Gravity/acceleration equivalent?

[ubavontuba]

Forum,

Okay, so Einstein says that to an internal observer a room being pulled under a constant 1g acceleration is indistinguishable (by experiment) from a room hanging from a tree by a rope.

Putting gravitational divergence aside, do you think this is true?

If I (as an inside observer) can demonstrate by internal experiment that the room is either accelerating or is at rest in a gravitational field, would this be important?

 

[JesseM]

Forum,

Okay, so Einstein says that to an internal observer a room being pulled under a constant 1g acceleration is indistinguishable (by experiment) from a room hanging from a tree by a rope.

Putting gravitational divergence aside, do you think this is true?

If I (as an inside observer) can demonstrate by internal experiment that the room is either accelerating or is at rest in a gravitational field, would this be important?

Yes, that would contradict the equivalence principle, provided the room is arbitarily small so that differences in gravity from one internal region to another (tidal forces, for example) become negligible. Also, I think you'd have to assume the time period you're looking at becomes arbitrarily brief--the equivalence principle only applies to arbitrarily small regions of spacetime, although offhand I don't know how gravity could be distinguished from acceleration even given an extended period of time to make your observations.

 

[bernhard.rothenstein]

acceleration, gravity

Forum,

Okay, so Einstein says that to an internal observer a room being pulled under a constant 1g acceleration is indistinguishable (by experiment) from a room hanging from a tree by a rope.

Putting gravitational divergence aside, do you think this is true?

If I (as an inside observer) can demonstrate by internal experiment that the room is either accelerating or is at rest in a gravitational field, would this be important?

A look at
Ling Tsai
The relation between gravitational mass, inertial mass and velocity
am.J.Phys. 54 340 (1986)
could be illuminating

 

[ubavontuba]

Yes, that would contradict the equivalence principle, provided the room is arbitarily small so that differences in gravity from one internal region to another (tidal forces, for example) become negligible. Also, I think you'd have to assume the time period you're looking at becomes arbitrarily brief--the equivalence principle only applies to arbitrarily small regions of spacetime, although offhand I don't know how gravity could be distinguished from acceleration even given an extended period of time to make your observations.

So if I can really do this, it would really be important? Does anyone disagree or wish to add qualifications?

A look at
Ling Tsai
The relation between gravitational mass, inertial mass and velocity
am.J.Phys. 54 340 (1986)
could be illuminating

I'll Google this since it sounds like something in the same vein as I'm thinking from the title, but do you have a link?

Edit: Okay, I found it on my own. I'll do a little reading and let you know if it's applicable to my feat of derring-do...

 

[dicerandom]

Yes, that would contradict the equivalence principle, provided the room is arbitarily small so that differences in gravity from one internal region to another (tidal forces, for example) become negligible. Also, I think you'd have to assume the time period you're looking at becomes arbitrarily brief--the equivalence principle only applies to arbitrarily small regions of spacetime, although offhand I don't know how gravity could be distinguished from acceleration even given an extended period of time to make your observations.

Even if your room was very small if you let enough time go by there would be a tidal effect which you could observe. Although I think this would only apply in the freefalling (or extremely low acceleration) case, since if you were accelerating at any decent rate your test particles would hit the floor of your box before you noticed any tidal effects.

 

[JesseM]

So if I can really do this, it would really be important?

Yes. As this page says:

The equivalence principle can be stated as "At every spacetime point in an arbitrary gravitational field, it is possible to chose a locally inertial coordinate system such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems.

Or as http://scholar.uwinnipeg.ca/courses/38/4500.6-001/Cosmology/Principle%20of%20Equivalence%20in%20Mathematical%20Form.htm puts it:

General relativity yields the special theory of relativity as an approximation consistent with the Principle of Equivalence. If we focus our attention on a small enough region of spacetime, that region of spacetime can be considered to have no curvature and hence no gravity. Although we cannot transform away the gravitational field globally, we can get closer and closer to an ideal inertial reference frame if we make the laboratory become smaller and smaller in spacetime volume. In a freely falling (non-rotating) laboratory occupying a small region of spacetime, the laws of physics are the laws of special relativity. Hence all special relativity equations can be expected to work in this small segment of spacetime.

So, a violation of this would certainly be important, although the fact that thousands of smart physicists have studied the issue and found no reason to doubt the equivalence principle should lead you to suspect there is very likely a flaw in your idea.

