If the clocks accelerate at the same rate in the same direction, they will remain motionless relative to each other, and there is no time dilation.

If they accelerate, say, in opposite directions but following symmetrical trajectories, there will be time dilation, but it will cancel out exactly when the two clocks meet again. To see this, I am attaching a simple but technically accurate diagram of two accelerating worldlines. In this diagram, the dots correspond to "ticks" at equal time intervals (as measured along the respective worldlines by the two atomic clocks); the dashed lines correspond to the constant-time hypersurfaces attached to each worldline at each "tick".

I think V.T.'s explanation is considered correct to first order.  But there is a large body of literature on the Bell Spaceship Paradox dealing exactly with this question, and there is not a complete consensus on it because of the ambiguity of "equal acceleration."  If an inertial observer sees equal acceleration, the spaceships see themselves as drifting apart and a string connecting them would break.  Born Rigid Motion, described in the early days of special relativity, would have the trailing spaceship accelerate a little faster to preserve the measured distance between them according to themselves (it becomes contracted for an inertial observer).  This only works well when one pauses after acceleration.  If the acceleration is continuous, the two spaceships can never agree among themselves that their acceleration is equal.  See Figure 3 and associated discussion in the paper below.  Obviously if there is ambiguity over equality of acceleration, then velocity and time dilation share that ambiguity.

Article Isotropy, equivalence and the laws of inertia

At low speeds, where relativistic effects of LORENTZ contraction are negligible, the effects illustrated in the Bell Spaceship Paradox, according to which distances cannot be maintained, are negligible. Regarding the measure of the acceleration,would be enough to put the clocks in the same space ship, that at low speeds guarantees that the distance between them is maintained, since  the Born rigid motion is well approximated by an ideal rigid motion.

In any cases it seems clear, also from what affirmed by V.T. Toth and Robert, that the time dilation factor between the clocks can not have the value of (1+gh/c2) where h is the distance measured between the spaceships and g is the constant acceleration measured by an external observer. The time dilation in accelerated frames does not depend on the values of the acceleration at low speeds.

Stefano, that's right but it's hard to quantify "low speeds."  At sufficiently low speed, clocks can be moved about without any loss of synchronization.  Swann pointed this out in 1960 in a paper in AJP.  So I think we were both kind of guessing at what you might be "really asking."  Clock synchronization and therefore time dilation changes from one moment to the next due to speed changes are the origin of "apparent" time dilation for objects lower in an accelerated frame, so it is dependent on the SR effect.  If you keep speeds low enough to not have any SR effects, then, well, you get rid of all things dependent on SR effects.  : )

Robert, I  wanted to point out that acceleration in itself does not produce any time dilation at all between two equally accelerated clocks. Here we are talking about real effects, I don't want even talk about "apparent effects". Real effects are the one measured in the experiment of Hafele and Keating for measuring the time dilation of two atomic clocks. 

This value (1+gh/c2) which should be valid as a first approximation, as the factor for time dilation in accelerated frames, calculated as such also by Schiff,  doesn't have any reason  to exist.

If it existed even at very low speeds, in the first instants of acceleration, with a suitable high acceleration "g" and distance "h" between the clocks , it would be able to produce a significative time dilation between the two equally accellerated clocks, which can't be true.

Stefano, that's fine.  You cannot use equivalence then.  You will have to derive gravity some other way.  The point of equivalence is often misunderstood.  It is not about "real" effects, but about the principle that any observer in a similar situation must make the same observations.  In other words, the observer in the front of a 1g accelerating spaceship in free space makes the same observations of a clock at the bottom, that an observer on the surface of the earth would over a similar distance.

Few people (some, but not me) think the mechanism is identical.  And that is the advantage of equivalence.  It offers mechanisms like Doppler shift that we can work out from kinematics that we don't readily see in a gravitational field.

By the way, according to Swann, the two clocks you mention are not part of a synchronized Einstein reference frame.  Nothing is driving them into sync.  Therefore SR doesn't have anything to say about the relation between them, only about them individually (which seems more or less to be your point).

I'd also like to point out the difference between a 1 on 2 and a 2 on 1 measurement.  All SR measurements are simultaneous and use at least 2 synchronized observers in the reference frame making the measurement.  Have you ever noticed that no one ever makes that kind of measurement when talking about equivalence, and rarely when measuring accelerated frames?  It is because most of the interesting effects go away when such a measurement is made, just as you have supposed.  It will actually take 4 observers (well, at least 3) to make such measurements of the two clocks, and indeed, they will be found to be ticking at the same rate.

