, one for each instant of time on the clock carried by the accelerating observer. This is how the rest frame of the observer is defined: it is a continuous family of instantaneously comoving inertial frames.
This is where I see a problem. No instantaneously comoving inertial frame can be said to be perfectly equivalent to the accelerating frame (at that instant) because the acceleration is still different in the two frames, even though they are instantaneous comoving.

So if acceleration has some effect in and of itself, you cannot equate these instantaneous frames.

pervect
Staff Emeritus
SR directly addresses only inertial frames. Can SR be applied to analyze accelerated frames?
The mathematical techniques needed to deal rigorously with arbitrary coordinate systems (which includes coordinate systems associated with an accelerated observer as a special case) are typically taught along with GR. But they are just mathematical techniques, there isn't any physical content to them, any more than there is physical content in going from cartesian (x,y,z) coordinates to spherical (r, theta, phi) coordinates.

Thus we have the situation that SR + the mathematical techniques needed to rigorously deal with arbitrary coordinate systems can handle accelerated frames.

However, a typical student who has been taught a typical SR course will be lacking the mathematical techniques to deal with arbitrary coordinates in a systematic manner. This falls in the general category of tensor analysis.

For example, consider a point on the rim of a rotating disc. I have see an analysis which makes claims by assigning an associated, instantaneous inertial frame comoving with the point at each position. My question is whether this can be used correctly since the frame of the point is always accelerating toward the center of the disc whereas any comoving inertial frame is not accelerating. Can you justify ignoring the acceleration in such an analysis?
As long as the quantities used are tensor quantities, the acceleration of the local frame will not affect the value of any tensor quantities. Therefore it is a perfectly valid technique when one analyzes the situation with tensors.

However, the exact definition of tensors, and also the defintion of local frames (as opposed to global ones) are part of the mathematical apparatus of tensor analysis, something that the typical SR student does have at their disposal.

One can find early treatments of some problem that don't use tensor analysis. Many of these can be useful to students who study and observe the techniques uses. For the student that is always asking "why" and wanting to do things their own way rather than study the literature and observe, if the student gets it wrong, the lack of rigor in the non-tensor analysis may not convice the sutdent. In the worst case the student makes a mistake, isn't convinced by the paper, and goes off on a more-or-less crackpot course of insisting that the published papers are all wrong.

Of course, things aren't always this simple, while published papers have a higher chance of being right than wrong, having gone through some sort of peer review process, there are still plenty of wrong published papers as peer review is far from perfect.

Thanks. Seeing that the general problem involves tensors (which I'm not familiar with), can you just answer this simpler question?

Suppose I have an accelerating clock observed from some IRF. Will the instantaneously observed rate of the clock depend only on the instantaneous velocity (using LT) or does the acceleration also enter into the clock's rate (relative to the observer)?

I ask in part because in GR gravitational acceleration is supposed to affect clock rates.

Nugatory
Mentor
Suppose I have an accelerating clock observed from some IRF. Will the instantaneously observed rate of the clock depend only on the instantaneous velocity (using LT) or does the acceleration also enter into the clock's rate (relative to the observer)?
If the clock is constructed in such a way that the force accelerating it won't disturb it (a pendulum clock would be a bad choice, an atomic clock probably pretty good) the acceleration will not affect it.

I ask in part because in GR gravitational acceleration is supposed to affect clock rates.
There is no gravitational acceleration in GR. A clock deeper in a gravity well will dilate relative to a clock not so deep, but that has nothing to do with acceleration. It's a result of spacetime curvature, which is a property of the spacetime you're in and not the frame (whether inertial or accelerated) that you use to assign coordinates to events in that spacetime.

If the clock is constructed in such a way that the force accelerating it won't disturb it (a pendulum clock would be a bad choice, an atomic clock probably pretty good) the acceleration will not affect it.
In other words, the LTs depend only on velocity so higher order derivatives are irrelevant. This is now obvious to me.