 

[JesseM]

Even if your room was very small if you let enough time go by there would be a tidal effect which you could observe. Although I think this would only apply in the freefalling (or extremely low acceleration) case, since if you were accelerating at any decent rate your test particles would hit the floor of your box before you noticed any tidal effects.

How would these tidal effects manifest in the freefalling case, for a very small room falling for an extended period of time?

 

[ubavontuba]

JesseM et al,

I need to make it clear that I'm referring to the hanging room case as per this paper written by Albert Einstein. I'm not saying I can make this determination in a free-falling room in space versus a free-falling room in a gravity field ("Except for the splat at the end... they're practically similar" -Tigger). Would this still be important?

I feel I should forewarn you that this concept really works, but it's also a bit mischevious. That is; I have found a loophole to the conundrum, but I don't think it has a lot of practical considerations.

Edit: I wish to add that I will start out with a simple and rather silly version of the test. You will then likely wish to qualify the experiment, so I strongly recommend you satisfy your qualifications now to the best of your abilities. I will accept later qualifications, but only if it is conceded that my first experiment works under the current treatise.

Any qualifications you make will be examined for merit and depending on their limiting factors I will refine the experimental concept to compensate. If the restrictions get severe, I may need to ascertain that the measurements are purely hypothetical (as they may be very small) and have this still be accepted as valid. Of course, this will only be in response to restrictions that are not specified in the Einstein paper.

Is everyone in accordance? Are you ready for the silliness to begin?

 

[dicerandom]

How would these tidal effects manifest in the freefalling case, for a very small room falling for an extended period of time?

The test particles would slowly separate from one another due to the extremely minute difference in the gravitational force (or geodesics, if you prefer) due to their differing heights above the planet. Even with a very small room this effect would become evident given enough time, provided you don't hit the ground first. This separation would not occur in the flat space room.

JesseM et al,

I need to make it clear that I'm referring to the hanging room case as per this paper written by Albert Einstein. I'm not saying I can make this determination in a free-falling room in space versus a room free-falling in a gravity field ("Except for the splat at the end... they're practically similar" -Tigger). Would this still be important?

I feel I should forewarn you that this concept really works, but it's also a bit mischevious. That is; I have found a loophole to the conundrum, but I don't think it has a lot of practical considerations.

Well don't keep us in suspense, let's hear it

 

[JesseM]

JesseM et al,

I need to make it clear that I'm referring to the hanging room case as per this paper written by Albert Einstein. I'm not saying I can make this determination in a free-falling room in space versus a free-falling room in a gravity field ("Except for the splat at the end... they're practically similar" -Tigger). Would this still be important?

I don't think it should matter--after all, a hanging room as seen by a freefalling observer passing it should look just like an accelerating room as seen by an inertial observer passing it. You could even imagine a small room in a box inside a freefalling lab, with the box using rockets to maintain a constant distance from the Earth even as the lab falls so the box approaches the lab's ceiling--this should appear just like a small room-in-a-box inside an inertial lab in deep space using rockets to accelerate in the direction of the lab's ceiling.

I feel I should forewarn you that this concept really works, but it's also a bit mischevious. That is; I have found a loophole to the conundrum, but I don't think it has a lot of practical considerations.

Again, the fact that this would violate such a basic principle should lead you to have some doubts about whether you have actually found such a loophole, I think. But anyway, let's hear it!

Edit: I wish to add that I will start out with a simple and rather silly version of the test. You will then likely wish to qualify the experiment, so I strongly recommend you satisfy your qualifications now to the best of your abilities. I will accept later qualifications, but only if it is conceded that my first experiment works under the current treatise.

OK, the best way I can put it is that if you compare the hanging room and the accelerating room, for any small but finite room over a finite time there will be some slight differences, but in the limit as the volume and time approach zero, the magnitude of these differences should also approach zero. I'd be very surprised if you could find something that violates this.

 

[pervect]

Forum,

Okay, so Einstein says that to an internal observer a room being pulled under a constant 1g acceleration is indistinguishable (by experiment) from a room hanging from a tree by a rope.

Putting gravitational divergence aside, do you think this is true?

If I (as an inside observer) can demonstrate by internal experiment that the room is either accelerating or is at rest in a gravitational field, would this be important?

The tidal forces (aka Riemann curvature tensor) are different for the observer in the room and the observer standing on a spherical planet.

This is essentially, however, a consequence of the gravitational divergence, so I'm not sure if you'd include it as a separate phenomenon or not.

