Understanding Time Dilation in Relativity: An Example

[squeeky100]

I have only a layman's understanding of Relativity, and I'm somewhat confused about the time dilation concept. The only way that I can think of to ask my question is to present it in an example.

A spaceship moving at a constant .9 c traveses the distance between two inertial reference frames that are .9 light years apart (about 5.4 trillion miles). An observer at a stationary reference point clocks the event at 1 year. But, due to the dilation effects, the ships clocks shows only .44 years. Assume for this example that reference frames are stationary. Since distance and velocity are constant here, how can can v = ds/dt be valid
from the ship's perspective ? Does the ship recognize that it has traveled the 5.4 trillion miles in only .44 light years. I realize that I'm really missing something here.

 

[JesseM]

A spaceship moving at a constant .9 c traveses the distance between two inertial reference frames that are .9 light years apart (about 5.4 trillion miles).

An inertial reference frame is a coordinate system which fills all of space and time, probably you mean that the ship travels between two objects which share the same inertial rest frame?

An observer at a stationary reference point clocks the event at 1 year. But, due to the dilation effects, the ships clocks shows only .44 years. Assume for this example that reference frames are stationary. Since distance and velocity are constant here, how can can v = ds/dt be valid
from the ship's perspective ? Does the ship recognize that it has traveled the 5.4 trillion miles in only .44 light years. I realize that I'm really missing something here.

Are you familiar with length contraction? If the two objects are 0.9 light years apart in their own rest frame, then in the frame of a ship moving at 0.9c relative to them, they will only be 0.9 * sqrt(1 - 0.9^2) = 0.39 light years apart in the rest frame of the ship.

 

[squeeky100]

Yes, I misstated the example, and you clearified it nicely. What prompted my inquiry was one of the TV program series Universe concerning light speed and future space travel. Does time dilation and contraction effects imply that near light speed travel might significantly reduce the elasped time for a traveler to distant places. ie, a 100 light year distance might actually be reached be reached in far less time ?

 

[JesseM]

Yes, I misstated the example, and you clearified it nicely. What prompted my inquiry was one of the TV program series Universe concerning light speed and future space travel. Does time dilation and contraction effects imply that near light speed travel might significantly reduce the elasped time for a traveler to distant places. ie, a 100 light year distance might actually be reached be reached in far less time ?

It might be reached in far less time from the crew's perspective, yes (although if they went there and returned to Earth they'd find that more than 200 years had passed on Earth). http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html has some tables showing the times that it'd take from the crew's perspective to get to different destinations if the ship was constantly accelerating at 1G for the first half of the trip and then decelerating at 1G for the second half (though this would require vast amounts of energy so it probably isn't realistic)--in theory the crew could get to the Andromeda galaxy, 2 million light years away, in only 28 years of their own subjective time!

 

[Makep]

I have only a layman's understanding of Relativity, and I'm somewhat confused about the time dilation concept. The only way that I can think of to ask my question is to present it in an example.

A spaceship moving at a constant .9 c traveses the distance between two inertial reference frames that are .9 light years apart (about 5.4 trillion miles). An observer at a stationary reference point clocks the event at 1 year. But, due to the dilation effects, the ships clocks shows only .44 years. Assume for this example that reference frames are stationary. Since distance and velocity are constant here, how can can v = ds/dt be valid
from the ship's perspective ? Does the ship recognize that it has traveled the 5.4 trillion miles in only .44 light years. I realize that I'm really missing something here.

Time dilation and length / space contraction are two physical phenomena that must happen to all moving bodies, especially accelerating bodies. A moving body will never realize these phenomena relative to itself. It will however, realize it relative to another object or observer outside of the system that is moving / under observation. In space, you do not know that you are moving until you understand and perceive the movement with respect to other objects around you. This is called the Theory of Relativity. It does not compare with its own inertial frame, but that of the other - with respect to the other, with regard to the other, relative to the other, hence relativity.

