# What I understand about Time Dilation

• I

## Main Question or Discussion Point

As I understand it, time dilation can occur under gravity or acceleration, which can be referred to as Gravity Time Dilation and can also appear to be happening when two things are in different frames of reference, which can be referred to as Velocity Time Dilation.

With Gravity Time Dilation, as I understand it there has been objective evidence that there is time dilation under gravity or acceleration.

With Velocity Time Dilation is the evidence here thought to be relative rather than objective?

Imagine for example two space ships were to pass each other. From Spaceship A's perspective, Spaceship B1 might be measured to have passed it at 0.2c in the +ve direction, and from Spaceship B1's perspective, Spaceship A1 might be measured to have passed it at 0.2c in the -ve direction. As I understand it, both would be stating that the clock of the other was going slower than its own clock. It would seem like a logical contradiction for that to objectively be the case, but it could be claimed that there is no objective truth on the matter, and that the truth about the situation is relative.

But supposing that unknown to the people on Spaceship A1 and Spaceship B1, Spaceship A1 was in the same frame of reference as Spaceship A2, and Spaceship B1 was in the same frame of reference as Spaceship B2, and that Spaceship A2 and Spaceship B2 were involved in a twin paradox experiment, where Spaceship B2 had accelerated away from Spaceship A2 and was going to go away and then back again.

As I understand it in the twin paradox experiment the clock on B2 would be thought to be objectively running slower than the clock on A2.

But if that was the case, then would the B1 not be running at the same rate as the B2 clock (whether it was in the same frame of reference as B2's outward journey unaccelerating velocity or B2's inward journey unaccelerating velocity) since they are in the same frame of reference, and likewise A1 be in the same frame of reference as A2. Such that while clock B1 could be thought to be running slower than the clock on A1 from A1's perspective, and the clock A1 could be thought to be running slower than the clock on B1 from B1's perspective, objectively the clock on B1 would be running slower than the clock on A1?

Is there any experiment not involving any Gravity Time Dilation which indicate that Velocity Time Dilation actually happens, rather than just appearing to?

• Parth

Related Special and General Relativity News on Phys.org
Spaceship A1 and Spaceship B1, Spaceship A1 was in the same frame of reference as Spaceship A2, and Spaceship B1 was in the same frame of reference as Spaceship B2, and that Spaceship A2 and Spaceship B2 were involved in a twin paradox experiment, where Spaceship B2 had accelerated away from Spaceship A2 and was going to go away and then back again.

As I understand it in the twin paradox experiment the clock on B2 would be thought to be objectively running slower than the clock on A2.
You may want to clarify what spaceship B2 actually does.
1) It can't be in one reference frame the whole way, because it must change speed at least once to get back to A2.
2) Its clock can't run at the same speed as B1's clock if it's running at different speed.
3) Are all 4 ships on the same line, or is B2 going in some random direction? I hope not because it would complicate things greatly.

In general, the twin paradox is resolved by understanding the discontinuity at the turning point.
Velocity time dilation is verified by muon's decay, which takes much longer for a moving muon compared to a standing or slow one.
There is a way to understand twin paradox through gravitational time dilation, but it's more of an expert exercise and little of an explanation.

PeroK
Homework Helper
Gold Member
Is there any experiment not involving any Gravity Time Dilation which indicate that Velocity Time Dilation actually happens, rather than just appearing to?
The experiments at CERN, for example, show that unstable particles live longer in the lab frame depending on their velocity. The faster they are moving the longer they live, in complete agreement with SR.

SR is 113 years old and very well tested and confirmed.

You may want to clarify what spaceship B2 actually does.
1) It can't be in one reference frame the whole way, because it must change speed at least once to get back to A2.
2) Its clock can't run at the same speed as B1's clock if it's running at different speed.
3) Are all 4 ships on the same line, or is B2 going in some random direction? I hope not because it would complicate things greatly.

In general, the twin paradox is resolved by understanding the discontinuity at the turning point.
Velocity time dilation is verified by muon's decay, which takes much longer for a moving muon compared to a standing or slow one.
There is a way to understand twin paradox through gravitational time dilation, but it's more of an expert exercise and little of an explanation.
1) B1 It won't be in the same frame of reference as B2 the whole way. But just pick either the frame of reference B2 is in as it leaves or the one it is in as it returns to A2. I don't think it makes a difference which one you pick.
2) It will be at the same velocity as B2. What velocity that would be would depend on whether you pick the point where B2 is leaving A2 or returning to A2.
3) The same line.

