Is acceleration the same as gravitational time dilation?

[Buckethead]

It seems that time dilation does not occur due to acceleration alone. It surprised me to learn this due to the equivalence principle as I was under the impression that one could not experimentally know the difference between being accelerated in space and being at rest in a gravitational field (other than tidal forces). The only experiment I'm aware of to verify this was done with muons in a centrifuge with a force of 10^18 g being applied (lots of g). My question: Is this definitive proof? Does the fact that a muon is a particle make this experiment any less valid than say putting a quartz watch inside a centrifuge? Isn't acceleration a form of space warping due to the differential time dilation between one part of an accelerated object and another? Thanks (no math in the answer please).

 

[dauto]

Does the fact that a muon is a particle make this experiment any less valid than say putting a quartz watch inside a centrifuge?

No. Why should it?

 

[Nugatory]

Isn't acceleration a form of space warping due to the differential time dilation between one part of an accelerated object and another?

No. The easiest way of seeing this is to consider two objects (subatomic particles or not, makes no difference) moving through the same point in spacetime but with different accelerations. For example, if one spaceship is accelerating from the left, the other from the right, and there's a point where they pass each other: the curvature/warping of spacetime is the same for both, as they're at the same point and there's only one way the curvature can be at that one point, yet they're experiencing very different accelerations.

Acceleration is what you experience when you're not traveling in a straight line (also called a geodesic) in spacetime.

 

[phinds]

Acceleration is what you experience when you're not traveling in a straight line (also called a geodesic) in spacetime.

Hm ... that doesn't seem like the right way to say it. Contrary to the implication of your statement, you CAN travel on a geodesic under acceleration if you choose to, it's more that if you are NOT traveling on a geodesic then you MUST be accelerating.

 

[Nugatory]

Hm ... that doesn't seem like the right way to say it. Contrary to the implication of your statement, you CAN travel on a geodesic under acceleration if you choose to, it's more that if you are NOT traveling on a geodesic then you MUST be accelerating.

What's an example of a geodesic trajectory through spacetime along which an accelerometer will register any value except zero?

 

[dauto]

What's an example of a geodesic trajectory through spacetime along which an accelerometer will register any value except zero?

I can't think of any. I think your statement was correct as stated.

 

[phinds]

What's an example of a geodesic trajectory through spacetime along which an accelerometer will register any value except zero?

Guess I have a misunderstanding. I don't understand why you cannot accelerate along a geodesic.

For example, in deep space a geodesic is more or less a Euclidian straight line and a spaceship traveling along that geodesic could accelerate. Likewise, with some trickly navigation, a spaceship could accelerate along a geodesic that Euclid would see as more of a curved line (in space). Why is this not correct?

 

[dauto]

Guess I have a misunderstanding. I don't understand why you cannot accelerate along a geodesic.

For example, in deep space a geodesic is more or less a Euclidian straight line and a spaceship traveling along that geodesic could accelerate. Likewise, with some trickly navigation, a spaceship could accelerate along a geodesic that Euclid would see as more of a curved line (in space). Why is this not correct?

That's not correct because we are talking about a geodesic in 4-D space-time, not a geodesic in 3-D space. When a ship accelerates it actually is changing geodesics (it's trajectory through space-time is different than what it would've been had it not accelerated). You cannot change how fast you move through a geodesic. You are always moving at the rate of 60 minutes per second.

 

[pervect]

It seems that time dilation does not occur due to acceleration alone. It surprised me to learn this due to the equivalence principle as I was under the impression that one could not experimentally know the difference between being accelerated in space and being at rest in a gravitational field (other than tidal forces). The only experiment I'm aware of to verify this was done with muons in a centrifuge with a force of 10^18 g being applied (lots of g). My question: Is this definitive proof? Does the fact that a muon is a particle make this experiment any less valid than say putting a quartz watch inside a centrifuge? Isn't acceleration a form of space warping due to the differential time dilation between one part of an accelerated object and another? Thanks (no math in the answer please).

Time dilation does occur due to an accelerated frame of reference, in fact Einstein's hypothetical accelerating elevator was one of the reasons time dilation due to gravity was expected.

I suspect from context that what you are referring to is something different, called the "clock postulate". The clock postulate says that if you use an inertial frame of reference, the acceleration of a particle doesn't matter to it's proper time.

This is different from saying that non-inertial accelerating frames of reference do not experience time dilation -they do.

 

[phinds]

That's not correct because we are talking about a geodesic in 4-D space-time, not a geodesic in 3-D space. When a ship accelerates it actually is changing geodesics (it's trajectory through space-time is different than what it would've been had it not accelerated). You cannot change how fast you move through a geodesic. You are always moving at the rate of 60 minutes per second.

OK, got it. Thanks.

 

[Buckethead]

Time dilation does occur due to an accelerated frame of reference, in fact Einstein's hypothetical accelerating elevator was one of the reasons time dilation due to gravity was expected.

I suspect from context that what you are referring to is something different, called the "clock postulate". The clock postulate says that if you use an inertial frame of reference, the acceleration of a particle doesn't matter to it's proper time.

This is different from saying that non-inertial accelerating frames of reference do not experience time dilation -they do.

I find it difficult to understand why an object in a centrifuge is considered to be in an inertial frame. What is the difference between such an object and an object accelerating in a linear way through space? Both experience a force equivalent to gravity. More importantly in an accelerating elevator in space a beam of light curves and isn't this why Einstein suspected that the acceleration due to gravity also curves light? This being the case, a beam of light passing through a room tied to a giant centrifuge would also appear to curve to a person standing in the room directly indicating that time for this person must be moving more slowly relative to a person outside the centrifuge to account for the longer path of the light beam. So why is one considered inertial and the other not?