 

[meteor9]

Jesse and Makep explanation are accurate and i have nothing against that , simply because Squeeky mentioned a TV series which stimulated his enquiry , here i mention another TV series which really puzzling me and generated some idea on which i 'm still pondering , it's about the BBC TV production 'Torchwood' which is an anagram of 'Doctor Who' series. Of course this topic could seem belonging to science fiction field , as it was said in another thread , but we should consider that all the exact and experimental sciences have been grown in the matrix of fiction, all the things which surround us today , the world in which we live with all its sophisticated technology , were at first a 'fiction' . Everything in every field which has skimmed across the human mind subjectively has became objectively concret , so i should say that i take more seriously the science fiction than the 'exact and experimental science' , because in the previous i notice much more blooming and lighting up of the thoughts. Anyway in that TV series , it occurs something they call 'time fault' or 'time failure' , where creatures belonging to other 'times' or 'galaxies' appear suddenly in our present , and people living at this moment can appear in some very distant past . What happened to 'time' ? What is this time failure ? I hope you will not say that 'ohh , this is just a fiction' , this is not a fiction , this is something which would happen or exists already , simply our 'logic mind' is not ready to see that . Do you have some formula for this ?

 

[Naty1]

Time dilation and length / space contraction are two physical phenomena that must happen to all moving bodies, especially accelerating bodies.

NO. This is not correct.

Velocity, not acceleration, causes time to slow and length to contract. If one wants to argue acceleration has an indirect affect on time and distance, which is not the usual accepted terminology, it would only be through the effect of different velocities at different times.

The only affect that acceleration can be argued to have is an approximate equivalence to gravity (via the "equivalence" principle of general relativity) and the emergence during acceleration of an event horizon at a distance behind the motion.

Aand here you have, I think ,the right idea but the wrong explanation:

"A moving body will never realize these phenomena relative to itself."
Of course it will:
You mean to say that an observer will measure these changes in time and distance in another inertial frame, but not their own. So if two rocketships pass each other, each observes identical time dilation and length contraction in the other rocket ship, but not their own.

 

[DrGreg]

Jesse and Makep explanation are accurate and i have nothing against that , simply because Squeeky mentioned a TV series which stimulated his enquiry , here i mention another TV series which really puzzling me and generated some idea on which i 'm still pondering , it's about the BBC TV production 'Torchwood' which is an anagram of 'Doctor Who' series. Of course this topic could seem belonging to science fiction field , as it was said in another thread , but we should consider that all the exact and experimental sciences have been grown in the matrix of fiction, all the things which surround us today , the world in which we live with all its sophisticated technology , were at first a 'fiction' . Everything in every field which has skimmed across the human mind subjectively has became objectively concret , so i should say that i take more seriously the science fiction than the 'exact and experimental science' , because in the previous i notice much more blooming and lighting up of the thoughts. Anyway in that TV series , it occurs something they call 'time fault' or 'time failure' , where creatures belonging to other 'times' or 'galaxies' appear suddenly in our present , and people living at this moment can appear in some very distant past . What happened to 'time' ? What is this time failure ? I hope you will not say that 'ohh , this is just a fiction' , this is not a fiction , this is something which would happen or exists already , simply our 'logic mind' is not ready to see that . Do you have some formula for this ?

There's a difference between science fiction that makes plausible predictions about how technology might develop in the future and science fiction that depicts (in the present day) scenarios that are completely outside all current scientific theories. In the latter case there's not much to be said except 'ohh, this is just a fiction'.

Would you also like some formula for how Santa Claus delivers all his presents in a few hours?

 

[meteor9]

I'm sure that you will find some formula for Santa Claus

 

[Makep]

NO. This is not correct.

Velocity, not acceleration, causes time to slow and length to contract. If one wants to argue acceleration has an indirect affect on time and distance, which is not the usual accepted terminology, it would only be through the effect of different velocities at different times.

The only affect that acceleration can be argued to have is an approximate equivalence to gravity (via the "equivalence" principle of general relativity) and the emergence during acceleration of an event horizon at a distance behind the motion.

Aand here you have, I think ,the right idea but the wrong explanation:

"A moving body will never realize these phenomena relative to itself."
Of course it will:
You mean to say that an observer will measure these changes in time and distance in another inertial frame, but not their own. So if two rocketships pass each other, each observes identical time dilation and length contraction in the other rocket ship, but not their own.

Movement of an object does not begin with an inherent velocity. The final velocity of a moving body is the result of the average acceleration of that body. Without an initial acceleration a body will not assume its final velocity. In this respect, acceleration, not velocity, is the process by which time dialtion and space / length contraction happen. The factor by which this happens is predicted, for a body with a constant velocity, by the Lorentz transforms, etc.