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The experiments at CERN, for example, show that unstable particles live longer in the lab frame depending on their velocity. The faster they are moving the longer they live, in complete agreement with SR.

SR is 113 years old and very well tested and confirmed.
In CERN is the particle not going around a circuit, and would that not involve some kind of acceleration (centripetal acceleration?)?

Also like with the A1, B1, A2, B2 example I gave, with A1 and B1 either one could appear to have a slower clock depending on which frame of reference it was measured from. Though given the A2, B2 situation, would the situation of which clock was going slower not have to objectively only be one way?

PeroK
Homework Helper
Gold Member
In CERN is the particle not going around a circuit, and would that not involve some kind of acceleration (centripetal acceleration?)?

Also like with the A1, B1, A2, B2 example I gave, with A1 and B1 either one could appear to have a slower clock depending on which frame of reference it was measured from. Though given the A2, B2 situation, would the situation of which clock was going slower not have to objectively only be one way?
The particles that crash into each other are accelerated but the resulting particles do not.

Also, note that, acceleration does not cause time dilation. Although there are those who would argue that point!

The basic paradox you refer to is explained by The fact that two clocks in relative motion can only pass once, unless one or both of them changes its inertial reference frame. It's changing your inertial reference frame, and with it your simultaneity convention, that is The key to these paradoxes, not acceleration.

PeroK
Homework Helper
Gold Member
@name123 there is a general issue that many students learn about time dilation and length contraction, but not the relativity of simultaneity. These students then present time dilation paradoxes that apparently cannot be explained. And, are often very reluctant to accept that SR includes the relativity of simultaneity, preferring to insist that the paradoxes debunk SR.

There's another thread entitled "resolving observations in two reference frames" that it may be worth looking at, where calculations involving time dilation were inadequate to solve the problem.

• Freixas
The particles that crash into each other are accelerated but the resulting particles do not.

Also, note that, acceleration does not cause time dilation. Although there are those who would argue that point!

The basic paradox you refer to is explained by The fact that two clocks in relative motion can only pass once, unless one or both of them changes its inertial reference frame. It's changing your inertial reference frame, and with it your simultaneity convention, that is The key to these paradoxes, not acceleration.
So the equivalence between gravity and acceleration doesn't include time dilation?

In the Twin Paradox, I thought that the idea was that when they meet one clock would be slower than the other (the one that accelerated away would be the one that went slower), and that it would be influenced by how long the departing twin travelled away for, and at what velocity. So I was assuming that for A2 and B2, B2's clock would be going slower than A2's both on the outward journey and the return journey. Am I mistaken?

PeroK
Homework Helper
Gold Member
So the equivalence between gravity and acceleration doesn't include time dilation?
This equivalence is not what you think. It's between an accelerating reference frame and a gravitational field.

If you have an accelerating rocket with clocks At the front and rear, then an external observer will measure the time dilation of each clock entirely based on its velocity. The magnitude of the acceleration has no effect, per se.

But, inside the rocket, an observer ( who is accelerating with the rocket) will measure the rear clock time dilated relative to the front clock. This time dilation is equivalent to that where the front clock is higher in a gravitational field.

The greater the acceleration, the greater the equivalent difference in gravitational potential across the rocket.

This is essentially the equivalence principle.

PeroK
Homework Helper
Gold Member
So the equivalence between gravity and acceleration doesn't include time dilation?

In the Twin Paradox, I thought that the idea was that when they meet one clock would be slower than the other (the one that accelerated away would be the one that went slower), and that it would be influenced by how long the departing twin travelled away for, and at what velocity. So I was assuming that for A2 and B2, B2's clock would be going slower than A2's both on the outward journey and the return journey. Am I mistaken?
The resolution to the twin paradox is not simple. There are perhaps hundred threads on here alone.

I prefer a version with no acceleration. Technically, the travelling twin is the one who changes inertial reference frame. That results in what is called differential ageing.

The time dilation itself is symmetric on both the outward and return journeys, but the change of reference frame causes a change in the simultaneity convention.

In any case, acceleration does not directky cause the time dilation to be asymmetric, it's fundamentally the change of reference frame that does that.

