Twin's Paradox: Q&A on Acceleration, Velocity & Time Dilation

[wildsman]

Before people start bashing me for not reading up on other threads, I did try and perhaps I am not physics-savvy enough to understand but I just found the answers to be contradictory and confusing.

So, here are two scenarios that confuse me:

Scenario 1: Two twins start off on earth. One experiences acceleration, leaves earth, orbits Earth at .8c, returns to find other twin much older.

Scenario 2: Two twins start off in different ships orbiting Earth at the same high constant velocity - one goes to Earth at time t, the other goes to Earth at time t+2 (light years). The second twin finds the first twin to be much older.

My questions:

1) What part does acceleration play in this beyond giving them separate frames of reference?
2) AFAIK it is the high relative velocity which causes time dilation not the acceleration. Given that either twin could see the other as traveling at the higher velocity, how does the acceleration required to shift frames of reference affect the problem?
3) Would each of them see the same thing looking at the other (once each of them reaches constant velocity - albeit vastly different velocities) given the symmetrical nature of relativity? How does this change when one of them goes to visit the other?

 

[Nugatory]

Before people start bashing me for not reading up on other threads, I did try and perhaps I am not physics-savvy enough to understand but I just found the answers to be contradictory and confusing.

So, here are two scenarios that confuse me:

...

The best answer I can give you:

Don't try working out the complicated and subtle rotating/orbiting/accelerating cases until you have the basic straight-line version of the twin paradox, the one described here, down cold. Start at the beginning, read through it until you find yourself stuck, then ask for help getting unstuck.

 

[ghwellsjr]

Before people start bashing me for not reading up on other threads, I did try and perhaps I am not physics-savvy enough to understand but I just found the answers to be contradictory and confusing.

I think maybe your point of confusion is that you are trying to think in terms of multiple frames of reference for the same scenario--one for each twin. Instead, just think in terms of a single frame of reference in which the twins can accelerate independently to any speed. Make sure this frame never accelerates so that it is an Inertial Reference Frame (IRF). It has its own Coordinate Time. Then the Time Dilation of each twin is easily calculated from their speed as referenced to the IRF using the gamma factor which at 0.8c is 1.667. So if a twin is traveling at 0.8c then his clock takes longer to tick out one second than the Coordinate Time by the factor of 1.667. But usually we like to think in terms of the tick rate of the twins' clocks so that if the Coordinate Time has progressed through, say, five years, then a twin's clock traveling at 0.8c will tick through three years and if a twin is stationary in the IRF, then his clock will tick through five years. With that in mind, your two scenarios should be very easy to analyze.

So, here are two scenarios that confuse me:

Scenario 1: Two twins start off on earth. One experiences acceleration, leaves earth, orbits Earth at .8c, returns to find other twin much older.

You can use any IRF you want but it makes sense to use the one in which the Earth is stationary. We pretend like the Earth is not rotating and not orbiting the sun in order to make the problem simpler. Now one twin is stationary in our chosen IRF so his clock ticks at the same rate as the Coordinate Time of the IRF. So now you just take the length of time that the traveling twin is orbiting at 0.8c and divide that by gamma, 1.667, and you have how much age he has accumulated during his trip. So if we use the numbers I gave earlier, we could say that while the stationary Earth twin aged by five years, the orbiting twin aged by three years. Is that still confusing?

Scenario 2: Two twins start off in different ships orbiting Earth at the same high constant velocity - one goes to Earth at time t, the other goes to Earth at time t+2 (light years). The second twin finds the first twin to be much older.

Whenever you specify a time, you have to be clear about which frame it is according to. I will assume that you mean the IRF of the earth. Then when you specify a time difference such as t+2, we assume that it's in the same IRF but note that light years are a measure of distance, not time, so I will assume that you mean two years.

You have both twins orbiting at 0.8c which means their clocks are both Time Dilated by the same factor and ticking slower than the Earth's IRF Coordinate Time. So at time t, the first twin becomes stationary and his clock starts ticking at the same rate as the Earth's Coordinate Time. Two years later the second twin becomes stationary but during that period of time, his clock has progress through 2/1.667 or 1.2 years so this twin finds the first twin to be 0.8 years older than himself.

