I What I understand about Time Dilation

[name123]

Because they can only check their clock rates unambiguously as they pass. At any other time they have to use some kind of complicated simultaneity criterion appropriate for their paths in a curved spacetime - and naive intuition from SR most definitely does not apply to this. The directly measured (with Doppler) rate will vary smoothly through the orbit, and the appropriate Doppler correction will also vary. The result will be a varying clock rate which will sum to the same elapsed time.

So you are suggesting that satellite A will have some local inertial frame from which it would observe satellite B's clock to be "ticking" faster than its own?

 

[name123]

Slower relative to their own local inertial frame. The qualifier is crucial.

Am I correct in thinking that a local intertial frame being considered to be a small part of the orbit, and that the orbit is being approximated somehow by a series of small straight lines?

No, it's just a reflection of the fact that each satellite has a different local inertial frame.

But whatever the local inertial frame, does the satellite not observe the other satellite's clock to be going slower?

 

[Nugatory]

So I think he did mean to state that the other satellites clock is observed to be running slower.

This is where we have to be vary careful with words like "see" and "appear".
Suppose I'm watching a strobe light that is designed to flash once per second when it's just sitting on the floor at my feet, not moving relative to me. Now we start the thing moving...

If it is moving towards me, I will see the flash rate to be greater than one flash per second; that is, flashes will reach my eyes less than one second apart. If it is moving away from me, I will see the flash rate to be less than one flash per second; there will be more than one second between successive flashes reaching my eyes.

But when I allow for light travel time, I will calculate that flashes are leaving the strobe more than one second apart no matter whether it is moving towards me or away from me. This is time dilation.

 

[PeterDonis]

Am I correct in thinking that a local intertial frame being considered to be a small part of the orbit, and that the orbit is being approximated somehow by a series of small straight lines?

Yes, but those straight lines don't all point in the same direction. That's why you can't "assemble" all of the local inertial frames into a single global inertial frame.

whatever the local inertial frame, does the satellite not observe the other satellite's clock to be going slower?

No. As @russ_watters has already pointed out, each satellite actually sees (as in, directly sees with eyes or instruments) a varying Doppler effect from light signals from the other satellite--sometimes it sees the other's clock running faster, sometimes slower. And each satellite can average over the varying Doppler effect it sees from those light signals to predict that, over one complete orbit, the other satellite's clock will have the same elapsed time as its own (because the "faster" and "slower" effects average out).

But each satellite can also, as has been discussed, "correct" the Doppler shifted light signals it sees for light travel time. But in doing that, it has to adopt a particular simultaneity convention. When we say that each satellite "sees the other satellite's clock running slow", what we really mean is that, if the satellite corrects the light signals it sees for light travel time according to the simultaneity convention of its own local inertial frame, it will come up with a clock rate for the other satellite that is slower than its own. But that correction will change as the satellite goes around its orbit; there is no way to "add up" all those local corrections to get a global answer, because there is no way to "add up" all the local inertial frames to build a single global frame.

 

[name123]

But when I allow for light travel time, I will calculate that flashes are leaving the strobe more than one second apart no matter whether it is moving towards me or away from me. This is time dilation.

So you are stating that the time dilation will be such that if both satellites had strobe clock's it would be calculated that the strobe clock on the other satellite is flashing at a slower rate?

Does that mean that if the orbit was big enough, or the strobes went off at smaller time intervals or some combination of the two, that there could be a discrepancy between the observed strobe flashes, and the count showing on the satellites clock when it passed?

 

[Janus]

But as I also wrote:
So if A's clock ticks are roughly in synch with C's on the outward journey (and then continue at the same rate) and B's clock ticks are roughly in synch with C's on the inward journey (and then continue at the same rate) then the amount of "ticks" on C's clock would be roughly equal to the amount A's clock ticked on the outward journey and the amount B's ticked on the inward journey. And since this is less than the clock ticked on Earth, why would it be wrong to conclude that like C the clocks on A and B were objectively ticking slower than those on Earth?

Because whether or not it is The Earth clock or the Ship clock that is ticking slower is frame dependent and no frame has preference over any other. Thus in the frame which ship A is at rest with respect to, The Earth clock always runs slower than the ship clock. In this frame it takes 2.5 yrs for the Earth to age 1 year The only reason C's clock accumulates even less time is that for part of the trip, the clock on C runs much slower than that. And this view is no less objectively true than the Earth frame's view that the Clock on ship C ran slow for the whole trip. You still end up with ship C aging less than the Earth, and even though for a time ship A and C age at the same rate, The Earth has always aged less than ship A You are trying to take the Earth frame view as being the "true objective reality" and force everything else to fit and this is not how things work.

