# Q & A: Gravity, Acceleration, and Time Dilation

Q:
Have you ever heard of the Twin Vials Paradox? It seems to be implied by Einsteins Equivalence Principle and the prediction from General Relativity that a clock in a strong gravitatinal field will run slower. Here is the situation: Two indentical specimens of radioactive material with known half lives are placed in two vials. One of these vials is put into an ultra centrifuge capable of producing a radial acceleration equivalent to half-a-million Gs, while the other vial remains on the bench top. The Equivalence Principle states that a force produced by a gravitating body and a force of equal magnitude due to an accelerating frame of reference are indistinguishable. Therefore, the vial in the ultra centrifuge should experience some degree of relativistic time dilation. The indentical radioactive specimen vials perform the function of a pair of naturally synchronized clocks. The time dilation will be detectable as a difference in the radioactive decay rates between the two identical specimens. When you consider that A. Eddingtons 1919 confirmatory evidence for the General Theory was produced by the Suns paltry 28 Gs, then some degree of time dilation should be easily observable in a 450,000 G field over and extended period of time. Do you know if this experiment has ever been attempted and what the results were? And by what factor would time slow down in a 450,000 G field? I look forward to your response. Thank you.
- Pat Dolan (age 42)
Seattle, WA, USA
A:
Hi Pat,

A deep and complicated question!

Fortunately there is no paradox here, although some explanation is in order, and a clarification.

The short answer is that you can figure out the time dilation using just Special Relativity from the point of view of the lab frame, in which one of the samples is moving quickly and the other is at rest. The time dilation factor is given by

T/Tprime = 1/sqrt(1-v**2/c**2) where v is the speed of the vial in the centrifuge, and c is the speed of light.

To calculate the relative decay rates from the point of view of the accelerating vials, you do need to use general relativity, but the ratio of the rates comes out to be the same as we just found, so there's no paradox.

The experiments which back this up are almost identical to the one you propose! They involve storing muons moving very close to the speed of light in a circular pipe with magnets all around to steer the muons around in a circle. Muons are heavy, unstable cousins of the electron. Muons decay with a precisely known lifetime, about 2.197 microseconds. The acceleration of the muons around the ring in the 1966 experiment I have access to in this context was 5x10**20 cm/sec**2, or 5x10^17 times that of gravity (a trillion times more than you suggest)! No effect was seen other than those predicted by special relativity (Farley, 1966). More precise measurements have been made since, and here at the University of Illinois, we have some of the world's experts. They have measured the time dilation factor due to the fact that the muons are moving to a few parts per million, with no evidence for any additional effect from the acceleration. An effect seen would have violated general relativity.

The time dilation in a gravitational field does not depend on the local strength of the field, but rather "how deep you are inside" one. If the gravitational field is nearly uniform, so that it is almost as strong way up high as it is near the ground, then there will still be gravitational redshift of light climbing up against gravity.

My source here is the fine book "Gravitation", by Misner, Thorne and Wheeler -- the specific case you mention is discussed on p. 1055.

Tom (and Mike)

(published on 10/22/2007)

## Follow-Up #1: twins

Q:
If this reasoning were true, the same logic should apply to the twins paradox - the 'resolution' of which claims the traveling(non-inertial) twin would be younger irrespective of the path he took. Looking forward to your answer.
- Dilip Rajeev (age 23)
Trivandrum, India
A:
Yes, that's right, assuming that gravitational effects aren't important.

When the traveling twin spends a lot of time farther 'up' in a gravitational field, General Relativistic effects can reverse that.

Mike W.

(published on 09/05/2008)

## Follow-Up #2: equivalence

Q:
I don't quite understand the answers here. Is there unambiguous proof that centrifugal acceleration does not have the same effect as gravitational acceleration? How should we in that case describe the principle of equivalence?
- Bob (age 46)
Norway
A:
Yes, the experiments Tom cited show the difference, as do many satellite experiments. Let's just discuss circular orbiting objects. Since they stay at a fixed distance, there's a fixed time to exchange signals, so both observers agree on which clock is faster. An object held in orbit by some force besides gravity has a slower clock just as expected from Special Relativity. If it's held in orbit by gravity, there's an additional General relativistic effect, of the opposite sign. In the case of geosynchronous satellites, used for GPS, the GR effect is actually larger.

The equivalence principal applies to uniform gravitational fields and uniformly accelerating reference frames. Gravitational fields around a planet or star are not uniform, because 'down' is different directions depending on where you are.

Mike W.

(published on 03/21/2009)

## Follow-Up #3: Acceleration and time dilation

Q:
Can I recast the question to focus on acceleration rather than gravity? It is generally stated that if a traveler undergoes constant acceleration then his clock slows relative to a stationary observer (and both agree on this point). This is usually stated in the context of a rocket traveling to a distant galaxy. For example, one online calculator shows that if you travel at a constant 5G acceleration then decelerate, for a total trip of 10 light years, 1.5 years pass for you, but on earth 10.4 years will pass. This is linear movement and acceleration. Question: why does centripetal acceleration in steady rotation not cause the same time dilation? It is acceleration and it is constant. Thanks.
- John (age 48)
NY, NY, USA
A:
The effect you refer to exists and is well measured.  In fact the GPS system of orbiting satellites show this effect.  If not for general relativistic corrections, the navigational errors would be too serious to allow the necessary accuracy we need and are used to.
See:    for details.