Hence we can use ICRFs to do path integrals to solve problems involving any motion.

I think I had the impression that acceleration had some effect on clocks from a misunderstanding of GR.

There is no gravitational acceleration in GR. A clock deeper in a gravity well will dilate relative to a clock not so deep, but that has nothing to do with acceleration. It's a result of spacetime curvature, which is a property of the spacetime you're in and not the frame (whether inertial or accelerated) that you use to assign coordinates to events in that spacetime.
Thanks for pointing out a misunderstanding. I have to learn the right way to look at it.

Einstein points out that an observer in a black box can't tell whether he is on the earth or accelerating at g somewhere far away. That gives the idea that the observer on earth is in fact accelerating wrt to some frame. Would that be the IRF of a free falling object near him?

Nugatory
Mentor
Einstein points out that an observer in a black box can't tell whether he is on the earth or accelerating at g somewhere far away. That gives the idea that the observer on earth is in fact accelerating wrt to some frame. Would that be the IRF of a free falling object near him?
Yes.

pervect
Staff Emeritus
Thanks. Seeing that the general problem involves tensors (which I'm not familiar with), can you just answer this simpler question?

Suppose I have an accelerating clock observed from some IRF. Will the instantaneously observed rate of the clock depend only on the instantaneous velocity (using LT) or does the acceleration also enter into the clock's rate (relative to the observer)?

I ask in part because in GR gravitational acceleration is supposed to affect clock rates.
There's a good FAQ on this at http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html, which also points out that my first post was , errr, wrong. It's not just a question of tensor analysis, there is some physical content to the question.

But lets get to the answer. The answer is that the acceleration of an ideal clock does not affect it's ticking rate as observed in an inertial frame.

FAQ said:
The clock postulate generalises this to say that even when the moving clock accelerates, the ratio of the rate of our clocks compared to its rate is still the above quantity. That is, it only depends on v, and does not depend on any derivatives of v, such as acceleration. So this says that an accelerating clock will count out its time in such a way that at any one moment, its timing has slowed by a factor (γ) that only depends on its current speed; its acceleration has no effect at all.
For some experimental tests of this see http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Clock_Hypothesis

The clock hypothesis states that the tick rate of a clock when measured in an inertial frame depends only upon its velocity relative to that frame, and is independent of its acceleration or higher derivatives. The experiment of Bailey et al. referenced above stored muons in a magnetic storage ring and measured their lifetime. While being stored in the ring they were subject to a proper acceleration of approximately 10^18 g (1 g = 9.8 m/s2). The observed agreement between the lifetime of the stored muons with that of muons with the same energy moving inertially confirms the clock hypothesis for accelerations of that magnitude.
I believe there are some remarks about this issue in the textbook "Gravitation" as well. I recall reading them but couldn't find the exact section to refresh my memory. My recollection is that the authors noted that even the enormous acceleration due to the strong nuclear force between protons in the atomic nucleus had no effects on timekeeping.

Note, however, that while acceleration of a clock does not affect it's rate as observed by an inertial observer, if we accelerate both the clock AND the observer, the result is no longer true. If we insist that our observer remain inertial, however, we can say that acceleration does not affect the rates of clocks.

Note, however, that while acceleration of a clock does not affect it's rate as observed by an inertial observer, if we accelerate both the clock AND the observer, the result is no longer true. If we insist that our observer remain inertial, however, we can say that acceleration does not affect the rates of clocks.
I'm not sure how that case is different. Naively, at any instant, the accelerating observer can assign a rest ICRF to himself and another ICRF to the clock, measure the velocity of the clock and then calculate the clock rate (as seen by the observer) at that instant using the LT. So, it shouldn't matter that the observer is accelerating.