My questions:

1) What part does acceleration play in this beyond giving them separate frames of reference?

Please don't think in terms of separate frames of reference. Use one IRF and describe how each twin accelerates. In other words, you want to specify the speed of each twin as a function of the Coordinate Time of the IRF. The acceleration is merely the process of a twin changing his speed. We usually assume that this acceleration is instantaneous because we want to keep the problem simple.

2) AFAIK it is the high relative velocity which causes time dilation not the acceleration.

That is correct as long as you specify the velocities as relative to a single IRF, not relative to another object or twin.

Given that either twin could see the other as traveling at the higher velocity, how does the acceleration required to shift frames of reference affect the problem?

This question is loaded with issues. We don't want to be concerned about what either twin sees of the other twin, only how they are actually traveling according to a specified IRF. Got it?

3) Would each of them see the same thing looking at the other (once each of them reaches constant velocity - albeit vastly different velocities) given the symmetrical nature of relativity? How does this change when one of them goes to visit the other?

These questions here are too general to give an overarching answer and as Nugatory said, they are much easier to deal with in an in-line scenario, rather than a circular one. In general, if the scenario is symmetrical, then they will see each other symmetrically but your two scenarios don't meet that requirement. If they did, they would have to end up at the same age after they were reunited.

 

[Meir Achuz]

Just one point about orbiting Earth. There is one altitude at which the gravitational red shift cancels the SR blue shift, and there is no differential aging.

 

[Nugatory]

Just one point about orbiting Earth. There is one altitude at which the gravitational red shift cancels the SR blue shift, and there is no differential aging.

This interaction between GR and SR is one of about 93 reasons why OP should start with the straight-line case - especially because there's a pretty decent FAQ ready and waiting.

 

[wildsman]

I think maybe your point of confusion is that you are trying to think in terms of multiple frames of reference for the same scenario--one for each twin. Instead, just think in terms of a single frame of reference in which the twins can accelerated independently to any speed.

I am deliberately trying to bring up the multiple frames of reference. I understand the concepts behind single frame of reference (independent of the two twins) but I want to know how it would affect the scenario if I treated the accelerating twin as the frame of reference (I realize that it is contradictory to say IRF and accelerating).

You can use any IRF you want but it makes sense to use the one in which the Earth is stationary. We pretend like the Earth is not rotating and not orbiting the sun in order to make the problem simpler. Now one twin is stationary in our chosen IRF so his clock ticks at the same rate as the Coordinate Time of the IRF. So now you just take the length of time that the traveling twin is orbiting at 0.8c and divide that by gamma, 1.667, and you have how much age he has accumulated during his trip. So if we use the numbers I gave earlier, we could say that while the stationary Earth twin aged by five years, the orbiting twin aged by three years. Is that still confusing?

Not at all. Like I said, I understand the Math with a stationary IRF - I just want to know how the Math changes if I pick the twins to be the IRFs.

Whenever you specify a time, you have to be clear about which frame it is according to. I will assume that you mean the IRF of the earth. Then when you specify a time difference such as t+2, we assume that it's in the same IRF but note that light years are a measure of distance, not time, so I will assume that you mean two years.

I actually wanted to see how the Math changes with different IRFs. And regarding the time, I meant years - not light years ('doh).

You have both twins orbiting at 0.8c which means their clocks are both Time Dilated by the same factor and ticking slower than the Earth's IRF Coordinate Time. So at time t, the first twin becomes stationary and his clock starts ticking at the same rate as the Earth's Coordinate Time. Two years later the second twin becomes stationary but during that period of time, his clock has progress through 2/1.667 or 1.2 years so this twin finds the first twin to be 0.8 years older than himself.

Please don't think in terms for separate frames of reference. Use one IRF and describe how each twin accelerates. In other words, you want to specify the speed of each twin as a function of the Coordinate Time of the IRF. The acceleration is merely the process of a twin changing his speed. We usually assume that this acceleration is instantaneous because we don't want to keep the problem simple.