In another conversation another scenario was considered, one in which there are 2 satellites in free fall orbit at the same velocity and altitude but in opposite directions around a sufficiently large glass sphere. They would observe each other's clocks to be going slower than their own (due to velocity time dilation) but when pass each other again they would observe that actually the clock measurements were objectively the same. So while there relative observations were different (when they both observed the other's clock to be going slower), they are able to objectively tell that those observations were did not imply that the other's clock was actually going slower. So while I can understand that A would think the clock on Earth was running slow, given that is was in synch with C's and C's was found to be have objectively ran slower than that on Earth, then I am not sure why you think it would be wrong to conclude that A's was running slower also.

The actual rate at which each clock would measure the other as running would depend on their relative positions in their orbits. While they would measure each other's clocks as running slow as they pass each other, they would measure each other as running fast when they are at some different points of their orbits. It is the combined effect of the other clock running slow and then fast that adds up to it accumulating the same amount of time as your satellite clock, which ran at a steady rate the whole time. You are conflating the end result of the journey with what happens during the journey.

For an analogy, assume we have two drivers taking a number of laps around a race track. One of them drives at a steady pace, but the other sometimes drives slower and sometimes drives faster.
Now it works out that they each take the same time to go from starting line to starting. Now while you can say that their average speed per lap was the same, you cannot say that at an given moment between the start and finish of a lap that they had completed the same portion of a lap or were driving at the same speed.

This is similar as what happens with accumulated time and clock tick rates in the satellite scenario. The difference is that with the satellites, each Satellite will say that its clock ticked at a steady rate while the other clock varied in tick rate over the course of its orbit. And again, this view is not any less "real" or objective than the view from the Earth frame which says the clocks on both satellites ticked at the same rate the whole time.

 

[PeterDonis]

Does that mean that if the orbit was big enough, or the strobes went off at smaller time intervals or some combination of the two, that there could be a discrepancy between the observed strobe flashes, and the count showing on the satellites clock when it passed?

No. See my post #65.

 

[Ibix]

So you are suggesting that satellite A will have some local inertial frame from which it would observe satellite B's clock to be "ticking" faster than its own?

There is no inertial frame that covers both satellites at once, except as they pass each other. That's why naive SR assumptions don't work in this case

 

[Nugatory]

Am I correct in thinking that a local intertial frame being considered to be a small part of the orbit, and that the orbit is being approximated somehow by a series of small straight lines?

No. That's not what a momentarily comoving inertial frame means; we would use the same concept for something accelerating in a straight line.

At any single moment, both satellites are at a single point in space, and they have some velocity vector. A moment later those velocities will be different, but that's beside the point - we're just talking about this one moment when each satellite is moving in a specific direction with a specific speed. Now we have the simple situation of special relativity - two objects in motion relative to each other - and we can calculate the time dilation between them at that moment. It's completely irrelevant that a moment later their relative speeds will be different; we're talking about right now, not a moment later.

 

[name123]

But each satellite can also, as has been discussed, "correct" the Doppler shifted light signals it sees for light travel time. But in doing that, it has to adopt a particular simultaneity convention. When we say that each satellite "sees the other satellite's clock running slow", what we really mean is that, if the satellite corrects the light signals it sees for light travel time according to the simultaneity convention of its own local inertial frame, it will come up with a clock rate for the other satellite that is slower than its own. But that correction will change as the satellite goes around its orbit; there is no way to "add up" all those local corrections to get a global answer, because there is no way to "add up" all the local inertial frames to build a single global frame.

If a satellite had two clocks on board, then could the simultaneity convention not be that their ticks are simultaneous? And presumably that will work all the way around the orbit. And if it corrects the light signals for light time travel according to that convention, it will always conclude that the clock rate for the other satellite is slower than its own. If so then it seems to me to a contradiction to claim that all the conclusions that the other satellites clock rate was slower than it's own clocks rate, even when the same amount of ticks had happened for each.

 

[Janus]

So you are stating that the time dilation will be such that if both satellites had strobe clock's it would be calculated that the strobe clock on the other satellite is flashing at a slower rate?

Does that mean that if the orbit was big enough, or the strobes went off at smaller time intervals or some combination of the two, that there could be a discrepancy between the observed strobe flashes, and the count showing on the satellites clock when it passed?

No, because the rate of strobe flashes that you visually perceive depends on the both the relative speed and direction of that speed relative to you, if you are receding from each other you see a slower strobe rate, if you are closing in on each other you see a faster strobe rate. Again, over the course of one orbit this will add up so that the number of flashes you saw matches the other satellites clock reading. (if you were visually watching the other clock during the orbit, you would also see its tick rate slow down and speed up to match the strobe rate you were seeing.)