As to the quote below:

You mean to say that an observer will measure these changes in time and distance in another inertial frame, but not their own. So if two rocketships pass each other, each observes identical time dilation and length contraction in the other rocket ship, but not their own.

I don't quite get you here. With regards to identical,it would be correct only if their initial acceleration or the average acceleration that brought them to their current constant velocity are equal. Otherwise, they would not be identical.

 

[JesseM]

Movement of an object does not begin with an inherent velocity. The final velocity of a moving body is the result of the average acceleration of that body.

What do you mean by "initial" and "final"? At every moment an object has a well-defined velocity in any given inertial coordinate system, velocity defined simply by the rate the position coordinate changes as the time coordinate changes. So, if in an inertial coordinate system an object is at position x=5 meters at time t=4 seconds, and it's at position x=13 meters at time t=6 seconds, then it's moved a distance of 13-5=8 meters in 6-4=seconds, so if it was moving at constant velocity during that time, the velocity must have been 8/4=2 meters/second in the +x direction. How the object may have accelerated prior to this is irrelevant to calculating the velocity during this period of time.

In this respect, acceleration, not velocity, is the process by which time dialtion and space / length contraction happen. The factor by which this happens is predicted, for a body with a constant velocity, by the Lorentz transforms, etc.

Time dilation in a given coordinate system depends only on the velocity, which as I said is simply based on how the position-coordinate changes with the time coordinate. On object whose velocity has a magnitude of v during a given time-interval will always have its clock slowed down by a factor of $$\sqrt{1 - v^2/c^2}$$.

I don't quite get you here. With regards to identical,it would be correct only if their initial acceleration or the average acceleration that brought them to their current constant velocity are equal. Otherwise, they would not be identical.

No, you're incorrect, past acceleration doesn't influence the current time dilation, it's only based on current velocity. And if ship A measures ship B to be moving at speed v in A's inertial rest frame, that always means that ship B measures ship A to be moving at speed v in B's rest frame, so each will measure the other's clocks to be slowed down by the same factor of $$\sqrt{1 - v^2/c^2}$$.

 

[Fredrik]

...past acceleration doesn't influence the current time dilation, it's only based on current velocity.

This is of course 100% correct if "time dilation" refers to how the coordinate time between two events on the time axis of one inertial frame is different in another inertial frame, but the existence of the phrase "gravitational time dilation" makes things a bit more complicated. Gravitational time dilation is essentially the same phenomenon as the one that makes two clocks attached to opposite ends of an accelerating solid rod in flat spacetime tick at different rates, and in this case their past acceleration is important. (Their current acceleration will tell you their relative ticking rates, but to know what times they're displaying, you need to know how they have accelerated in the past).

 

[JesseM]

This is of course 100% correct if "time dilation" refers to how the coordinate time between two events on the time axis of one inertial frame is different in another inertial frame, but the existence of the phrase "gravitational time dilation" makes things a bit more complicated. Gravitational time dilation is essentially the same phenomenon as the one that makes two clocks attached to opposite ends of an accelerating solid rod in flat spacetime tick at different rates, and in this case their past acceleration is important. (Their current acceleration will tell you their relative ticking rates, but to know what times they're displaying, you need to know how they have accelerated in the past).

Yeah, Makep was talking about the twin paradox so I was mainly just thinking of SR time dilation. However, I don't think it's really right to say that gravitational time dilation depends on acceleration, I think it just depends on the curvature of spacetime at the points the worldline travels through; it's true that locally, gravitational time dilation is *equivalent* to the time dilation experienced by clocks experiencing Born rigid acceleration, but of course, the time dilation of clocks accelerating in SR is calculated solely based on their velocity as a function of time v(t), so it only "depends on acceleration" in the sense that the acceleration determines this velocity function.