Can you explain what you mean?

jbriggs444
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I'm not sure how that case is different. Naively, at any instant, the accelerating observer can assign a rest ICRF to himself and another ICRF to the clock, measure the velocity of the clock and then calculate the clock rate (as seen by the observer) at that instant using the LT. So, it shouldn't matter that the observer is accelerating.
The synchronization convention in use by the observer's ICRF at one instant and by the ICRF at the next instant is constantly changing due to the acceleration. This affects the sequence of time coordinates ascribed to the ticks of the observed clock.

The observed clock's "tick rate" in the accelerating frame is judged based on the delta between the time coordinates ascribed for each tick.

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Yes, but 1) the clock hypothesis is not violated, and 2) the apparent changes in the behavior of the clock versus an observer in an IRF are entirely due to the observer's acceleration, right? Moreover, the observer knows his own acceleration vector at every instant. So, it seems that he can still calculate the clock's behavior without violating the assumption that the clock's rate is independent of it's own acceleration (the clock hypothesis), just like an observer in an IRF.

It's just harder to analyze, right?

Say at ##t_0## (in the observer's frame) the observer selects his instantaneous ICRF as a base IRF. He can thereafter measure his own motion relative to that base IRF, using an accelerometer. Can't he then transform his measurements of the clock's path and rate to that IRF and realize that all variations in the clock rate are due to both his motion and the clock's motion in that base IRF?

From that perspective, I don't understand why we should claim that there is anything special about the case where the observer is accelerating versus when he is in an IRF, except that the calculations are different because the observer's velocity is changing.

pervect
Staff Emeritus
I'm not sure how that case is different. Naively, at any instant, the accelerating observer can assign a rest ICRF to himself and another ICRF to the clock, measure the velocity of the clock and then calculate the clock rate (as seen by the observer) at that instant using the LT. So, it shouldn't matter that the observer is accelerating.

Can you explain what you mean?
If you are careful to always use inertial frames in your analysis (as you did with your MCIF), then there is in fact no problem. So the caution would be unnecessary.

In the FAQ I quoted earlier, what I was trying to say would be addressed by the following section.

But what about the Equivalence Principle?

Sometimes people say "But if a clock's rate isn't affected by its acceleration, doesn't that mean the Equivalence Principle comes out wrong? If the Equivalence Principle says that a gravitational field is akin to acceleration, shouldn't that imply that a clock isn't affected by a gravitational field, even though the textbooks say it is?"

No, the Equivalence Principle is fine. Again, the confusion here is the same sort of thing as above where we spoke about the wind chill factor. Let's try to see what is happening. Imagine we have a rocket with no fuel. It sits on the launch pad with two occupants, a couple of astronauts who can't see outside and who believe that they are accelerating at 1 g in deep space, far from any gravity.
A minor note of clarification. The astronauts who can't see outside are applying the equivalence principle when they say that they think they are accelerating at 1g in deep space. They are saying that their observations should be "equivalent". This is fairly obvious from context but wasn't explicitly stated.

One of the astronauts sits at the base of the rocket, with the other at its top, and they both send a light beam to each other. Now, general relativity tells us that light loses energy as it climbs up a gravitational field, so we know that the top astronaut will see a redshifted signal. Likewise, the bottom astronaut will see a blueshifted signal, because the light coming down has fallen down the gravitational field and gained some energy en route.

How do the astronauts describe what is going on? They believe they're accelerating in deep space. The top astronaut reasons "By the time the light from the bottom astronaut reaches me, I'll have picked up some speed, so that I'll be receding from the light at a higher rate than previously as I receive it. So it should be redshifted—and yes, so it is!" The bottom astronaut reasons very similarly: "By the time the light from the top astronaut reaches me, I'll have picked up some speed, so that I'll be approaching the light at a higher rate than previously as I receive it. So it should be blueshifted—and yes, so it is!"

As you can see, they both got the right answer, care of the Equivalence Principle. But their analysis only used their speed, not their acceleration as such. So just like our wind chill factor above, applying the Equivalence Principle to the case of the rocket doesn't depend on acceleration per se, but it does depend on the result of acceleration: changing speeds!