 

[name123]

There is no inertial frame that covers both satellites at once, except as they pass each other. That's why naive SR assumptions don't work in this case

I am not sure what you mean. I was thinking that they would be in different local inertial frames at all times.

 

[Nugatory]

So you are stating that the time dilation will be such that if both satellites had strobe clock's it would be calculated that the strobe clock on the other satellite is flashing at a slower rate?

Yes. You've worked hard to make the situation symmetrical, so any other answer (implying an asymmetry) would be surprising.

Does that mean that if the orbit was big enough, or the strobes went off at smaller time intervals or some combination of the two, that there could be a discrepancy between the observed strobe flashes, and the count showing on the satellites clock when it passed?

No, and until this is completely unsurprising to you, you might want to work with the less tricky traditional form of the twin paradox. This situation is covered in the "Doppler explanation" section of the FAQ that I linked to earlier.

 

[name123]

No, because the rate of strobe flashes that you visually perceive depends on the both the relative speed and direction of that speed relative to you, if you are receding from each other you see a slower strobe rate, if you are closing in on each other you see a faster strobe rate. Again, over the course of one orbit this will add up so that the number of flashes you saw matches the other satellites clock reading. (if you were visually watching the other clock during the orbit, you would also see its tick rate slow down and speed up to match the strobe rate you were seeing.)

Do you agree that the time dilation will be such that if both satellites had strobe clock's it would be calculated that the strobe clock on the other satellite is flashing at a slower rate?

 

[russ_watters]

Do you agree that the time dilation will be such that if both satellites had strobe clock's it would be calculated that the strobe clock on the other satellite is flashing at a slower rate?

No. I don't mean to sound rude, but this is starting to get repetitive. It feels like you have a slight misconception that you are trying very hard not to let go of. You heard that 'the other guy's clock' is always slower than yours, but that just isn't the case.

 

[PeterDonis]

If a satellite had two clocks on board, then could the simultaneity convention not be that their ticks are simultaneous?

A simultaneity convention has to cover events on both satellites if you're going to use it to calculate one satellite's clock rate in the other's frame. And since the satellites are in relative motion, there is no way to construct a simultaneity convention that has all ticks of clocks on both satellites simultaneous.

 

[name123]

No, and until this is completely unsurprising to you, you might want to work with the less tricky traditional form of the twin paradox. This situation is covered in the "Doppler explanation" section of the FAQ that I linked to earlier.

I read the http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_doppler.html part of a link you supplied to me before. Is that what you meant?

I had a slight problem with it. Terrence is supposed to calculate that Stella's clock is running slow, and putting out a flash once every 7 seconds.
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Terence computes that Stella's clock is really running slow by a factor of about 7 the whole time
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And that she is traveling for 14 years (according to Terrence).
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Here's the itinerary according to Terence:

Start Event
Stella flashes past. Clocks are synchronized to 0.
Outbound Leg
Stella coasts along at (say) nearly 99% light speed. At 99% the time dilation factor is a bit over 7, so let's say the speed is just a shade under 99% and the time dilation factor is 7. Let's say this part of the trip takes 7 years (according to Terence, of course).
Turnaround
Stella fires her thrusters for, say, 1 day, until she is coasting back towards Earth at nearly 99% light speed. (Stella is the hardy sort.) Some variations on the paradox call for an instantaneous turnaround; we'll call that the Turnaround Event.
Inbound Leg
Stella coasts back for 7 years at 99% light speed.
Return Event
Stella flashes past Terence in the other direction, and they compare clocks, or grey hairs, or any other sign of elapsed time.
According to Terence, 14 years and a day have elapsed between the Start and Return Events; Stella's clock however reads just a shade over 2 years.
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So I would expect the amount of flashes Terrence sees to be equal to the number of seconds in 2 years.

But it then states that because of the doppler effect he sees a flash every 14 seconds on the way out, and 14 flashes per second on the way back.
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On the Outbound Leg, Terence sees a flash rate of approximately one flash per 14 seconds; on the Inbound Leg, he sees her clock going at about 14 flashes per second.
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Which seems to me to be equal to the number of seconds in 1/2 year on the way out and the number of seconds in 98 years on the way back. But the clock never flashed that many times. So I guess I have misunderstood.

 

[name123]

No. I don't mean to sound rude, but this is starting to get repetitive. It feels like you have a slight misconception that you are trying very hard not to let go of. You heard that 'the other guy's clock' is always slower than yours, but that just isn't the case.