 

[Fredrik]

However, I don't think it's really right to say that gravitational time dilation depends on acceleration, I think it just depends on the curvature of spacetime at the points the worldline travels through;

The fact that clocks on different floors in the same building on Earth tick at different rates is said to be due to gravitational time dilation. The reason why they tick at different rates is that they are accelerating by different amounts in their respective local inertial frames. (The proper time in a local inertial frame is $\sqrt{dt^2-dx^2-dy^2-dz^2}$, just as in a global inertial frame in flat spacetime, so different proper accelerations means differently sized spatial displacements and therefore different proper times). The reason why they experience different proper accelerations is that the bottom end of the building is being accelerated by an external force, while the internal forces are striving to keep a constant distance between infinitesimally nearby points in the co-moving local inertial frames.

This is exactly what causes the desynchronization of clocks attached to opposite ends of an accelerating rigid rod in flat spacetime. So in this particular case, I think it's definitely correct to say that gravitational time dilation depends on acceleration.

In what other situations is the term "gravitational time dilation" used? I don't know a definition that explains when to use it. If I had to invent a definition myself, I'd make sure that the term "time dilation" is only used when the relative ticking rates of two clocks is well-defined (and I suspect that it isn't in complicated GR scenarios because of difficulties associated with comparing clocks at different locations). Now that I think about it, I realize that I don't know any situation where the term "gravitational time dilation" is appropriate except for the one involving clocks at different constant "altitudes" in a Schwarzschild spacetime.

...the time dilation experienced by clocks experiencing Born rigid acceleration, but of course, the time dilation of clocks accelerating in SR is calculated solely based on their velocity as a function of time v(t), so it only "depends on acceleration" in the sense that the acceleration determines this velocity function.

Hm, let's see. It may be true, but it's not obviously true. The relative ticking rates of the two clocks should be defined as

$$\frac{d \tau_1}{dt}\bigg/\frac{d \tau_2}{dt}=\frac{\gamma_2}{\gamma_1}$$

OK, I agree.

 

[JesseM]

The fact that clocks on different floors in the same building on Earth tick at different rates is said to be due to gravitational time dilation. The reason why they tick at different rates is that they are accelerating by different amounts in their respective local inertial frames. (The proper time in a local inertial frame is $\sqrt{dt^2-dx^2-dy^2-dz^2}$, just as in a global inertial frame in flat spacetime, so different proper accelerations means differently sized spatial displacements and therefore different proper times). The reason why they experience different proper accelerations is that the bottom end of the building is being accelerated by an external force, while the internal forces are striving to keep a constant distance between infinitesimally nearby points in the co-moving local inertial frames.

This is exactly what causes the desynchronization of clocks attached to opposite ends of an accelerating rigid rod in flat spacetime. So in this particular case, I think it's definitely correct to say that gravitational time dilation depends on acceleration.

But again, if gravitational time dilation is understood to be equivalent to the differential time dilation of clocks undergoing Born rigid acceleration (and this equivalence should only work precisely in the case where you look at a small region of spacetime where curvature is negligible, or when you look at the 'uniform gravitational field' seen in the non-inertial Rindler coordinate system in flat spacetime), then since the differential time dilation of clocks undergoing Born rigid acceleration is understood in terms of their different velocities as seen in an inertial coordinate system, wouldn't we say the same about gravitational time dilation?

In what other situations is the term "gravitational time dilation" used? I don't know a definition that explains when to use it. If I had to invent a definition myself, I'd make sure that the term "time dilation" is only used when the relative ticking rates of two clocks is well-defined (and I suspect that it isn't in complicated GR scenarios because of difficulties associated with comparing clocks at different locations). Now that I think about it, I realize that I don't know any situation where the term "gravitational time dilation" is appropriate except for the one involving clocks at different constant "altitudes" in a Schwarzschild spacetime.

I was imagining that anytime you start two clocks at the same position, give them different paths through curved spacetime, then bring them together again, any calculation of how much time elapsed on each one would involve "gravitational time dilation". But that was just an assumption, I haven't seen a formal definition of the term, you could be right that it's only a well-defined notion in Schwarzschild spacetime. On the other hand, Chris Hillman's post #4 on this thread seems to be saying that anytime one observer is sending signals to another in curved spacetime, and the proper time interval between the signals being sent on the first worldline is different from the proper time interval between signals being received, then this would be said to involve "gravitational time dilation":

The so-called "gravitational time dilation" is a straightforward curvature effect. In any curved manifold, initially parallel geodesics will in general converge (positive curvature) or diverge (negative curvature) as you run along one of them. Near the exterior of the event horizon of a black hole (in the simplest case, this situation is modeled by the Schwarzschild vacuum solution of the Einstein field equation of gtr, or EFE for short), two radially outgoing null geodesics corresponding to signals sent from an infalling observer will diverge. That means that when the signals are received by our distant static observer, the time between the two, as measured by an ideal clock carried by this static observer, will be larger than the time between the emission of the two signals, as measured by an ideal clock carried the infalling observer.