OK, I think I understand you correctly now. An observer's acceleration means his velocity is changing, so what he measures about some independent clock is affected by his own motion. What threw me off was this:

If we insist that our observer remain inertial, however, we can say that acceleration does not affect the rates of clocks.
If I understand correctly, whether acceleration of a clock (which is absolute) affects the rate of the clock is not observer dependent; it is just a fact that acceleration doesn't affect the clock (only relative velocity alters the clock's rate from any observer's point of view). An accelerating observer who thinks that the clock's own acceleration does affect the observed clock rate, is mistaken and hasn't analyzed the problem correctly.

Do you agree?

Your response above seemed to make some exception to the clock hypothesis, but i don't think that was your intention.

You have left me wondering about the issue of clock rates in a declining gravitational field such as at higher altitudes above the earth. A clock farther from the earth (e.g. on a tower) runs faster than one on the earth? It is true that their accelerations are different, but that shouldn't matter. Moreover we know that the difference in rates is constant. So the only SR conclusion I can see is that ICRFs of the clocks are in constant motion relative to one another.

Perhaps this constant relative motion is the same as the velocity of an object dropped from the tower when it reaches the earth?

But that doesn't seem to be working very well. Where is the symmetry of clock rates here? Do observers in the tower and on earth agree on which clock is ticking faster?...

I need to sleep. If you have any hints I'd appreciate them.

pervect
Staff Emeritus
OK, I think I understand you correctly now. An observer's acceleration means his velocity is changing, so what he measures about some independent clock is affected by his own motion. What threw me off was this:

If I understand correctly, whether acceleration of a clock (which is absolute) affects the rate of the clock is not observer dependent; it is just a fact that acceleration doesn't affect the clock (only relative velocity alters the clock's rate from any observer's point of view). An accelerating observer who thinks that the clock's own acceleration does affect the observed clock rate, is mistaken and hasn't analyzed the problem correctly.

Do you agree?
A quick summary of my position: proper time is observer independent, and coordinate time isn't. It is perhaps unclear what is meant by "the rate of the clock" without talking about this distinction. I make certain unstated assumptions by what I mean, but you, as a reader , may be making different unstated assumptions :(. So I think we need to disentangle the two sorts of time.

Proper time is the sort of time an idealized clock measure. As a non-idealized realization of proper time, we can use the NIST (National institute of standards) defiintion of the second. The second is defined as a certain number of a specified atomic transition regardless of any of these other factors such as where you are. what the gravitational potential is. how fast you are moving, or how much you are accelerating. The proper second, or its realization via the NIST standard second, are independent of the observer.

Coordinate time is something that we build up on top of proper time. To define coordinate time, we need to define the notion of simultaneity. And as Einstein's train thought experiment shows (I hope you are familiar with this - I don't want to digress to explain it, but let me know if explanation is needed) the notion of simultaneity is observer dependent. This observer dependence creeps into the notion of coordinate time, which is also observer dependent.

Given a reference clock, and a notion of simultaneity, we can define a coordinate time for any specified event covered by our coordinate system as the proper time reading on our reference clock which is simultaneous with our specified event.

If two clocks are not at the same location in space-time, we need to use coordinate time to compare them, because we need the notion of simultaneity to define what "now" is for the distant clock. This means that observer dependence is introduced (subtly) when we talk about "observers" and/or "frames". The observer or frame has a proper clock (which is independent of the observer), but it also has a notion of simultaneity, which is dependent on the observer.

If we use only proper time, there isn't any way to meaningfully compare clocks at different locations, we are lacking the necessary definition of simultaneity. Using proper time we can only say "all clocks tick at 1 second per second", which we operationally define as xxxx transitions of the yyyyy atomic transition, so there isn't any concept of clocks having "different rates".

Thus, when we talk about comparing clocks, I implicitly assume that we are talking about comparing them via the observer dependent coordinate time.

Your response above seemed to make some exception to the clock hypothesis, but i don't think that was your intention.
Defintely not my intention, I hope this new response clarifies things.