So, what you have supposed is in complete agreement with orthodox theory I believe.  Under a measurement by an Einstein synchronized frame, the two clocks accelerated identically will tick identically.  And this is independent of the direction in which the clocks move, which is basically what V.T. was saying.

Now, you might find an occasional physicist who will argue the effects of equivalence are "real," but I cannot defend that position.  I only use them as apparent.  If you wish to have such a discussion, perhaps my comment has made it clear enough that the thread will now attract a debater for you.  : )

Robert, I don't talk about Einstein equivalence principle here, but about real effects on time dilation of acceleration.

By the way for me exists valid only the WEP which was tested by the experiments. The extension perfomed by Einstein as Synge said, and I firmly believe, does not make sense. It is an arbitrary derivation  proved useful to "guess" on some results of real phenomena about gravitation, like gravitational redshift or time dilation, which Einstein himself refrained to use even in his light bending derivation in his GR book.

If you don't want to use the equivalence principle, and you will only make Einstein measurements, i.e. you will measure both the accelerated clocks simultaneously in your frame, and their velocities are equal at that time in your frame, their ticking rate will be the same.  The acceleration doesn't matter.  I'm fudging a little here, because you can't actually measure their ticking "rate" at an instant in time, but as long as you measure the start and stop at each clock simultaneously with the other one, you get the result you expect.

BTW, due to conservation of energy, the WEP implies the EEP, see Cliff Will's book for more discussion.  It's not a proof but a good argument.  We can calculate the free space version very well based on SR, and lots of GR tests confirm that those results carry over to gravity, so the EEP is very well verified.  However, if you want to debate this, I cannot, I do not have the data and you can read Will's book for yourself, but you can of course post here and maybe someone else will answer.

Robert if we are talking about "apparent" effects there is no energy conservation for them because the energy is real for "real effects".

Time dilation is one, and is the one which is measured by comparing elapsed times of two atomic clocks firts together and the rebrought together. No other definitions are accepted othewise we are talking about different things.

I can see your point of view, you make the hypothesys on the reference systems, and on the apparent to real phenomena,

apparent effects in acc frames + WEP--> real effects of g field

and

apparent effects on g field + WEP --> real effects on acc. frames???

it could work then, but it is impossible to demostrate it, it is a Principle and that's it, while the WEP is a law.

Thanks Robert for having insisted in this point and I would say that if it is not an apparent effect on the first part, the principle does not hold.

It takes energy to get up and down in the accelerated elevator exactly as in a gravitational field.

I have no further information than that, Stefano, and I do not wish to debate.  I have referred you to Will's book if you wish to examine the arguments.  I am unfollowing now as I'm working under a deadline at the moment.  Good luck!

Yes Robert,

so the principle could not be applied if there  is not a rule

but in a G field the Redshift of photons is an apparent energy loss, while in accelerated frames it is real for the doppler effect.

In any cases the answer to the initial question is NO. There is no time dilation between equally accelerated clocks.

I would also consider the case when acceleration is 0. Two inertial systems can be accelerated to the same speed heading on to one another in the opposite direction prior to the experiment.

The synchronisation t=0, t'=0, t''=0 is accomplished when their origins coincide. From that point onward they diverge. At symmetric distances from the stationary origin there are a mechanisms to stop the moving clocks when coinciding.

While from the logical point of view time dilation should be the same for both from the point of view of the stationary system, their respective relative time dilation would vary in accordance with Lorentz formulae.

There can only be one timer state after clocks had been stopped while covering the same distance, the same duration.

There seems to be that common logic is contradicted by Lorentz transformation between the two moving systems in this travelling twins paradox.

Yes Andrew but this is another story... The mean acceleration is 0 but inside there is motion.

A correction has been done in the paper

Deleted research item The research item mentioned here has been deleted

What do you mean by a different clock rate? For this you will need to define an observer. if the clocks go along exactly identical path in space-time, they are indistinguishable and the question is meaningless. On the other hand if the clocks are separated by a fixed distance in space-time then it all depends on the location and velocity of the observer. Special relativity work all the time not at only high relative speeds.

Yes but who said that SR is capable to face that prediction?