That is correct as long as you specify the velocities as relative to a single IRF, not relative to another object or twin.


This question is loaded with issues. We don't want to be concerned about what either twin sees of the other twin, only how they are actually traveling according to a specified IRF. Got it?

These questions here are two general to give an overarching answer and as Nugatory said, they are much easier to deal with in an in-line scenario, rather than an circular one. In general, if the scenario is symmetrical, then they will see each other symmetrically but your two scenarios don't meet that requirement. If it did, they would have to end up at the same age after they were reunited.

Thanks! The thing is I understand it when it's a stationary IRF. The problem occurs when I try to shift them. You are advising me not to shift them and I completely understand that. I am not doing it to find the answer - I am trying to understand how it works if I shift the IRF.

Let me show you how I am looking at the other IRFs. Let us name the twin who lands first as Alan and the one who lands second (t+2) as Bob.

If I pick Alan as the IRF, both twins start off stationary (if they are at the same velocity). When Alan slows down to a stop, it would seem like Bob is moving away faster. Thus, because of time dilation, Bob's clock looks like it has slowed down. Then, finally after two years, when Bob lands he has already gained a head start (so to speak) of .8 years. So, Alan is .8 years older than Bob by the end.

Now, if I pick Bob as the IRF, both twins start off stationary. When Alan slows down, Bob should see Alan as accelerating in the negative direction. Thus, because of time dilation, Alan's clock should look slowed down. Then, finally after two years when Bob catches up to Alan, Alan should have gained a head start of .8 years. So, Alan is .8 years younger than Bob by the end.

And, here we go - obviously I made a mistake somewhere. I am just interested in learning where I went wrong as I may be missing a bigger picture here.

 

[Nugatory]

Let me show you how I am looking at the other IRFs. Let us name the twin who lands first as Alan and the one who lands second (t+2) as Bob.

If I pick Alan as the IRF...
Now, if I pick Bob as the IRF, ...

As you've described the setup, neither is ever using an inertial reference frame. You cannot choose either as an IRF.

 

[wildsman]

As you've described the setup, neither is ever using an inertial reference frame. You cannot choose either as an IRF.

Well, I put Alan and Bob in orbit, but what if you put them in a straight line at a constant velocity - it would work as an IRF and my setup would still work.

 

[wildsman]

This interaction between GR and SR is one of about 93 reasons why OP should start with the straight-line case - especially because there's a pretty decent FAQ ready and waiting.

You are right of course - I should have started with the straight line case. And, FYI, I did go through the FAQ you suggested. And, although it didn't directly address the issues I raised, it was informative.

 

[Nugatory]

Well, I put Alan and Bob in orbit, but what if you put them in a straight line at a constant velocity - it would work as an IRF and my setup would still work.

Have you worked through the link in post #2 of this thread? It most certainly will not work as you described in the straight-line case.

[edit: this post and post #9 crossed. We'll stop talking past one another in a moment]

 

[wildsman]

Have you worked through the link in post #2 of this thread? It most certainly will not work as you described in the straight-line case.

[edit: this post and post #9 crossed. We'll stop talking past one another in a moment]

Are you referring to the Equivalence Principle Analysis? I don't see how that entirely applies to scenario 2.

And, even with regards to the traditional straight line Twins Paradox, I didn't entirely understand why "Stella is far 'down' in the potential well; Terence higher up" in the example given for the EP Analysis.

 

[Dale]

Like I said, I understand the Math with a stationary IRF - I just want to know how the Math changes if I pick the twins to be the IRFs.

The math doesn't change in any inertial reference frame. However, you cannot pick the traveling twin or any other non-inertial observer to "be the IRF" since they are non-inertial. If you base a reference frame on a non-inertial object then that reference frame will be non-inertial.

If you use a non-inertial frame then the math does change. The math which works in any reference frame (inertial or non-inertial) and in any spacetime (flat or curved) is as follows. The time recorded on any clock is given by:
$$\int d\tau$$
where $d\tau^2=-ds^2/c^2$ and $ds^2$ is the metric (inertial or non-inertial, flat or curved) with a (-+++) signature.