You have left me wondering about the issue of clock rates in a declining gravitational field such as at higher altitudes above the earth. A clock farther from the earth (e.g. on a tower) runs faster than one on the earth? It is true that their accelerations are different, but that shouldn't matter. Moreover we know that the difference in rates is constant. So the only SR conclusion I can see is that ICRFs of the clocks are in constant motion relative to one another.
If you have an actual gravitational field, like that of the Earth, there isn't any coordinate system that represents "an inertial observer". So it's unclear how to analyze the problem without using relativity, because the prescription "use an inertial observer" doesn't have any clear meaning in GR.

Assuming, then that we are willing to use GR, if we wish to be precise, we also need to specify exactly what coordinate system we are using to compare clocks. I will use the TAI coordinate time standard, also known as "atomic time", as described in the wiki http://en.wikipedia.org/wiki/International_Atomic_Time, without going into the details of why this choice is a good one, and how many/most other common choices will yield equivalent results.

Then, using the TAI coordinate time standard, higher clocks definitely tick faster than the lower clocks. In GR the important issue for coordinate clock rate is the gravitational potential. Clocks deeper in a gravity well tick slower than clocks higher in a gravity well according to TAI time.

If you recast the problem using the principle of equivalence to an accelerating spaceship in flat space-time, you do not have to use GR. Inertial observers are clearly available, and you can use them to solve the problem without getting involved in GR at all.

Now I understand better what you meant about accelerating observers. Proper time has no relativity of rates and is by definition independent of motion. Observation of rate changes happens only for observers not moving with the clock. The clock hypothesis has to do with other observers of the clock. However, observed acceleration is absolute for all observers (because they can measure the effects of their own acceleration) and therefore the clock hypothesis is indeed observer independent, but the hypothesis is a statement about observers. Maybe I'm rambling here.

Perhaps we can think about accelerated frames by imagining that we have an initial inertial frame in which to synchronize clocks. (It seems difficult to do so in an accelerating frame.) Then as that frame accelerates the Coordinate clocks of the frame remain synchronized. (But not in rotation because the acceleration is not uniform in the frame.)

I'll just accept the effects of a gravitational gradient for now. I am not prepared to understand the mathematics of GR.

BTW, is there a name for uniformly accelerated frames?

PeterDonis
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Then as that frame accelerates the Coordinate clocks of the frame remain synchronized.
No, they don't. More precisely, there is *no* way to have a frame in which all of the following are true:

(1) There is a family of observers all of whom are "at rest" in the frame (i.e., they are all at rest relative to each other);

(2) All of the observers have nonzero proper acceleration;

(3) All of the observers's clocks stay synchronized.

This is true even if all of the observers are accelerating linearly in the same direction. The (non-inertial) coordinates in which a family of such observers is at rest are called Rindler coordinates; see here:

http://en.wikipedia.org/wiki/Rindler_coordinates

Note that, although all of the observers are at rest in Rindler coordinates, their proper accelerations are not all the same; the ones "higher up" (i.e., further along in the direction of acceleration) have smaller proper accelerations than the ones "lower down". Also, the rates of their clocks are different: the clocks "higher up" run faster than the clocks "lower down". (This is similar qualitatively to what happens in a gravity well, but the variation of clock rate with "height" is different.) This is why their clocks can't stay synchronized.

By proper acceleration, I suppose you mean the acceleration that an observer measures with an accelerometer (e.g. by measuring the instantaneous acceleration of a body to which no forces are applied)?

Perhaps this is tangential, but I ask myself, in the case of a rigid body undergoing uniform acceleration at some point in the body, are the proper accelerations of all points in the body constant and equal?

From the point of view of some IRF, the body contracts continuously in the direction of acceleration because it is rigid in its rest frame. So it seems that in an IRF, the coordinate accelerations of all points in a rigid body cannot be the same. (If the velocities in the IRF were the same at all points in the body at all moments, the body could not contract.)