These two "ideal clocks" are assumed to be absolutely identical and in particular, by definition they always "run at the same rate" under any circumstances (a real clock, even an atomic clock, will be affected by acceleration and so on); the "relativity" in gtr can be taken to refer to the fact that when we compare identical ideal clocks located at different "places", we must expect discrepancies, depending upon the details of the ambient gravitational field, the relative motion of the observers, and the method by which the comparison is made (typically, lightlike signals, but these can in general take more than one path and there are other complications we probably don't want to get into here).

 

[Fredrik]

But again, if gravitational time dilation is understood to be equivalent to the differential time dilation of clocks undergoing Born rigid acceleration (and this equivalence should only work precisely in the case where you look at a small region of spacetime where curvature is negligible, or when you look at the 'uniform gravitational field' seen in the non-inertial Rindler coordinate system in flat spacetime), then since the differential time dilation of clocks undergoing Born rigid acceleration is understood in terms of their different velocities as seen in an inertial coordinate system, wouldn't we say the same about gravitational time dilation?

If there's a natural coordinate system we can use (like Schwarzschild coordinates), the definition of relative ticking rates that I included near the end of my previous post should work in curved spacetime too, but the right-hand side would be more complicated. It wouldn't contain second-order derivatives (acceleration components) but it would depend on velocity components in a complicated way.

I was imagining that anytime you start two clocks at the same position, give them different paths through curved spacetime, then bring them together again, any calculation of how much time elapsed on each one would involve "gravitational time dilation".

Unfortunately this would mean that we would end up using the term "gravitational time dilation" in some situations where gravity doesn't really have anything to do with it, e.g. if you travel around the world on a train and your twin sits on a chair by the tracks waiting for you to get back.

On the other hand, Chris Hillman's post #4 on this thread seems to be saying that anytime one observer is sending signals to another in curved spacetime, and the proper time interval between the signals being sent on the first worldline is different from the proper time interval between signals being received, then this would be said to involve "gravitational time dilation":

DrGreg suggested that too in another thread. I thought it sounded reasonable then, and I still think it does.

 

[DrGreg]

DrGreg suggested that too in another thread. I thought it sounded reasonable then, and I still think it does.

I don't remember saying that exactly, but my memory sometimes fails...

...er...

...what was I saying?

I think this could be resolved by doppler shifts (Bondi's k-calculus technique): let A send a radio signal to B who reflects it back to A.

If A dectects no "two-way" doppler shift for the round trip signal, then A and B can be regarded as "stationary" relative to each other. But, in those circumstances, if B measures a different frequency than A, i.e. a "one-way" doppler shift, then there's a "gravitational" time dilation between them.

I think I'd call it "gravitational time dilation" even if A and B are undergoing Born-rigid acceleration in flat spacetime.

This characterisation by Doppler shift is something I've just thought of rather than read in a book, but I can't see anything wrong with it. This is a coordinate-free characterisation as it involves only the proper-time clocks of A and B to measure frequency.

Of course this works only if it is possible for light to travel A-B-A, but in those cases where it is not (e.g. crossing an event horizon) then the notion of time dilation may not make much sense anyway?

In general, when A and B are in relative motion, you could regard the doppler shift as having two components: a SR-like factor k_s due to relative motion and a GR factor k_g due to a difference in proper acceleration. For A-to-B the overall doppler factor would be k_sk_g and for B-to-A it would be k_s / k_g (giving a round-trip factor of k_s²). Again I stress this is something I've just thought up for myself, so there could be some logical flaw in my reasoning.

 

[Naty1]

Makep:
In your post, # 10, every sentence in the first paragraph is incorrect. I'm going to explain each one not to be mean but to advise. It's difficult to get direct feedback on how your ideas and language match usual scientific terminology...I know because I went through it...many do!

Movement of an object does not begin with an inherent velocity.