Mister T
Gold Member
Imagine for example two space ships were to pass each other. From Spaceship A's perspective, Spaceship B1 might be measured to have passed it at 0.2c in the +ve direction, and from Spaceship B1's perspective, Spaceship A1 might be measured to have passed it at 0.2c in the -ve direction. As I understand it, both would be stating that the clock of the other was going slower than its own clock.
This is not a complete way to state the situation. If each had only one clock they would not be able to reach that conclusion. A needs two clocks, separated along the line of relative motion, to be able to determine that B's clock is running slow. But B will not agree that A has synchronized his two clocks correctly, and can attribute A's conclusion to the way A synchronized his two clocks.

Likewise for B.

Until you explore the above and attempt to understand it you will not be able to sort out your misunderstanding. It's the relationship between proper time and coordinate time that first needs to be understand before the symmetry of time dilation can be understood.

It would seem like a logical contradiction for that to objectively be the case, but it could be claimed that there is no objective truth on the matter, and that the truth about the situation is relative.
There is an abundance of evidence indicating that it happens. It only seems like a logical contradiction if you don't understand it.

If it didn't happen, engineers who design particle accelerators would be creating entirely different designs. Likewise for the engineers responsible for maintaining the GPS.

Mister T
Gold Member
In the Twin Paradox, I thought that the idea was that when they meet one clock would be slower than the other
No. One clock will show that less proper time has elapsed on it than the other. That's a comparison of two proper times.

Time dilation (what you are calling velocity time dilation) is a comparison of a proper time to a coordinate time, not a comparison of two proper times.

Janus
Staff Emeritus
Gold Member
As I understand it, time dilation can occur under gravity or acceleration, which can be referred to as Gravity Time Dilation and can also appear to be happening when two things are in different frames of reference, which can be referred to as Velocity Time Dilation.

With Gravity Time Dilation, as I understand it there has been objective evidence that there is time dilation under gravity or acceleration.

With Velocity Time Dilation is the evidence here thought to be relative rather than objective?

Imagine for example two space ships were to pass each other. From Spaceship A's perspective, Spaceship B1 might be measured to have passed it at 0.2c in the +ve direction, and from Spaceship B1's perspective, Spaceship A1 might be measured to have passed it at 0.2c in the -ve direction. As I understand it, both would be stating that the clock of the other was going slower than its own clock. It would seem like a logical contradiction for that to objectively be the case, but it could be claimed that there is no objective truth on the matter, and that the truth about the situation is relative.

But supposing that unknown to the people on Spaceship A1 and Spaceship B1, Spaceship A1 was in the same frame of reference as Spaceship A2, and Spaceship B1 was in the same frame of reference as Spaceship B2, and that Spaceship A2 and Spaceship B2 were involved in a twin paradox experiment, where Spaceship B2 had accelerated away from Spaceship A2 and was going to go away and then back again.

As I understand it in the twin paradox experiment the clock on B2 would be thought to be objectively running slower than the clock on A2.

But if that was the case, then would the B1 not be running at the same rate as the B2 clock (whether it was in the same frame of reference as B2's outward journey unaccelerating velocity or B2's inward journey unaccelerating velocity) since they are in the same frame of reference, and likewise A1 be in the same frame of reference as A2. Such that while clock B1 could be thought to be running slower than the clock on A1 from A1's perspective, and the clock A1 could be thought to be running slower than the clock on B1 from B1's perspective, objectively the clock on B1 would be running slower than the clock on A1?

Is there any experiment not involving any Gravity Time Dilation which indicate that Velocity Time Dilation actually happens, rather than just appearing to?
What you've run into is a common stumbling block for those first learning Relativity. It is the distinction between "time dilation" and "total difference in accumulated time".
Let's change your set up a bit. Let's consider three spaceships A, B and C and the Earth. Spaceship A is traveling away from the Earth at a constant 0.6c(relative to the Earth) and Spaceship B is traveling towards the Earth in the opposite direction at 0.6c. Spaceship C starts at the Earth, travels away at 0.6c, then turns around and returns to the Earth at the same speed (ship C follows the basic twin paradox scenario).
During the outbound leg of ship C:
Earth will measure the clock on ship's A, B and C as ticking 0.8 as fast as its own.
Ship A will measure the clock on Ship C as ticking at the same rate as its own, the Earth clock ticking at a rate of 0.8 and the clock on ship B ticking at a rate of 0.47
Ship B will measure the Earth clock ticking at a rate 0.8 of its own, and the clocks on ships A and C ticking at a rate 0.47 of its own
Ship C will measure Ship A's clock as ticking at the same rate as its own, the Earth clock ticking at a rate of 0.8, and Ship B's clock ticking at a rate of 0.47

During the return leg of ship C:
Earth will still measure the clocks on all three clocks as ticking at a rate of 0.8
Ship A will measure the Ship Clock as ticking at a rate of 0.47, the Earth clock ticking at a rate of 0.8, and Ship B's clock ticking at the same rate as his own.
Ship B will measure the Earth clock as ticking at 0.8, the Ship C clock as ticking at the same rate as his own, and Ship A's clock ticking at 0.47
Ship C will measure ship A's clock as ticking at 0.47, The Earth clock at 0.8 and Ship B's clock ticking at the same rate as his own.