Taking acceleration as absolute (in any IRF), this implies that the proper accelerations cannot be uniform throughout the body. (These differences in proper accelerations are very small for accelerations like g.)

Since there are different velocities among the particles in the body (due to increasing contraction) in the IRF, their clocks go out of sync as well in the IRF (very slightly and only along the direction of the acceleration).

Then, what happens in the Rindler frame of a particle in this accelerating rigid body? If we take an ICFR for that Rindler frame, we are in an instantaneous rest frame for the body. So, no contraction is seen and the clocks remain synchronized. This seems rather nice because if we are in an accelerating spaceship we don't see any distortions of shape nor in the synchronization of clocks (I think). However we can measure a very slight difference in the proper accelerations of the nose of the ship versus the tail! Very strange.

---

In the other case, there is a group of independent objects with the same proper accelerations. Viewed from an IRF (S), the objects have coordinates at some time t0. Because the proper accelerations are equal, the coordinate accelerations are also equal in the IRF (S). Thus the group of objects moves as a rigid configuration (is translated) in the IRF (without any contraction of the group) and their velocities are uniform in IRF (S) at any instant. The clocks on the objects remain in sync by symmetry in S.

Now consider one of these bodies in it's Rindler frame and refer back to the IRF (S) above. Take an ICRF in the Rindler frame at a moment when it is not at rest with S. From this ICRF for the Rindler frame, we see the group of objects as contracted because we are moving relative to S. In the ICRF frame the contraction increases with time as the configuration moves faster. So, the objects are moving because the group is contracting in any ICRF and therefore the clocks aren't synchronized in the Rindler frame.

The two cases of rigid body acceleration versus a uniformly accelerating rigid group appear complementary. It's seems that the changes in proper acceleration in the one case and clock desynchronization in the second case are "second order" effects which are very small except at extreme accelerations.

I've tried to think this through in a qualitative sense without doing all the calculations. Have I've gone off the tracks?

Nugatory
Mentor
By proper acceleration, I suppose you mean the acceleration that an observer measures with an accelerometer?
yes.
(Note that by this definition, an observer free-falling in the earth's gravity is not undergoing proper acceleration - he's inertial)

Perhaps this is tangential, but I ask myself, in the case of a rigid body undergoing uniform acceleration at some point in the body, are the proper accelerations of all points in the body constant and equal?
Good question, and the answer is "no". Google for "Born rigidity" and "Born rigid motion". Also search here for "Bell Spaceship Paradox".

PeterDonis
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2019 Award
By proper acceleration, I suppose you mean the acceleration that an observer measures with an accelerometer (e.g. by measuring the instantaneous acceleration of a body to which no forces are applied)?
Yes.

in the case of a rigid body undergoing uniform acceleration at some point in the body, are the proper accelerations of all points in the body constant and equal?
Constant, yes. Equal, no.

From the point of view of some IRF, the body contracts continuously in the direction of acceleration because it is rigid in its rest frame. So it seems that in an IRF, the coordinate accelerations of all points in a rigid body cannot be the same.
Yes.

Taking acceleration as absolute (in any IRF), this implies that the proper accelerations cannot be uniform throughout the body. (These differences in proper accelerations are very small for accelerations like g.)
Yes.

Since there are different velocities among the particles in the body (due to increasing contraction) in the IRF, their clocks go out of sync as well in the IRF (very slightly and only along the direction of the acceleration).
The clocks going out of sync is not frame-dependent; it can be verified by direct measurement (for example, by having the clocks exchange light signals and each clock timing how long it takes light to make the round trip).

If we take an ICFR for that Rindler frame, we are in an instantaneous rest frame for the body.
But only for an instant, because the instantaneous rest frame is constantly changing.