 

[wildsman]

The math doesn't change in any inertial reference frame. However, you cannot pick the traveling twin or any other non-inertial observer to "be the IRF" since they are non-inertial. If you base a reference frame on a non-inertial object then that reference frame will be non-inertial.

If you use a non-inertial frame then the math does change. The math which works in any reference frame (inertial or non-inertial) and in any spacetime (flat or curved) is as follows. The time recorded on any clock is given by:
$$\int d\tau$$
where $d\tau^2=-ds^2/c^2$ and $ds^2$ is the metric (inertial or non-inertial, flat or curved) with a (-+++) signature.

So, if I understand you correctly, an Inertial Reference Frame means that the object remains in inertia throughout the problem? I think I misunderstood that part. I thought that it had start in inertia - but I honestly, now that I think about it I don't know why I thought that.

 

[ghwellsjr]

Just one point about orbiting Earth. There is one altitude at which the gravitational red shift cancels the SR blue shift, and there is no differential aging.

But there is no altitude at which a freely orbiting body will travel at 0.8c around the earth. This means that if we were considering reality (as opposed to all the pretending that I mentioned in post #3), a body would have to be firing an exceeding large rocket force to keep the body in "orbit", in which case the gravitational aging would be so insignificant as to be pointless to consider, don't you think?

 

[ghwellsjr]

So, if I understand you correctly, an Inertial Reference Frame means that the object remains in inertia throughout the problem? I think I misunderstood that part. I thought that it had start in inertia - but I honestly, now that I think about it I don't know why I thought that.

No, it doesn't mean that at all and I thought you understood this when you said in post #8:

I understand the concepts behind single frame of reference (independent of the two twins)

Please go back and read the beginning of my post #3. Can you see that it's the frame that's always inertial, not necessarily any objects that are described according to the frame? You can have all objects constantly accelerating or momentarily accelerating or never accelerating--whatever you want. But as I said later in the post, we usually want our objects to accelerate instantly, just to keep the calculations simple, which means they are always inertial, except for those instants when they are changing speed. But we don't have to keep it simple, an Inertial Reference Frame can handle any type of acceleration for all objects, just not for itself.

 

[Dale]

So, if I understand you correctly, an Inertial Reference Frame means that the object remains in inertia throughout the problem? I think I misunderstood that part. I thought that it had start in inertia - but I honestly, now that I think about it I don't know why I thought that.

An inertial reference frame is one in which inertial objects move in a straight line at a constant velocity. Inertial objects are objects with 0 proper acceleration, which can be expressed mathematically as having 0 covariant derivative or experimentally as an ideal accelerometer reading 0. So, experimentally, an inertial frame is one where accelerometers that read 0 move along straight lines at contstant velocity. (A line that starts out straight and later bends is not a straight line.)

Hope that helps.

 

[Nugatory]

Are you referring to the Equivalence Principle Analysis? I don't see how that entirely applies to scenario 2.

I was suggesting that you apply the Doppler shift analysis and spacetime diagram analysis to the straight-line case with both twins accelerating.

 

[ghwellsjr]

Well, I put Alan and Bob in orbit, but what if you put them in a straight line at a constant velocity - it would work as an IRF and my setup would still work.

Good idea. We'll do that with your scenario:

Let me show you how I am looking at the other IRFs. Let us name the twin who lands first as Alan and the one who lands second (t+2) as Bob.

With your indulgence, I'm going to modify your scenario slightly. Instead of Bob landing after 2 years (or 24 months), I'm going to have him land one month later at 25 months. This makes the calculations come out a lot cleaner.

Here is a spacetime diagram depicting your scenario. Alan's progress is shown with the thick blue line and Bob's progress is shown with the thick red line. The dots show one-month increments of time for each twin. I start in an IRF with both twins traveling at 0.8c. At the Coordinate Time of 0, Alan comes to a stop while Bob continues on until the Coordinate Time of 25 months when he stops. Note that at the very beginning of the scenario in the lower left corner, both Alan (blue) and Bob (red) are colocated so I show a blue line with red dots:

If I pick Alan as the IRF, both twins start off stationary (if they are at the same velocity). When Alan slows down to a stop, it would seem like Bob is moving away faster. Thus, because of time dilation, Bob's clock looks like it has slowed down. Then, finally after two years, when Bob lands he has already gained a head start (so to speak) of .8 years. So, Alan is .8 years older than Bob by the end.