 

[PAllen]

I find it difficult to understand why an object in a centrifuge is considered to be in an inertial frame. What is the difference between such an object and an object accelerating in a linear way through space? Both experience a force equivalent to gravity. More importantly in an accelerating elevator in space a beam of light curves and isn't this why Einstein suspected that the acceleration due to gravity also curves light? This being the case, a beam of light passing through a room tied to a giant centrifuge would also appear to curve to a person standing in the room directly indicating that time for this person must be moving more slowly relative to a person outside the centrifuge to account for the longer path of the light beam. So why is one considered inertial and the other not?

An object in a centrifuge is moving non-inertially. You can study what happens from any frame you want. If you analyze in an inertial frame, the clock postulate tells you that you only need to worry about speed. You can also choose analyze from coordinates in which object has fixed spatial coordinates. Now the clock hypothesis does not apply, and the metric will no longer be diag (-1,1,1,1). The metric will demonstrate pseudogravity time dilation where clock rates depend on radial position in these coordinates.

 

[Buckethead]

An object in a centrifuge is moving non-inertially. You can study what happens from any frame you want. If you analyze in an inertial frame, the clock postulate tells you that you only need to worry about speed. You can also choose analyze from coordinates in which object has fixed spatial coordinates. Now the clock hypothesis does not apply, and the metric will no longer be diag (-1,1,1,1). The metric will demonstrate pseudogravity time dilation where clock rates depend on radial position in these coordinates.

As an amateur I'm having difficulty deciphering what you are saying, but are you saying that from outside the centrifuge only the velocity of the object matters so would experience time dilation due to special relativity (and this is the clock postulate?), but from inside the centrifuge speed is not relevant but the clock will slow due to pseudogravity?

 

[A.T.]

As an amateur I'm having difficulty deciphering what you are saying, but are you saying that from outside the centrifuge only the velocity of the object matters so would experience time dilation due to special relativity (and this is the clock postulate?), but from inside the centrifuge speed is not relevant but the clock will slow due to pseudogravity?

Yes, different frames will explain the time dilation differently.

 

[PAllen]

As an amateur I'm having difficulty deciphering what you are saying, but are you saying that from outside the centrifuge only the velocity of the object matters so would experience time dilation due to special relativity (and this is the clock postulate?), but from inside the centrifuge speed is not relevant but the clock will slow due to pseudogravity?

Basically yes. I could quibble with wording, but that is the general idea.

 

[Buckethead]

Since it's possible to have high acceleration and low velocity (small radius centrifuge for example) or low acceleration/high velocity (long arm centrifuge), it appears to me that the two perspectives are at odds. For example I could be in a centrifuge with a short arm or a long arm and my velocities would be different even though my accelerations could be matched. So from outside the time dilation of the two would be different (due to velocity related effect only), but from inside the centrifuge the time dilation would be the same in both cases due to the accelerations being the same.

 

[PAllen]

Since it's possible to have high acceleration and low velocity (small radius centrifuge for example) or low acceleration/high velocity (long arm centrifuge), it appears to me that the two perspectives are at odds. For example I could be in a centrifuge with a short arm or a long arm and my velocities would be different even though my accelerations could be matched. So from outside the time dilation of the two would be different (due to velocity related effect only), but from inside the centrifuge the time dilation would be the same in both cases due to the accelerations being the same.

The resolution is simple. All forms of gravitational or pseudo-gravitational time dilation don't depend on the 'force' or acceleration; they depend on potential difference. Thus, for a small and rapid centrifuge, you have high gee but only a small height (distance between center, which is inertial, and edge). The potential difference is small because of the small distance (integral (centrifugal acceleration) dr ). For a very large, low acceleration centrifuge, you have large potential difference from the inertial center due the large distance.

 

[pervect]

Since it's possible to have high acceleration and low velocity (small radius centrifuge for example) or low acceleration/high velocity (long arm centrifuge), it appears to me that the two perspectives are at odds. For example I could be in a centrifuge with a short arm or a long arm and my velocities would be different even though my accelerations could be matched. So from outside the time dilation of the two would be different (due to velocity related effect only), but from inside the centrifuge the time dilation would be the same in both cases due to the accelerations being the same.

Time dilation is proportional to acceleration * distance rather than acceleration. So the time dilation wouldn't be the same in the two cases you decribe.

Treating acceleration a as gravity g:

Time dilation is proportional to gravitational potential g*h when g is constant (or more generally, the integral of g*dh if g varies). Writing the integral as a differential, one could say that the rate of change of time dilation with distance is equal to the acceleration.

 

[Buckethead]

Thanks everyone for all the help. I think I finally have a clearer understanding of this. I was stuck on thinking force was affecting time because I was looking at this the wrong way. Force is an effect, not a cause. If I keep it straight in my head that (in the case of gravity) it is mass that causes curvature and curvature that causes time dilation and then apply that to a centrifuge it is easier to see that likewise it is not the force that causes time dilation, but the curvature created by the centrifuge which is related to it's speed and radius that causes the time dilation and the force is just an artifact of the speed and curvature. Is all of this accurate?

 

[PAllen]

No, not really. Curvature only affects time dilation between a pair of observers as a second order correction to SR Doppler. Looking first at SR, what we call pseudo-gravitational time dilation is just the way ordinary SR Doppler manifests for a family of non-inertial observers meeting certain conditions (e.g. for the family of observers we call a rocket frame; or the family moving with a centrifuge). There is no curvature affect - it is ultimately just relative velocity Doppler modified by the non-inertial coordinates. In GR, for two very nearby observers in relative motion, you have the same Doppler as SR; if they are further apart, the effect is modified by curvature. Gravitational time dilation in GR (just like SR) is still (to first order) a manifestation of SR Doppler for a non-inertial family of observers; curvature provides a second order correction.