So, no contraction is seen
You can't really make this comparison, because the accelerated object will have internal stresses, whereas an object that is moving inertially will have none.

the clocks remain synchronized.
No, you can't say that, because saying clocks remain synchronized for a single instant doesn't really mean anything.

if we are in an accelerating spaceship we don't see any distortions of shape
Compared to what? As I noted above, the accelerating spaceship will have internal stresses; that means its shape will not be the same as that of an identical spaceship moving inertially. For ordinary accelerations like 1 g, the difference will be very small, but it will be there.

nor in the synchronization of clocks (I think).
No, there will be a difference in clock rates. However, there is one sense in which the clocks remain "in sync" (though that's not really the right term): they all agree on which events are simultaneous (i.e., a pair of events that happens at the same time according to one clock will happen at the same time according to all the clocks--they may not agree on the numerical value of that time, but they will all agree that both events happen at the same time).

However we can measure a very slight difference in the proper accelerations of the nose of the ship versus the tail! Very strange.
Not quite so strange once you realize that the clock rates differ too.

In the other case, there is a group of independent objects with the same proper accelerations. Viewed from an IRF (S), the objects have coordinates at some time t0. Because the proper accelerations are equal, the coordinate accelerations are also equal in the IRF (S). Thus the group of objects moves as a rigid configuration (is translated) in the IRF (without any contraction of the group) and their velocities are uniform in IRF (S) at any instant.
This is ok, except that this use of the word "rigid" is not standard. The standard usage of "rigid" refers to the objects' motion relative to each other; i.e., it's not coordinate-dependent. See below.

The clocks on the objects remain in sync by symmetry in S.
No, they don't, because the clocks are moving relative to S, so events which are simultaneous in S (which is what you are using to define what "synchronized" means) are not simultaneous for the clocks. Or, to put it another way, this usage of "synchronized" is non-standard, like your usage of "rigid" above--the standard usage refers to the clocks themselves, not to an arbitrary ICRF, so "synchronized" in the standard usage is not coordinate-dependent.

Now consider one of these bodies in it's Rindler frame and refer back to the IRF (S) above. Take an ICRF in the Rindler frame at a moment when it is not at rest with S. From this ICRF for the Rindler frame, we see the group of objects as contracted because we are moving relative to S. In the ICRF frame the contraction increases with time as the configuration moves faster. So, the objects are moving because the group is contracting in any ICRF and therefore the clocks aren't synchronized in the Rindler frame.
More precisely, the clocks are not synchronized relative to each other, and in fact they don't even share a common simultaneity (which is different from the previous case, as I noted above, where all the clocks do share a common simultaneity, even though their rates differ), because they are moving relative to each other.

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pervect
Staff Emeritus
By proper acceleration, I suppose you mean the acceleration that an observer measures with an accelerometer (e.g. by measuring the instantaneous acceleration of a body to which no forces are applied)?

Perhaps this is tangential, but I ask myself, in the case of a rigid body undergoing uniform acceleration at some point in the body, are the proper accelerations of all points in the body constant and equal?
No, as you correctly argue below. This is closely related to the so-called Bell spaceship paradox.

From the point of view of some IRF, the body contracts continuously in the direction of acceleration because it is rigid in its rest frame. So it seems that in an IRF, the coordinate accelerations of all points in a rigid body cannot be the same. (If the velocities in the IRF were the same at all points in the body at all moments, the body could not contract.)

Taking acceleration as absolute (in any IRF), this implies that the proper accelerations cannot be uniform throughout the body. (These differences in proper accelerations are very small for accelerations like g.)

Since there are different velocities among the particles in the body (due to increasing contraction) in the IRF, their clocks go out of sync as well in the IRF (very slightly and only along the direction of the acceleration).

Then, what happens in the Rindler frame of a particle in this accelerating rigid body? If we take an ICFR for that Rindler frame, we are in an instantaneous rest frame for the body. So, no contraction is seen
True
and the clocks remain synchronized.
False :(.