Do you disagree with the first answer Nugatory gave in post #73? Or do you think I have misunderstood it?

 

[PeterDonis]

That's not what a momentarily comoving inertial frame means

I was thinking that they would be in different local inertial frames at all times.

I think there are some items that need clarification.

A local inertial frame is a set of inertial coordinates (i.e., coordinates that work just like standard inertial coordinates in special relativity) that covers a small patch of spacetime--a small region in space over a small interval of time.

A "momentarily comoving inertial frame" means we pick a particular event on the worldline of some object, like a satellite, and construct a local inertial frame in which the event we picked is the origin and the object we picked is at rest at that event. This amounts to approximating a small segment of that object's worldline as a straight line, if the object is moving in free fall, as the satellites are. But @Nugatory correctly points out that you can also construct a local inertial frame around an event on the worldline of an object that is not in free fall. If you do that, the worldline of the object will not be a straight line, even in the local inertial frame.

If we pick different events on the worldline of the same object, such as different events along the orbit of a satellite, the local inertial frames at these different events each cover only a small region of space and a small interval of time around those different chosen events. And if either the object is accelerating (i.e., nonzero proper acceleration, acceleration that is felt) or spacetime is curved, then these local inertial frames will not "line up", in the sense that the straight lines in one of them will be "at an angle" to the straight lines in the other. But this is a separate matter from the question of what the straight lines in each local inertial frame represent--in particular, whether the worldline of our chosen object is a straight line in any of these local inertial frames (see above).

If we have two objects that are passing each other, then we can construct two local inertial frames centered on the event where they pass. In each of these frames, one of the objects will be at rest (at least momentarily), and the other will be moving; but both objects will be "in" both frames (their worldlines will appear in both frames). And if both objects are in free fall (as the satellites are), then both of their worldlines will be straight lines in either local inertial frame; they just won't be straight lines in the same direction. So each object will only be at rest in one of the two inertial frames; but they will still appear in the other, just not at rest.

 

[name123]

A simultaneity convention has to cover events on both satellites if you're going to use it to calculate one satellite's clock rate in the other's frame. And since the satellites are in relative motion, there is no way to construct a simultaneity convention that has all ticks of clocks on both satellites simultaneous.

Wouldn't the time dilation be calculated by observing the flashes from the other satellite and then taking account of the doppler effect, and then comparing that rate to the on board clocks

 

[PeterDonis]

Which seems to me to be equal to the number of seconds in 1/2 year on the way out and the number of seconds in 98 years on the way back.

No. Terence does not see Stella's turnaround at 7 years elapsed on his clock; he sees it at close to 14 years elapsed on his clock. The article goes into all that.

So Terence sees 1 year's worth of flashes coming in from Stella's outbound leg during close to 14 years of his time--or one flash every 14 seconds. He then sees 1 year's worth of flashes coming in from Stella's inbound leg during 1/14th of a year, or 14 flashes per second.

 

[russ_watters]

Do you disagree with the first answer Nugatory gave in post #73? Or do you think I have misunderstood it?

I believe that part of his response was limited to when the satellites are moving away from each other. Otherwise, over the entire orbit, the number of flashes sent and received would not match.

Everyone else is telling you this too...

 

[name123]

No. Terence does not see Stella's turnaround at 7 years elapsed on his clock; he sees it at close to 14 years elapsed on his clock. The article goes into all that.

So Terence sees 1 year's worth of flashes coming in from Stella's outbound leg during close to 14 years of his time--or one flash every 14 seconds. He then sees 1 year's worth of flashes coming in from Stella's inbound leg during 1/14th of a year, or 14 flashes per second.

Doh! Thanks :)

 

[PeterDonis]

Wouldn't the time dilation be calculated by observing the flashes from the other satellite and then taking account of the doppler effect, and then comparing that rate to the on board clocks

"Taking account of the doppler effect" requires adopting a simultaneity convention. If the satellite adopts the simultaneity convention of its own local inertial frame, then it gets the result that the other satellite's clock is running slow. But, as I've already explained, there is no way to "add up" these calculations from all the different local inertial frames around the satellite's orbit to conclude that the other satellite's clock will have less elapsed time around one complete orbit. That's because each of those local inertial frames has a different simultaneity convention, so you can't combine their calculations.

I have already explained this at least once. Other people have also explained points to you multiple times. Yet you keep asking the same questions. At this point no new questions are being asked and all of the questions asked have already been answered. So I am closing this thread. All of the answers you seek are here. You just need to take the time to carefully think through what has been said here.