These are all time dilation measurements, or one to one comparisons between tick rates of given clocks at any particular moment.

However, to work out the total accumulated time difference between Earth and Ship C for Ship C entire trip, you have to consider more than just time dilation. There are other relativistic effects that come into play.
For instance, length contraction.

Let's assume that the distance that ship C travels from the Earth before turning around is 0.6 light years. Thus according to the Earth ship C spends 1 year going away and 1 year coming back. But, for ship's A, B and C, for which the Earth has a relative velocity of 0.6c, the same distance is only measured as being 0.48 ly.
Thus during the outbound leg, both Ship A and C will conclude that it took 0.48ly/0.6c = 0.8 years by their clocks to reach the point of turnaround.
and during the Inbound leg, Both Ships B and C will conclude that it took 0.48ly/0.6c = 0.8 years by their clocks to reach the point of turnaround.

Ergo, Ship C will conclude that a total of 1.6 years will have elapsed for it during the trip, which is the same conclusion that the Earth came to.

Its a bit more complicated to work out what ship A concludes for the whole trip.
First, we know that It measures 0.8 years as passing on its own clock and Ship C's clock during the Outbound leg
Second, the Earth clock is measured as ticking off 0.8*0.8 yr = 0.64 yr during this time.
The return leg is a bit more tricky. We have to use relativistic velocity addition to determine that Ship A measures Ship C's relative velocity to be ~0.882c (it was this relative velocity I used to get the 0.47 factor for time dilation I gave above.
This gives a difference of velocity between Earth and Ship C of ~0.28c according to Ship A. We also know that Ship A measures the distance between Earth and the turn around as being 0.48 ly . Thus by Ship A's clock it should take 0.48 ly/ 0.282 c = ~1.7 yrs for Ship to complete the return leg. During which time the Earth clock accumulates 1.7yrs*0.8 1.36 yrs, which when added to the 0.64 yrs it accumulated during the outbound leg of ship C equals a total of 2 yrs which agrees with what the Earth itself measured.
Ship C's clock will accumulate 1.7y*0.47 = 0.8 yrs during the return leg. Added to the 0.8y accumulated during the outbound leg, gives 1.6 yrs total for the trip, agreeing with both the Earth's and Ship C's measurement.

Note something here. While the Earth measured that it aged 1 year during both legs, this isn't what Ship A measured. Ship A measured that the Earth aged 0.64 years during the outbound leg, and 1.36 years during the return leg. Same total time for the two legs but different leg times.

we can work out the same thing for Ship B and it will also agree as to the total time accumulated by Earth and ship C.

All that;s left is to work out things from Ship C's perspective.

Outbound leg takes 0.8 yrs (due to length contraction) and inbound take the same, for a total trip time of 1.6 years
Earth ages 0.8y*0.8 = 0.64 yrs during outbound leg and the same during the inbound leg for a total of 1.28 yrs.
This however, does not jive with the two years the Earth and the other two ships got for an answer. So what went wrong?

The answer lies in the fact that Ship C had to change its velocity between outbound and return legs. and this is a whole different kettle of fish than when dealing with unchanging velocities. When something is moving at constant speed it is in an inertial frame, But while it is changing velocity, it is in a non-inertial frame. and the rules for dealing with measurements made from within non-inertial frames are different than those for dealing with inertial frames.

For instance, if you are under an acceleration, You will measure clocks in the direction that you are accelerating as running fast, even if they are not moving with respect to you. The further away a clock is, the faster it will run as measured by you. If the clock is moving relative to you, this add an additional factor you need to deal with.

So what this means for ship C is that, as it changes velocity from going away to returning to Earth, it is undergoing an acceleration towards the Earth, and since Earth is quite a distance away in this direction, ship C will measure* the clock on Earth as running fast, and will the Earth clock will accumulate 0.72 yrs during this period, which when added to the 1.28 yrs accumulated during the coasting legs of the trip brings the total accumulated time for the Earth up to the proper 2 years.