Sure it can...how do quantum fluctuations produce particle antiparticle pairs: they appear instantaneously moving apart. How is a photon "born": at the speed of light and remains at the speed until it is reabsorbed.

The final velocity of a moving body is the result of the average acceleration of that body.

No the final velocity of a moving body depends on it's initial velocity plus to tal acceleration. It depend on "average" acceleration when the acceleration is uniform.

Without an initial acceleration a body will not assume its final velocity.

This is hard to interpret...but consider my photon explanation above to refute it. The final velocity of a body or particle is generally duer to it's initial velocity, all accelerations together with the net effect of all forces acting on it.

(The above comments have nothing to do with relativity, just classical velocity and acceleration. Acceleration is the rate of change of velocity with time; velocity is the rate of change of distance(position) with time.)

In this respect, acceleration, not velocity, is the process by which time dialtion and space / length contraction happen

.

Not according to Einstein, as explained above by others.

The factor by which this happens is predicted, for a body with a constant velocity, by the Lorentz transforms, etc.

No, the velocity does not have to be "constant". The lorentz tranforms work at the instantaneous velocity at the moment of measurement. I think some also explained this above.

 

[Naty1]

With regards to identical,it would be correct only if their initial acceleration or the average acceleration that brought them to their current constant velocity are equal. Otherwise, they would not be identical.

No. It makes no difference what their past accelerations happened to be.
Furthermore, their current velocities are irrelevant...only the relative velocities matter...each sees the other identically...thats afundamental tenet of special relativity...there is no absolute velcocity against which we can measure other velocities...all are relative...

If one rocket ship is going .7C and another in the opposite direction at .9C each measures the other identically...for example, each measures the other's clock as slower than their own clock...and by an identical amount...And if the speeds of the respective rocketships were interchanged relative to some inertial frame, relative to each other they would see everything identically to the prior case...each would still measure the clock in the other ship ticking slower by the same amount as the first case.

 

[phyti]

squeeky forgot to ask if he only needs 39% of the fuel, since the distance is only 39% of that listed in his star chart.

 

[Makep]

Makep:
In your post, # 10, every sentence in the first paragraph is incorrect. I'm going to explain each one not to be mean but to advise. It's difficult to get direct feedback on how your ideas and language match usual scientific terminology...I know because I went through it...many do!




Sure it can...how do quantum fluctuations produce particle antiparticle pairs: they appear instantaneously moving apart. How is a photon "born": at the speed of light and remains at the speed until it is reabsorbed.



No the final velocity of a moving body depends on it's initial velocity plus to tal acceleration. It depend on "average" acceleration when the acceleration is uniform.



This is hard to interpret...but consider my photon explanation above to refute it. The final velocity of a body or particle is generally duer to it's initial velocity, all accelerations together with the net effect of all forces acting on it.

(The above comments have nothing to do with relativity, just classical velocity and acceleration. Acceleration is the rate of change of velocity with time; velocity is the rate of change of distance(position) with time.)

.

Not according to Einstein, as explained above by others.



No, the velocity does not have to be "constant". The lorentz tranforms work at the instantaneous velocity at the moment of measurement. I think some also explained this above.

Thank you for you correction on 'average' acceleration. Concerning the other poinst I find it hard to understand how the Law of Conservation of Energy and Mass, as predicted by SR and the Lorentz transforms would operate. Please explain.

 

[Makep]

No. It makes no difference what their past accelerations happened to be.
Furthermore, their current velocities are irrelevant...only the relative velocities matter...each sees the other identically...thats afundamental tenet of special relativity...there is no absolute velcocity against which we can measure other velocities...all are relative...

If one rocket ship is going .7C and another in the opposite direction at .9C each measures the other identically...for example, each measures the other's clock as slower than their own clock...and by an identical amount...And if the speeds of the respective rocketships were interchanged relative to some inertial frame, relative to each other they would see everything identically to the prior case...each would still measure the clock in the other ship ticking slower by the same amount as the first case.

The usage of the term 'identical' does not go well with me.Please explain what you mean by it.

 

[Naty1]

Fredrik,Jesse...Great discussion!

When I first reread my post # 7, and it was too late to edit, I realized as Fredrik implied I had not explained my thinking very clearly...but now I'm glad it WAS a bit muddled as you guys did a great job clarifying. Thanks...