In the Rindler frame, there is a difference in clock rates between the upper and lower clock of (1+a*h/c^2). I've been looking for a good reference that explains this in more detail so far I haven't found one I cold recommend :(

The general approach that comes to my own mind is considering a case where you have two clocks that are synchronized in some inertial frame, each accelerates at the same coordinate acceleration for a short, identical time period in said inertial frame (and by symmetry for the same proper acceleration for the same proper time) ending up with the same velocity relative to the initial inertial frame. Because the clocks were moving, we expect each of them to become unsynchronized from a coordinate clock in the initial inertial frame due to their motion- by the clock postulate, there is no effect from their acceleration. By symmetry, we expect the amount of time lost to be the same for both clocks, so that both clocks become unsynchorinzed from coordinate clocks, but because the amount of desynchronization is the same for both clocks, they can be said to "remain synchronized" in the initial inertial frame.

[add]Correction clarification. The moving clocks run at a lower rate than the coordinate clocks, but if you multiply the moving clocks by the doppler correction factor, they difference between the moving clocks and the coordinate clocks will be constant.

i.e. we can write the proper time ##\tau## as a function of coordinate time

$$\tau(t) = \int \sqrt{1-(v(t)/c)^2}dt$$

and for ##t>t_{max]## v(t) = constant = ##v_{max}## we can write
$$\tau(t) =t \sqrt{1-(v_{max}/c)^2} + C$$ where C is some constant, and this expression works with identical values of C for both clocks, because v(t) is identical.

But the ending inertial frame is not the same as the initial inertial frame, and by the relativity of simultaneity, the clocks that are synchronized in the initial inertial frame are no longer synchronized in the ending inertial frame.

This follows directly from the Lorentz transform:

t' = gamma(t - v x / c^2)

If we consider two clocks at t=0 , x=0 and t=0, x=1 in the unprimed inertial frame, they are synchronized. In the primed inertial frame, the first clock has t'=0, but the second clock has t' = -v/c^2, and they are no longer synchronized.

I think the key thing you are missing is the relativity of simultaneity, which I tried to explain above as a consequence of the Lorentz transform. The issue is that "synchronized in a stationary frame" is a DIFFERENT condition than "synchronized in a moving frame". And when you acclelerate, you change frames! The clocks remain synchronized in their initial inertial frame (though they tick at a different rate because they move). As a consequence they cannot remain syncrhonized in the final moving frame.

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CKH
From the point of view of some IRF, the body contracts continuously in the direction of acceleration because it is rigid in its rest frame. So it seems that in an IRF, the coordinate accelerations of all points in a rigid body cannot be the same.
Peter: Yes.

CKH
Taking acceleration as absolute (in any IRF), this implies that the proper accelerations cannot be uniform throughout the body. (These differences in proper accelerations are very small for accelerations like g.
Peter: Yes.

I see a problem here. There can be only one proper acceleration at each point in this rigid body. But I arrived at these non-uniform proper accelerations using a particular IRF in which the body was contracting at some particular time and therefore at some particular rate. If at the same moment I took another IRF in which the object is moving at nearly c, I think I'd see a very low rate of contraction and get different proper accelerations from that frame.

Something's wrong with this picture because the proper accelerations should not depend on how I measure them at a given instant. Can you help?

Nugatory
Mentor
CKH
Something's wrong with this picture because the proper accelerations should not depend on how I measure them at a given instant. Can you help?
The proper acceleration at any event along the worldline of any piece of the body will be the same for all observers - attach an accelerometer to that piece and the position of its needle on the dial as it passes through that point must be independent of the observer's frame.

However,different observers moving at different speeds relative to one another will have different notions of "at the same time", and therefore will come to different conclusions about the positions, current speeds, length contractions, and changes of speed of the various parts of the accelerating body at any moment. For a less cluttered example, consider the Bell spaceship paradox (already mentioned a few times in this thread): In one frame the string breaks because it is length-contracted while the distance between the ships remains the same; in another frame the string breaks because the distance between the ships increases while the length of the string remains the same.