Again, while ship C also agrees with everyone else as to the total accumulated time for the Earth, it measures it as having gotten there by a different route.

* It is important here to note that throughout this post, when I use the word "measure", I do not mean "visually perceive". What you would visually see is complicated by things such as aberration, Doppler shift, light propagation delays, etc. (for example, the Earth would measure the Outbound leg of Ship C as taking 1 year, however, an Earth observer wouldn't visually see ship C reach its turnaround point until 1.6 yrs after it left. This is due to the extra time it takes light to travel from that point to the Earth. ) So by "measure", I mean what's left over after we account for these other effects.

• Bandersnatch
russ_watters
Mentor
With Gravity Time Dilation, as I understand it there has been objective evidence that there is time dilation under gravity or acceleration.

With Velocity Time Dilation is the evidence here thought to be relative rather than objective?
Both effects are observed simultaneously in GPS satellites.
Is there any experiment not involving any Gravity Time Dilation which indicate that Velocity Time Dilation actually happens, rather than just appearing to?
This is not a reasonable or necessary request. Scientific instruments are so sensitive that the gravity effects are noticeable if you place two clocks on different shelves, so it would be very difficult to do an experiment with no gravitational impact. But you can easily calculate the two effects, to account for them separately.

[experiments with high energy particles notwithstanding -- I prefer ones with real clocks]

Nugatory
Mentor
Is there any experiment not involving any Gravity Time Dilation which indicate that Velocity Time Dilation actually happens, rather than just appearing to?
The muon lifetime observations (take a look at the sticky thead at the top of this forum: https://www.physicsforums.com/threads/faq-experimental-basis-of-special-relativity.229034/) show time dilation in the absence of any significant gravitational effect.

However, it's not necessary to do this to confirm the effects of relative velocity.

Back up a moment to a much simpler theory, Newton's law of gravity. What was so special about it? It predicts that if you drop something, it will fall - but people have known that forever, probably even before they were people. Every monkey knows that, and watching my cat play at pushing objects off a shelf, I'm pretty sure that she knows it too. No, the big deal with Newton's laws is that they predict the exact speed and trajectory of the dropped object; we can compare these quantitative predictions with observations and conclude that we really have learned something important about the way the universe works.

Likewise, we can calculate the exact effect of gravitational time dilation and velocity-based time dilation in those situation where both are present. This is a very wide range of very different situations (the things referenced by the FAQ I linked above, and everything mentioned by other posters above) with very different contributions from gravity and velocity - but the results always match. So it is fair to say that both effects are well and thoroughly confirmed even without experiments that isolate each one.

This equivalence is not what you think. It's between an accelerating reference frame and a gravitational field.

If you have an accelerating rocket with clocks At the front and rear, then an external observer will measure the time dilation of each clock entirely based on its velocity. The magnitude of the acceleration has no effect, per se.

But, inside the rocket, an observer ( who is accelerating with the rocket) will measure the rear clock time dilated relative to the front clock. This time dilation is equivalent to that where the front clock is higher in a gravitational field.

The greater the acceleration, the greater the equivalent difference in gravitational potential across the rocket.

This is essentially the equivalence principle.
So from the outside, the time dilation is as you would expect given the velocities. But the time dilation with respect velocity is normally relative, but with the acceleration it is objective as can be observed from the inside of the rocket (if acceleration stopped and the clocks were bought together, one would have "ticked" less than the other). With CERN, where the particles undergo acceleration due to moving in a circle, can one tell if the time dilation relative or objective?

What you've run into is a common stumbling block for those first learning Relativity. It is the distinction between "time dilation" and "total difference in accumulated time".
Let's change your set up a bit. Let's consider three spaceships A, B and C and the Earth. Spaceship A is traveling away from the Earth at a constant 0.6c(relative to the Earth) and Spaceship B is traveling towards the Earth in the opposite direction at 0.6c. Spaceship C starts at the Earth, travels away at 0.6c, then turns around and returns to the Earth at the same speed (ship C follows the basic twin paradox scenario).
During the outbound leg of ship C:
Earth will measure the clock on ship's A, B and C as ticking 0.8 as fast as its own.
Ship A will measure the clock on Ship C as ticking at the same rate as its own, the Earth clock ticking at a rate of 0.8 and the clock on ship B ticking at a rate of 0.47
Ship B will measure the Earth clock ticking at a rate 0.8 of its own, and the clocks on ships A and C ticking at a rate 0.47 of its own
Ship C will measure Ship A's clock as ticking at the same rate as its own, the Earth clock ticking at a rate of 0.8, and Ship B's clock ticking at a rate of 0.47