One should be able to suppress the tidal forces in theory by making the planet a huge disk rather than a huge sphere. The metric for an infinite flat plane should be equivalent to two Rindler metrics (the metric of an accelerating spaceship) "glued together" at the z=0 plane, as I've mentioned in another thread.

Even with a finite disk rather than an infinite one, the supression of tidal forces should be good (but not perfect).

 

[rbj]

You could even imagine a small room in a box inside a freefalling lab, with the box using rockets to maintain a constant distance from the Earth even as the lab falls so the box approaches the lab's ceiling--this should appear just like a small room-in-a-box inside an inertial lab in deep space using rockets to accelerate in the direction of the lab's ceiling.

I had never heard that version of the EP thought experiment. (only the two basic ones, accelerating room in space vs. one sitting on the planet and free-falling room vs. one drifting freely in outer space, and in both comparisons what light or other free objects would be observed to do.) thanks for bringing it up.

 

[JesseM]

I had never heard that version of the EP thought experiment. (only the two basic ones, accelerating room in space vs. one sitting on the planet and free-falling room vs. one drifting freely in outer space, and in both comparisons what light or other free objects would be observed to do.) thanks for bringing it up.

I just made that thought-experiment up, to illustrate why it seems to me that the equivalence of free-falling frames and inertial frames would automatically imply the equivalence of an accelerating frame and a frame at rest relative to a gravitational field.

 

[yogi]

OK ubavontuba - Let's hear the revelation

 

[rbj]

I just made that thought-experiment up, to illustrate why it seems to me that the equivalence of free-falling frames and inertial frames would automatically imply the equivalence of an accelerating frame and a frame at rest relative to a gravitational field.

well, it's a good one. i'll have to remember to credit "JesseM" if i ever use it.

 

[ubavontuba]

How to test for gravity in the equivalence thought experiment

Forum,

Okay. Beginning with the treatise that the room is either hanging from an earthbound structure or an accelerating vehicle, I'd like to clarify one important aspect. That being that other than the fact that constant acceleration at the stated rate is currently impossible, all structures and systems obey all currently known laws of physics and that the structures are of reasonable parameters for either condition. Is this agreeable?

The occupant begins by being completely unaware of the conditions of the room and he must perform all tests within the room. He cannot look outside (through a window or some such). Nor may he use devices that can perceive exterior conditions directly through the walls (like an X-ray observatory).

Also, all effects of divergence are disallowed. That meaning that I will not fall on the effects of divergence as being a method to test for gravity (including tidal effects).

Lastly, I will reitterate that this is not meant to be a serious examinination of equivalence. Rather this is simply an exercise in creative thinking. Enjoy!

How to Test for Gravity in the Equivalence Thought Experiment

Let's examine one of the most basic tenants of relativity... that being the equivalence principal.

The equivalence principal basically states that it is impossible in a sealed room to conduct an experiment that could distinguish the difference between gravity and constant acceleration.

Of course there is spherical divergence to consider (immeasureably small), but let's place that aside for now and look at this in a more fundamental way.

Let's say I am in a sealed, hanging room (you'd like that, wouldn't you?) and that I want to find out if my room is part of an accelerating rocketship, or is situated comfortably on Earth. How might I easily determine this? Let's make this even harder by stipulating that initially, I don't even know that the room is hanging, or floating, or stable, or whatever.

Remember, Einstein's equivalence principal states that I shouldn't be able to tell by using any experiment in the room. I'd have to look outside to tell. Here goes:

To accomplish my feat, I need some very specialized equipment.

I need a snifter of brandy (make it full to the brim) and myself.

Let's proceed:

Step 1. Drink half the brandy. "Ah... good stuff."

Step 2. Place brandy snifter on the floor, but to the side a bit.

Step 3. Begin leaping laterally (from side to side) in the room.

Step 4. Observe the brandy.

If the brandy sloshes, I am in a relatively low mass room and therefore must be accelerating, or at least be separated from the Earth (suspended). If sloshes, go to the next step. If not, then you are in a fixed room on a heavy mass. You are feeling gravity (end of experiment).

Step 5. Start leaping from side to side. Build up as much pendulum acceleration as you can (swing the room like a child on a swing). Stop. Does the room continue swaying normally? Then you are hanging over a heavy mass and are experiencing gravity. If it stops rather suddenly, or it has an unusual and increasing resonance, then you are accelerating (Oh no! It's gone out of control!). In the latter case, disaster is soon to follow.