You should say, "If I pick Alan as the non-IRF (or non-inertial reference frame), then both twins start off stationary..." That part is easy, the rest of the scenario has to be measured and calculated by Alan, he just can't know magically what is happening to Bob in order to construct his non-IRF and there is no standard way to make a non-IRF like there is in SR to make an IRF. My favorite way is to use the radar method with Doppler observations which actually works identically to the standard way in SR for IRF's as you'll soon see.

In the radar method, Alan sends out radar signals that travel at the speed of light and reflect off Bob. At the moment of reflection, the image of Bob's clock is transmitted back to Alan along with the radar reflection. Alan notes the time according to his clock when he sent the radar signal and when he received it. Then he applies Einstein's second postulate that the signal took the same time to propagate to Bob as it did to reflect back to him. This allows him to assign the moment of the reflection as the average of the sent and received times. He can also calculate the distance to Bob at that moment as one-half the difference in the times (we are using units where c=1). Then Alan builds up a table showing his measurements and calculated results. Here is a diagram in the original IRF showing his sent radar signals as thin blue lines and the reflected and Doppler signals as thin red lines:

And here is the table showing his measurements and calculations:

Radar   Radar   Calculated   Calculated    Red's
Sent    Rcvd      Time        Distance     Time
 -2      -2       -2             0          -2
 -1      -1       -1             0          -1
  0       0        0             0           0
  1       9        5             4           3
  2      18       10             8           6
  3      27       15            12           9
  4      36       20            16          12
  5      45       25            20          15
  6      46       26            20          16
  7      47       27            20          17

Now from the data in the table, Alan can construct a spacetime diagram showing his non-IRF including his radar signals, reflections and superimposed Doppler signals:

As I mentioned before, this IRF looks exactly like the IRF of the first diagram (with the exception of the period prior to his stopping). Your analysis of how Alan observes Bob is correct, except that I have modified the time difference to be 10 months instead of 0.8 years (which is 9.6 months).

Now, if I pick Bob as the IRF, both twins start off stationary. When Alan slows down, Bob should see Alan as accelerating in the negative direction. Thus, because of time dilation, Alan's clock should look slowed down. Then, finally after two years when Bob catches up to Alan, Alan should have gained a head start of .8 years. So, Alan is .8 years younger than Bob by the end.

And, here we go - obviously I made a mistake somewhere. I am just interested in learning where I went wrong as I may be missing a bigger picture here.

Yes, you did make a mistake. You assumed (without doing any measurements or calculations) that Bob would see Alan's clock ticking slower than his own for the entire two-years of Coordinate Time (actually 25 months in my modification). But let's see what happens if Bob does exactly what Alan did to make those measurements and calculations. First we look back at the original IRF with the radar and Doppler signals that Bob sends and observes. Now Bob is sending out the red signals and observing the blue signals:

Here is the table that Bob makes:

Radar     Radar   Calculated   Calculated    Blue's
Sent      Rcvd      Time        Distance     Time
 -2        -2       -2             0          -2
 -1        -1       -1             0          -1
  0         0        0             0           0
  0.333     3        1.667         1.333       1
  0.667     6        3.333         2.667       2
  1         9        5             4           3
  1.333    12        6.667         5.333       4
  1.667    15        8.333         6.667       5
  2        16        9             7           6
  3        19       11             8           9
  4        22       13             9          12
  5        25       15            10          15
  6        28       17            11          18
  7        31       19            12          21
  8        34       21            13          24
  9        37       23            14          27
 10        40       25            15          30
 11        43       27            16          33
 12        46       29            17          36
 13        49       31            18          39
 14        52       33            19          42
 15        55       35            20          45
 16        56       36            20          46
 17        57       37            20          47
 18        58       38            20          48

And here is the diagram that Bob constructs:

Now I think you can see that your assumption that the Time Dilation that Bob determines of Alan's clock does not apply for the entire 25-month interval but only for the first 5 months of Alan's time or just over 8 months of Coordinate Time and Bob's time. After that, he will determine in his non-IRF that Alan's clock is ticking more rapidly than his own or that of the Coordinate Time (which are the same), the net result being that at the end, Alan's clock is 10 months ahead of Bob's clock, just like in all the other reference frames that we have considered in this post.