During the return leg of ship C:
Earth will still measure the clocks on all three clocks as ticking at a rate of 0.8
Ship A will measure the Ship Clock as ticking at a rate of 0.47, the Earth clock ticking at a rate of 0.8, and Ship B's clock ticking at the same rate as his own.
Ship B will measure the Earth clock as ticking at 0.8, the Ship C clock as ticking at the same rate as his own, and Ship A's clock ticking at 0.47
Ship C will measure ship A's clock as ticking at 0.47, The Earth clock at 0.8 and Ship B's clock ticking at the same rate as his own.

These are all time dilation measurements, or one to one comparisons between tick rates of given clocks at any particular moment.

However, to work out the total accumulated time difference between Earth and Ship C for Ship C entire trip, you have to consider more than just time dilation. There are other relativistic effects that come into play.
For instance, length contraction.

Let's assume that the distance that ship C travels from the Earth before turning around is 0.6 light years. Thus according to the Earth ship C spends 1 year going away and 1 year coming back. But, for ship's A, B and C, for which the Earth has a relative velocity of 0.6c, the same distance is only measured as being 0.48 ly.
Thus during the outbound leg, both Ship A and C will conclude that it took 0.48ly/0.6c = 0.8 years by their clocks to reach the point of turnaround.
and during the Inbound leg, Both Ships B and C will conclude that it took 0.48ly/0.6c = 0.8 years by their clocks to reach the point of turnaround.

Ergo, Ship C will conclude that a total of 1.6 years will have elapsed for it during the trip, which is the same conclusion that the Earth came to.

Its a bit more complicated to work out what ship A concludes for the whole trip.
First, we know that It measures 0.8 years as passing on its own clock and Ship C's clock during the Outbound leg
Second, the Earth clock is measured as ticking off 0.8*0.8 yr = 0.64 yr during this time.
The return leg is a bit more tricky. We have to use relativistic velocity addition to determine that Ship A measures Ship C's relative velocity to be ~0.882c (it was this relative velocity I used to get the 0.47 factor for time dilation I gave above.
This gives a difference of velocity between Earth and Ship C of ~0.28c according to Ship A. We also know that Ship A measures the distance between Earth and the turn around as being 0.48 ly . Thus by Ship A's clock it should take 0.48 ly/ 0.282 c = ~1.7 yrs for Ship to complete the return leg. During which time the Earth clock accumulates 1.7yrs*0.8 1.36 yrs, which when added to the 0.64 yrs it accumulated during the outbound leg of ship C equals a total of 2 yrs which agrees with what the Earth itself measured.
Ship C's clock will accumulate 1.7y*0.47 = 0.8 yrs during the return leg. Added to the 0.8y accumulated during the outbound leg, gives 1.6 yrs total for the trip, agreeing with both the Earth's and Ship C's measurement.

Note something here. While the Earth measured that it aged 1 year during both legs, this isn't what Ship A measured. Ship A measured that the Earth aged 0.64 years during the outbound leg, and 1.36 years during the return leg. Same total time for the two legs but different leg times.

we can work out the same thing for Ship B and it will also agree as to the total time accumulated by Earth and ship C.

All that;s left is to work out things from Ship C's perspective.

Outbound leg takes 0.8 yrs (due to length contraction) and inbound take the same, for a total trip time of 1.6 years
Earth ages 0.8y*0.8 = 0.64 yrs during outbound leg and the same during the inbound leg for a total of 1.28 yrs.
This however, does not jive with the two years the Earth and the other two ships got for an answer. So what went wrong?

The answer lies in the fact that Ship C had to change its velocity between outbound and return legs. and this is a whole different kettle of fish than when dealing with unchanging velocities. When something is moving at constant speed it is in an inertial frame, But while it is changing velocity, it is in a non-inertial frame. and the rules for dealing with measurements made from within non-inertial frames are different than those for dealing with inertial frames.

For instance, if you are under an acceleration, You will measure clocks in the direction that you are accelerating as running fast, even if they are not moving with respect to you. The further away a clock is, the faster it will run as measured by you. If the clock is moving relative to you, this add an additional factor you need to deal with.