You can qualify things by adding shock absorbers and whatnot to a suspended room or make the accelerating room's ship unusually massive, but this just defeats the spirit of the experiment and makes it so in that particular room it is hard to distinguish between gravity and acceleration. This wouldn't be applicable to a supposedly universal principal.

Even so, you could still tell because a shock absorbed room must take time to settle (it's just quicker) Whereas an accelerating room and frame in free space is ALWAYS settled on it's center of mass (laterally) regardless of where you go in the room. The system will always stop moving laterally when you do. A massive room would just require more sensitive equipment.

Therefore, you can determine the difference between acceleration and gravity in a sealed room, right?

 

[DaveC426913]

You are trying to move the room. While technically not actually peeking out the window, you are still gleaning information about what is outside the room from within. You are not using the sensations of gravity or acceleration to do so. That's effectively cheating.

There are umpteen ways in practice to determine the difference, especially if you allow the sort of real-world engineering effects (such as the fact that the room can swing).

The effect holds in principle, if not in practice.

 

[JesseM]

Actually, I disagree that your methods would even produce any different results depending on whether the box was being accelerated or held still in a gravitational field. It is crucial, though, that we assume the force on the box is coming from the same point in both cases--for example, if the box in Earth gravity is being held up by a cable which is attached to a crane sitting on the surface of the earth, then we should assume the box in deep space is being pulled from a cable attached to its top which is attached to a crane which is being accelerated by a rocket whose inertia is just as great as the Earth's (but assume in this case we are dealing only with the laws of SR so this rocket doesn't distort spacetime). Likewise, if the box is sitting on the top of the earth, then in the accelerating case it should be pushed from below by a rocket whose inertia is as great as the earth's. And if the box is being kept at a constant height from the Earth by rockets attached to its bottom, then the box in space should be accelerating by means of identical rockets attached to its bottom. As long as we keep things analogous in this way, I don't think you would detect any difference in your brandy experiment.

 

[dicerandom]

I can jump up and down in my apartment and get a half-full snifter of brandy to slosh, it isn't built so well

Cute though.

 

[ubavontuba]

You are trying to move the room. While technically not actually peeking out the window, you are still gleaning information about what is outside the room from within. You are not using the sensations of gravity or acceleration to do so. That's effectively cheating.

There are umpteen ways in practice to determine the difference, especially if you allow the sort of real-world engineering effects (such as the fact that the room can swing).

The effect holds in principle, if not in practice.

Too late! You didn't prequalify this.

Besides, the dichotomy of the experiment clearly states that no test conducted within the room might distinguish the difference. It is not qualified as in; no experiment except___ (fill in the blank) can tell.

It's a fundamental statement. It's not an; "He found a solution, so his solution can't count!" That's like kids saying, "Not it!" after they've been tagged already.

 

[ubavontuba]

Actually, I disagree that your methods would even produce any different results depending on whether the box was being accelerated or held still in a gravitational field. It is crucial, though, that we assume the force on the box is coming from the same point in both cases--for example, if the box in Earth gravity is being held up by a cable which is attached to a crane sitting on the surface of the earth, then we should assume the box in deep space is being pulled from a cable attached to its top which is attached to a crane which is being accelerated by a rocket whose inertia is just as great as the Earth's (but assume in this case we are dealing only with the laws of SR so this rocket doesn't distort spacetime). Likewise, if the box is sitting on the top of the earth, then in the accelerating case it should be pushed from below by a rocket whose inertia is as great as the earth's. And if the box is being kept at a constant height from the Earth by rockets attached to its bottom, then the box in space should be accelerating by means of identical rockets attached to its bottom. As long as we keep things analogous in this way, I don't think you would detect any difference in your brandy experiment.

Wait a minute! "whose inertia is just as great as the earth's... so this rocket doesn't distort spacetime!?" If we must fantasize away the fundamentals of physics to get the result we want, then I want to fantasize away the limitations of conservation so that we can have unlimited energy and while we're at it, let's imagine there is no cosmic speed limit either (warp drive!).

The experiment becomes invalid on a fundamental level... if it has no basis in physics to begin with. Einstein didn't prequalify it thusly (he merely dismissed the importance of the accelerating device).

 

[ubavontuba]

I can jump up and down in my apartment and get a half-full snifter of brandy to slosh, it isn't built so well

Cute though.