LeeH

Another point: The relative clock rates do not depend just on the magnitude of the acceleration but on the direction of the acceleration and on the displacement between the clocks. The stay-at-home ages faster because the traveler is accelerating toward home while at a distance. As Lee points out, that's true for both a linear trip or an orbital one.
Also, to be formal I should point out that in relativity the concept of 'stationary observer' is not meaningful.

Mike W.

(published on 11/29/2009)

## Follow-Up #4: relativistic time

Q:
I am not expert on SR or GR, but I like them very much, I understand them a little, and I don't believe in either of them. So please feel free to ignore my question, as it might not even be of educational importance. Well, this all (explanations and follow-up) seems a little vague. Also, there are no experiments other than muon decay, which is quite different than centrifuge system. It has not been explained anywhere how muons are "stored", I mean how can you stop them from decaying! From wikipedia Gravitational Time Dilation page... "On a rotating disk when the base observer is located at the center of the disk and co-rotating with it (which makes their view of spacetime non-inertial), the equation is , Td=1/root(1-(r**2xω**2/c**2))where r is the distance from the center of the disk (which is the location of the base observer), and ω is the angular velocity of the disk. (It is no accident that in an inertial frame of reference this becomes the familiar velocity time dilation Td=1/root(1-(v**2/c**2)))." And, as we can see, no reference has been made to any change in distance between two clocks (r = radius of the centrifuge arm is constant). This suggests that a stationary observer either at the center of the centrifuge observing the clock or outside the centrifuge but stationary with respect to the center should observe the time dilation on gravitating clock. Also, the effects of decrease in earth's gravitation at higher altitudes are noted, then, if we consider the whole earth as a big centrifuge, according to the wikipedia article, the effects of centrifugal gravitation must be observable. Thank you. Mitesh
- Mitesh Patel (age 27)
A:
Of course it's not just muons that last longer, but a large collection of different types of unstable particles all of which show exactly the same form of lifetime prolongation when kept in circular motion at high energies.

I had trouble following the rest of your question, although I can see that you have some specific point. However, I don't understand certain phrases, such as "centrifugal gravitation". Could you re-write it to make sure we understand the question?

Mike W.

(published on 01/31/2010)

## Follow-Up #5: Relativity in a centrifuge

Q:
Assume you actually did the vial-in-a-centrifuge experiment. The vial would have a leading side (the side toward the direction of rotation) and a trailing side. According to your answer, the clock rate on the leading side is the same as the clock rate on the trailing side. Can we generalize to the following rule: clock rates are the same at the front and the rear of all uniformly accelerating objects (whether acceleration is due to change of direction or change of magnitude)? This contradicts many web sites that claim that clocks run faster at the front and slower at the rear of accelerating objects. The web sites seem to analogize the accelerating object to an object in a gravity well. But acceleration is not like gravity over any extended space or time. So it should not affect the clock rate at the front differently from the clock rate at the rear like a gravity well. Einstein's example of the accelerating chest in his book "Relativity: the Special and General Theory" stipulates that the chest undergoes "a uniformly accelerated motion." Therefore the chest's lid and floor should have uniform velocity, so based on your answer a clock at the top should run at the same rate as the clock on the bottom. Similarly, someone within the chest hanging from a rope attached to the lid would feel the same gravity if the rope were short (so he was near the lid) or long (so he was near the floor). And light that enters at the top would curve down exactly the same as light that enters near the bottom (again, because of the uniform acceleration). So it seems that the top and bottom clocks should run at the same rate, using Einstein's analogy. Who is right -- you or those web sites? Obviously I think that you are. Or is there some other feature of relativity that changes the answer for an object that undergoes linear acceleration (change in magnitude) like the chest rather than change in direction acceleration like the muons in the Farley experiment?
- John (age 51)
NY, NY USA
A:
Let me describe the general rule first.
What shows up in the clock rates is the difference between the actual acceleration a and the free-fall gravitational acceleration g. The rule is the same in all cases: to lowest order in  a-g the clock rate compared to a reference clock is multiplied by (1+(a-g).r/c2), where r is the displacement from the reference clock and c is the speed of light. (Apologies for the misplaced dot in the dot product.)

Now let's apply that rule to your first case. In the spinning centrifuge, the inward acceleration a is at right angles to the displacement r between the leading and trailing edges, so the dot product is zero. Therefore their clocks run at the same rate.
In the rocket the acceleration is along the length of the rocket, so (a-g).r is not zero. The clock in the nose runs faster than the one in the tail in the leading  part run faster than those in the tail.

So both our answer and what it sounds like those sites say are true. The reasoning about applying the Lorentz time dilation factor to uniformly accelerating objects breaks down due to effects involved in changing signal transmission times. For circular motion, the time to send signals to and from the middle doesn't change, so the arguments are simpler. For the uniform case, it's true that the clock the acceleration points toward runs faster, despite the apparently similar experiences of the front and rear clocks.

Mike W.

(published on 03/20/2012)