So what this means for ship C is that, as it changes velocity from going away to returning to Earth, it is undergoing an acceleration towards the Earth, and since Earth is quite a distance away in this direction, ship C will measure* the clock on Earth as running fast, and will the Earth clock will accumulate 0.72 yrs during this period, which when added to the 1.28 yrs accumulated during the coasting legs of the trip brings the total accumulated time for the Earth up to the proper 2 years.

Again, while ship C also agrees with everyone else as to the total accumulated time for the Earth, it measures it as having gotten there by a different route.

* It is important here to note that throughout this post, when I use the word "measure", I do not mean "visually perceive". What you would visually see is complicated by things such as aberration, Doppler shift, light propagation delays, etc. (for example, the Earth would measure the Outbound leg of Ship C as taking 1 year, however, an Earth observer wouldn't visually see ship C reach its turnaround point until 1.6 yrs after it left. This is due to the extra time it takes light to travel from that point to the Earth. ) So by "measure", I mean what's left over after we account for these other effects.
So once returned the proper time that has passed for ship C is less than the proper time that has passed for Earth. Its clocks have objectively "ticked" less than those on Earth, regardless of how it appeared to ship C. Not quite clear on how from C's perspective the Earth's clocks suddenly seem to have been ticking faster after all, and not have done 0.8 the amount of "ticks" that had happened on C. If you can explain that, that would be great, if not, it doesn't matter.

But if for ship C its clocks have objectively ticked less than those on Earth's. Then its clock was ticking slower as it had appeared to the observer on Earth.

Given that C's journey consisted of an outward leg, during which it's clocks were in synch with those of A and an inward leg during which it's clocks were in synch with those of B, then can one not conclude that like C the clocks on A and B were objectively ticking slower than those on Earth?

Supposing A goes out along side C, and passes B at the point C turns around, and as B passes A it sets its clock to the time on A's and comes back along side C. If one were to assume almost instantaneous acceleration and deceleration for C, then presumably its clock would be roughly in synch with A's on the outward journey and in synch with B's on the inward journey, and the time on B's, like C's would be less than the clock it passes on Earth as C returns. Unlike C, A and B would be in inertial frames.

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Both effects are observed simultaneously in GPS satellites.
With satellites are they not undergoing acceleration due to a circular orbit around the Earth? If so are you suggesting that if one were to return to Earth that even taking the acceleration and gravity effects into account, there will be another objective effect? If so then how are such effects relative?

russ_watters
Mentor
With satellites are they not undergoing acceleration due to a circular orbit around the Earth? If so are you suggesting that if one were to return to Earth that even taking the acceleration and gravity effects into account, there will be another objective effect? If so then how are such effects relative?
You can literally calculate the effects separately. They have different equations. I'm not sure what you mean by "how are such effects relative" though.

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

[note; acceleration is rolled-up with gravity]

The muon lifetime observations (take a look at the sticky thead at the top of this forum: https://www.physicsforums.com/threads/faq-experimental-basis-of-special-relativity.229034/) show time dilation in the absence of any significant gravitational effect.

However, it's not necessary to do this to confirm the effects of relative velocity.

Back up a moment to a much simpler theory, Newton's law of gravity. What was so special about it? It predicts that if you drop something, it will fall - but people have known that forever, probably even before they were people. Every monkey knows that, and watching my cat play at pushing objects off a shelf, I'm pretty sure that she knows it too. No, the big deal with Newton's laws is that they predict the exact speed and trajectory of the dropped object; we can compare these quantitative predictions with observations and conclude that we really have learned something important about the way the universe works.

Likewise, we can calculate the exact effect of gravitational time dilation and velocity-based time dilation in those situation where both are present. This is a very wide range of very different situations (the things referenced by the FAQ I linked above, and everything mentioned by other posters above) with very different contributions from gravity and velocity - but the results always match. So it is fair to say that both effects are well and thoroughly confirmed even without experiments that isolate each one.
But gravity or acceleration time dilation is objective regarding the slowing of the proper time. But velocity time dilation is supposed to be relative is it not? I have presumed that it is and that there are no objective results regarding the slowing of proper time for velocity time dilation. With the muon experiments are the muons not in a circular orbit, and thus undergoing acceleration?

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You can literally calculate the effects separately. They have different equations. I'm not sure what you mean by "how are such effects relative" though.

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

[note; acceleration is rolled-up with gravity]
In the article it states:
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Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion .

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.
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It seems as though the velocity was used in the Special Relativity calculation, and the Earth mass in the General Relativity calculation. But the velocity in the case of satellites are not in a straight line. They are in circular orbit. Thus I thought acceleration. And it seems from what the article states, that this was ignored and that the time dilation was calculated as though they were travelling in a straight line, though it seems you disagree. The article doesn't seem to specifically mention that it has been taken into account that the satellites are in circular orbit.

As I understand it with Special Relativity, from the satellites' perspective clocks on Earth should be ticking more slowly (ignoring gravity time dilation effect). I also thought that with Special Relativity the truth of which are ticking more slowly due to velocity time dilation was supposed to be relative. That there was no objective truth about the matter. But from what you seem to be saying is that while it may seem relative, the clocks can be brought together and it can be objectively observed which of the clocks had actually been ticking slower due to the velocity time dilation effect found in Special Relativity. Such that the truth of the matter might have seemed relative, but actually wasn't.

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russ_watters
Mentor
Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.
---

It seems as though the velocity was used in the Special Relativity calculation, and the Earth mass in the General Relativity calculation. But the velocity in the case of satellites are not in a straight line. They are in circular orbit. Thus I thought acceleration. And it seems from what the article states, that this was ignored and that the time dilation was calculated as though they were travelling in a straight line, though it seems you disagree. The article doesn't seem to specifically mention that it has been taken into account that the satellites are in circular orbit.
It isn't ignored, it's in there. Gravity and acceleration are related, so gravitational time dilation while in orbit has it's own equation:

https://en.m.wikipedia.org/wiki/Gravitational_time_dilation

As I understand it with Special Relativity, from the satellites' perspective clocks on Earth should be ticking more slowly (ignoring gravity time dilation effect). I also thought that with Special Relativity the truth of which are ticking more slowly due to velocity time dilation was supposed to be relative. That there was no objective truth about the matter. But from what you seem to be saying is that while it may seem relative, the clocks can be brought together and it can be objectively observed which of the clocks had actually been ticking slower due to the velocity time dilation effect found in Special Relativity. Such that the truth of the matter might have seemed relative, but actually wasn't.
Right. If the activities of the observers are not identical/symmetrical, then there is an objective difference that can be used to tell which clock is faster/slower.

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Nugatory
Mentor
But gravity or acceleration time dilation is objective regarding the slowing of the proper time. But velocity time dilation is supposed to be relative is it not? I have presumed that it is and that there are no objective results regarding the slowing of proper time for velocity time dilation.
It is relative ("frame-dependent" would be a much better way of saying it), but that does not preclude objective experimental results. Different observers moving at different speeds relative to one another will measure different lifetimes for the muon (or whatever other phenomenon we're considering), but the lifetime that any given observer will measure is an objective quantity that we can calculate from what we know of that observer's speed relative to the muon. We can, if necessary, describe the lifetime without any reference to the rate of time - we can note where the hands of a clock are pointing when the muon is created and where they are pointing when the muon decays.
With the muon experiments are the muon's not in a circular orbit, and thus undergoing acceleration?
Not. They are shooting pretty much straight down from the sky above us. They're in free fall so not accelerating, and gravitational effects are well and thoroughly negligible (affects their speed by maybe one part in one billion - but don't take my word for it, calculate for yourself!). We see similar situations when particles are coasting after being created in a collision.

It isn't ignored, it's in there. Gravity and acceleration are related, so gravitational time dilation while in orbit has it's own equation:

https://en.m.wikipedia.org/wiki/Gravitational_time_dilation
I understand that, what I meant was that I had presumed they had not taken it into account in their calculations. That they had used the special relativity equations for the velocity time dilation, and had used the general relativity equations with a 0 velocity just to calculate the amount of time dilation that would have been expected given the gravity. One reason (apart from the way it was written) is that I would have thought if you used the General Relativity equations, and had included the velocity of the circular orbit, then you would have come to the correct answer. There would be no need to then calculate the special case equation and modify the answer from the General Relativity equation.

Right. If the activities of the observers are not identical/symmetrical, then there is an objective difference that can be used to tell which is faster/slower.
So there would be experiments to show that despite velocity time dilation appearing relative, actually it isn't. One would be shown to be correct and the other wrong.

[note; acceleration is rolled-up with gravity]
Acceleration of a clock has no effect on the ticking rate of said clock. That is called the "clock hypothesis".