Aha! A person of good cheer! That's the spirit!

 

[yogi]

Shucks - thought we were going to get some new insight - we skeptics always look for weird stuff - all you are doing is evaluating the structure - I want my money back

 

[JesseM]

Wait a minute! "whose inertia is just as great as the earth's... so this rocket doesn't distort spacetime!?" If we must fantasize away the fundamentals of physics to get the result we want

It's not fantasizing new laws of physics, it's simply subtracting out GR and imagining a universe where SR holds exactly. That's what the equivalence principle is all about, that locally a freefalling frame in GR is exactly like an inertial frame in SR.

In any case, you could always imagine the crane had thrusters or something so that it would be just as resistant to sideways acceleration as a crane sitting on the earth. And if you keep the box above the Earth in a way that does not allow its movements to push or pull on the earth--say, by rockets attached to the bottom of the box--then you don't have to worry about the inertia of the earth. But if the box is attached to the Earth in such a way that momentum applied to the box internally (by punching the wall, say) is dispersed into the earth, then it's no good to imagine a situation in space where the momentum is dispersed into a mass much less resistant to acceleration than the earth, these are non-equivalent physical situations just as much as if you imagine the box accelerated from the bottom through space but held up from the top on earth.

 

[ubavontuba]

Shucks - thought we were going to get some new insight - we skeptics always look for weird stuff - all you are doing is evaluating the structure - I want my money back

No refunds!

Besides, technically the structure is important to the experiment. It is the key to the fundamental difference between gravity and acceleration, that I am exploiting.

Remember, I said it was a "loophole," not a "fundamental discovery."

This loophole is applicable on all sorts of scales though. Therefore it really is a fundamental difference (even if it's not a paradigm changing consideration).

Frankly, I'm surprised that Einstein didn't address it. Apparently it didn't occur to him (or anyone else?) in regards to his paper... or he didn't think it worthy of mention.

 

[JesseM]

No refunds!

Besides, technically the structure is important to the experiment. It is the key to the fundamental difference between gravity and acceleration, that I am exploiting.

Remember, I said it was a "loophole," not a "fundamental discovery."

This loophole is applicable on all sorts of scales though. Therefore it really is a fundamental difference (even if it's not a paradigm changing consideration).

Frankly, I'm surprised that Einstein didn't address it. Apparently it didn't occur to him (or anyone else?) in regards to his paper... or he didn't think it worthy of mention.

Well, the fundamental form of the equivalence principle is all about freefalling frames in GR being compared to inertial frames in SR, and in both cases no external structure is needed. But like I said, if you can imagine a smaller box inside your inertial/freefalling box, which accelerates/stays at rest in a gravitational field using some structure external to itself but internal to the larger box (say, using rockets on its bottom), then it's a trivial extension of the "everything must look the same inside an inertial/freefalling box" principle to show that everything must look the same inside these smaller boxes (since they are part of the inside of the larger boxes, both accelerating towards the ceiling at the same rate and in the same way). But the principle of equivalence is really fundamentally about comparing the freefalling frame to the inertial frame, it only says that the at-rest-in-a-gravitational field lab must be equivalent to the accelerating-in-space lab to the extent that you can show why this follows from the freefalling/inertial equivalence.

Of course, the larger freefalling box need not have actual physical walls, it's just supposed to be a local region moving along with freefalling objects, so you can imagine falling past something like a smaller box on a rope attached to a crane and drawing an imaginary box around you as you fall. But then you'd have to ignore everything outside the imaginary freefalling box, and fix the boundary conditions on the imaginary walls, and then reproduce those same boundary conditions on the imaginary walls of an imaginary inertial box moving past an accelerating box-on-a-crane in empty space. So however the shaking of the small box (due to a tiny gnome inside dancing around, perhaps) affects the parts of the crane on the boundary of the freefalling box, you have to make sure the parts of the accelerating crane on the boundary of the inertial box are moving in exactly the same way. One way to do this would might be to imagine attaching the crane to an object of equal inertial mass as the Earth but ignore the gravitational effects of this mass as I suggested, another way would be to have thrusters on the sides of the crane on the portions outside the imaginary box which insure that the parts of the crane on the boundary move the same way as in the gravitational field. As long as you fix those boundary conditions, everything that happens inside the imaginary box should look the same whether the box is freefalling past a crane attached to the Earth or whether it's moving inertially past a similar crane that's accelerating in empty space.