Does this all make sense to you? Does it answer all your questions?

 

[ghwellsjr]

Thus, because of time dilation, Bob's clock looks like it has slowed down.
...
Thus, because of time dilation, Alan's clock should look slowed down.

As I pointed out in the previous post, Time Dilation is something that an observer can determine by sending radar signals, making measurements and observations, assuming the propagation time is the same for the outgoing radar signals as it is for their echoes, and then doing a bunch of calculations. Time Dilation is different in different reference frames but what each observer sees of the other ones clock slowing down does not depend on the different reference frames and that is what I want to show you now. This is the subject of relativistic Doppler in case you want to do some more reading about it.

I want to go back to the original reference frame that you've seen over and over again, except this time I want to show you what each twin actually sees. We'll start with what Alan sees of Bob's clock:

Note that Alan (in blue) sees Bob's clock running at 1/3 the rate of his own clock up until his clock reaches 45 months (and he sees Bob's clock at 15 months). After that, Alan sees Bob's clock running at the same rate as his own, just always at 30 months earlier than his own.

Now let's see how Bob views Alan's clock in the same reference frame:

Note that Bob (in red) sees Alan's clock also running at 1/3 the rate of his own clock even though it is Bob's clock that is Time Dilated in this reference frame. Unlike Alan, Bob only sees a difference in their two clocks up until his clock reaches 15 months (and he sees Alan's clock at 5 months). After that, Bob sees Alan's clock running at the same rate as his own, just always 10 months earlier than his own.

Even in Bob's non-IRF, we will note that both twins will see the same thing. Here's Bob's non-IRF showing what Alan sees of Bob's clock:

This shows that Alan again sees Bob's clock running at 1/3 the rate of his own for 45 months and then at the same rate.

Next we look at how Bob sees Alan's clock running in his own non-IRF:

This also shows the same results as Alan's IRF.

Now I want to use the Lorentz Transformation process on Alan's IRF to see what the scenario looks like in a frame where both twins are traveling at the same speed, 0.5c:

Again, even though both twins' clocks are Time Dilated to the same extent in this IRF, Alan still sees Bob's clock running at 1/3 of his own up to 45 months.

And the same IRF shows what Bob sees of Alan's clock:

And Bob still sees Alan's clock running at 1/3 of his own for the first 15 months and then the same thereafter.

 

[wildsman]

No, it doesn't mean that at all and I thought you understood this when you said in post #8:

Please go back and read the beginning of my post #3. Can you see that it's the frame that's always inertial, not necessarily any objects that are described according to the frame? You can have all objects constantly accelerating or momentarily accelerating or never accelerating--whatever you want.

I thought I understood it as well but it seems a little more confusing than I thought originally especially because of this response by DaleSpam:

An inertial reference frame is one in which inertial objects move in a straight line at a constant velocity.

But as I said later in the post, we usually want our objects to accelerate instantly, just to keep the calculations simple, which means they are always inertial, except for those instants when they are changing speed. But we don't have to keep it simple, an Inertial Reference Frame can handle any type of acceleration for all objects, just not for itself.

So, if I understand this correctly, it's the frame that isn't accelerating - any objects within it can accelerate at will. This may have been the source of my confusion all along. Thanks!

 

[Dale]

So, if I understand this correctly, it's the frame that isn't accelerating - any objects within it can accelerate at will.

That is correct. You can analyze accelerating objects in terms of an inertial reference frame. A non inertial object will undergo coordinate acceleration in an inertial reference frame, meaning that it will not travel in a straight line at a constant speed in those coordinates. But that doesn't prohibit the analysis.