 

[TheAntiRelative]

F. J. M. Farley, J. Bailey, R. C. A. Brown, M. Giesch, H. J¨ostlein, S. van der Meer, E. Picasso and M. Tannenbaum, Nuovo Cimento 45, 281-286 (1966), "The anomalous magnetic moment of the negative muon"

Above, they measured muon decay under many thousands of "G" from acceleration in a storage ring and there were no time effects unlike well-known decay rates of muons from the upper atmosphere. This, in my opinion, breaks equivelence. Below are more related studies I haven't been able to review yet though...



Bailey, J., Borer, K., Combley, F., Drumm, H., Eck, C., Farley, F.J.M., Field, J.H., Flegel, W., Hattersley, P.M., Krienen, F., Lange, F., Lebée, G., McMillan, E., Petrucci, G., Picasso, E., Rúnolfsson, O., von Rüden, W., Williams, R.W., and Wojcicki, S., “Final report on the CERN muon storage ring including the anomalous magnetic moment and the electric dipole moment of the muon, and a direct test of relativistic time dilation”, Nucl. Phys. B, 150, 1-75, (1979).

Carey, R.M. et al., “New Measurement of the Anomalous Magnetic Moment of the Positive Muon”, Phys. Rev. Lett., 82, 1632-1635, (1999).

 

[Mike2]

What may be of interest is how the Unruh affect applied to accelerating frames of reference might be applied to small regions of space at a fixed distance from a gravitating body.

 

[leandros_p]

Forum,

How to Test for Gravity in the Equivalence Thought Experiment

Let's examine one of the most basic tenants of relativity... that being the equivalence principal.

...
Remember, Einstein's equivalence principal states that I shouldn't be able to tell by using any experiment in the room. I'd have to look outside to tell. Here goes:
...
To accomplish my feat, I need some very specialized equipment.
...
I need a snifter of brandy (make it full to the brim) and myself.

Let's proceed:

Step 1. Drink half the brandy. "Ah... good stuff."

Step 2. Place brandy snifter on the floor, but to the side a bit.

Step 3. Begin leaping laterally (from side to side) in the room.

Step 4. Observe the brandy.

Step 5. Start leaping from side to side. Build up as much pendulum acceleration as you can (swing the room like a child on a swing). Stop. Does the room continue swaying normally? Then you are hanging over a heavy mass and are experiencing gravity. If it stops rather suddenly, or it has an unusual and increasing resonance, then you are accelerating (Oh no! It's gone out of control!). In the latter case, disaster is soon to follow.

You can qualify things by adding shock absorbers and whatnot to a suspended room or make the accelerating room's ship unusually massive, but this just defeats the spirit of the experiment and makes it so in that particular room it is hard to distinguish between gravity and acceleration. This wouldn't be applicable to a supposedly universal principal.

Hi ubavontuba,

Why don't you call a demolition unit to blow up the "damn" room, while keeping your eyes closed? Then, you should keep your promise not to look outside of the room and at the same time you would find out the status of the room. .

Now that's what I call "explosive" science, my friend.

Leandros

 

[pervect]

F. J. M. Farley, J. Bailey, R. C. A. Brown, M. Giesch, H. J¨ostlein, S. van der Meer, E. Picasso and M. Tannenbaum, Nuovo Cimento 45, 281-286 (1966), "The anomalous magnetic moment of the negative muon"

Above, they measured muon decay under many thousands of "G" from acceleration in a storage ring and there were no time effects unlike well-known decay rates of muons from the upper atmosphere. This, in my opinion, breaks equivelence. Below are more related studies I haven't been able to review yet though...

Right now, I have no idea of what you believe the problem is.

The lifetime of a muon in its rest frame is 1.56 us
http://hyperphysics.phy-astr.gsu.edu/HBASE/relativ/muon.html

http://www.g-2.bnl.gov/hepex0401008.pdf
tells us that the lifetime of a muon is 64.4 us with a momentum of 3.09 Gev/c under the conditions when it's magnetic moment is measured.

So there is *significant* time dilation, as one would expect. However, this time dilation will be entirely due to the velocity of the muon, not the acceleration of the muon:

dtau^2 = dt^2 - dx^2, the flat space formula for the Lorentz interval, implies that

(dtau/dt)^2 = 1 - (dx/dt)^2

Thus in the laboratory frame, there is no time dilation effect from the acceleration of the muon, only from its velocity.