Yet another Twin Paradox thread

[sylas]

Then what of the equivalence between gravity and acceleration? I thought since the gravitational effect caused spacetime dilation then acceleration would also cause spacetime dialation. An object at rest in a gravitational field will have its clock slow down.

A good example of this is a spaceship under a continuous constant acceleration (measuring acceleration at a given point on board gives a fixed acceleration) and which remains a fixed length (as measured by the people on board).

Using special relativity, you can calculate some consequences that may seem surprising at first glance.

Clocks at the front and the back of the spaceship are running at different rates. The clock at the front runs faster than the one at the rear. Also, the acceleration experienced at the front and back is different. There is a greater acceleration experienced at the rear than at the front. Any given point on the ship will have a fixed acceleration; but different points on board will show different fixed values, if the length remains constant for those on board.

[in edit, I rephrased the above to be clearer, I hope. Acceleration experienced on board varies with how far "forward" the measurement is taken, but does not vary with when the measurement is taken.]

You can calculate this using GR, in a suitably defined non-inertial frame. It is equivalent to the case of clocks at the top and the bottom of a tower in a gravitational field.

You can calculate this using SR. In this case, you should pick an inertial frame (not the spaceship itself), and calculate the world lines for the front and back, and calculate the proper time in the standard way... using the velocities, as I explained previously. Its not acceleration that counts for calculating the time dilations, but the velocity.

You will have to excuse me, because I consider myself a professional laymen, since I have read about 60 books on the subject but haven't really gotten my hands dirty with the mathmatics, even so I find it hard to see how I could have got this confused.

No problem, and welcome to the club. In my experience, we ALL get confused on this as we learn about it, even as as we read books.

Books alone can help; and a good teacher can help better as you learn what the books are describing. I'm not claiming to be a good teacher myself; but I do think people here will be able to help you sort this out better. And certainly for me, when I was learning this, just the books was not enough; I benefited greatly from the help of talking with people about what I was reading. Still goes on for me. I'm fine with SR but have a long way to go on GR.

But, say you had a light clock set up around a planet. The path of the light would curve and then the clock would read a slower time. Then say you set up a light clock in an accelerating ship, the light in the clock would curve giving an increasingly longer measurment of time. In SR the light clock only has straight lines, I would think adding the curvature would cause the clock to run increasingly slower due to the curvature itself.

A clock set up "around" a planet? You mean on a satellite? Which orbit? Or clocks placed on the surface around a planet?

Clock would read a slower time? Slower in comparison to what? Satellites will run either faster or slower than a clock on the surface, depending on the height of the orbit. A low orbit clock runs slower. A high orbit clock runs faster. The surface clock is not in an orbit, of course.

You can't say a clock runs "slower" unless you indicate what it is is compared with. In some cases (like clocks in orbit or on the surface) you can get an unambiguous answer as to which is running faster and which is running slower. In other cases (like clocks moving past one another in unaccelerated motions in flat space) you get different answers from the perspective of each clock.

I think we are diverging from the original topic, however. The main point for the original topic is that the time dilations follow from velocities, not from accelerations.

Cheers -- sylas

 

[Al68]

Now here is my problem. A person on Earth at some early time in this 4 year (earth time) time span and after 1 additional year for the light to reach this person, he sees the two ships a light year away making ship to ship contact. And in order to match reality, he will see something much much stranger. For about the next 3 years, he will see that both ships have not moved one single inch, they are frozen in time litereally waiting for the 3 additional year passage of time to pass on Earth and resolve the paradox. After the 3 years of looking, suddently the observer on Earth will once again see the ships traveling on their merry way, one of which will reach the Earth in about a half year time and of course when this happens, the person will have aged 4 years.

This would be very strange, but it's not what the person on Earth observes. He observes the first ship move away from Earth for two years, the ships pass, then the second ship move toward Earth for 2 years.

Since according to SR there is complete symmetry during the trip out, and again during the trip back, this means that at the instant of signal transfer between the two ships, something profound must happen such as a jump into the future.

The "magical" thing that happened is simply that the ship is moving toward, instead of away, from earth. This is true whether there are two ships, or just one that turns around. Because "earth time" is deduced by subtracting light transit time from observations of Earth time.

When a single ship turns around, Earth's clock "jumps ahead" for the exact same reason it would by changing perspective from one ship to another: The ship is moving toward Earth instead of away from it. Just consider a single signal sent from Earth and received by both ships as they pass (or a single ship at turnaround). Adjusting observed "earth time" for the signal delay yields a different "corrected" Earth time depending on which direction the ship is moving. And it's this adjusted "earth time" that jumps ahead, not actual observations of Earth time.

 

[ghwellsjr]

A couple weeks ago, here in the USA, we went off Daylight Saving Time. I stayed up until 2AM just to see the clock on my computer jump back an hour. It really did. I lost that extra hour's sleep though. In a few more months, I can stay up til 2AM again and see my clock jump ahead one hour but I don't think I will because I'll lose a lot of sleep.

Please read post #2 on this thread if you want to understand the Twin Paradox without all this nonsense about time jumping around. It's real simple.

 

[michelcolman]

Then what of the equivalence between gravity and acceleration? I thought since the gravitational effect caused spacetime dilation then acceleration would also cause spacetime dialation. An object at rest in a gravitational field will have its clock slow down.

Accelleration and gravity have similar, but not quite equal effects.

The rate at which a clock ticks, depends on its potential in the gravitational (or "accelerational"(?)) field.

For gravity, the amount of potential at increasing distances is limited because gravity quickly decreases at large distances from the object. Potential energy is lowest at the center of the object and increases asymptotically to some limit value at larger distances up to infinity. It does not keep increasing without limit. That's why there's an (almost) fixed difference between a clock close to a massive body, and the clocks "far away". If Earth was the only source of gravity, you would hardly observe any difference between a clock one light year away, and one twice as far, since both points have almost exactly the same gravitational potential in Earth's field which is minuscule at those distances. Potential is force times distance, and the force of gravity at those distances is negligible.

For accelleration, the situation is different since, if you consider yourself to be stationary, the entire universe is accellerating. The "force" (accelleration) does not drop off at a distance, but remains constant. This means that the potential energy in this field will keep increasing indefinitely without limit. That means that clocks ahead of you (in your direction of accelleration) are ticking very, very quickly, especially those that are really far away. There's no limit to the potential energy in this "field", so there's no limit to how fast clocks can go. Just walking back and forth will "move" the clocks in very distant galaxies back and forth by many years, in your reference frame.

Note that, in the second case, you will not actually "see" these clocks ticking abnormally fast. It's just that, as you accellerate towards them, distances contract and the speed of light relative to the surroundings appears to change (since it remains constant for you). This means that you will interpret the images you see as being from longer ago, so that the local time right now must be later. But it's not just an artifact of calculations, since the time difference will indeed be confirmed on arrival.

By the way, you can choose whether to use General Relativity (and an accelleration-"field") or Special Relativity (just considering the change in speed, length contraction, etc.) and both will yield the same result. In fact, GR was pretty much derived from SR in this way, by looking at what happens to distant clocks when accellerating, thanks to length contraction and time dilation caused by the different speed before and after the accelleration.

 

[sylas]

For accelleration, the situation is different since, if you consider yourself to be stationary, the entire universe is accellerating. The "force" (accelleration) does not drop off at a distance, but remains constant.

We are drifting from topic here, but it is interesting so I'll comment further with two points.

First... you don't so much see the rest of the universe "accelerating" as see it "falling" in the pseudo-gravitational field you experience by virtue of your own acceleration.

Second... surprisingly, the force does drop off with distance, in one quite objective sense.

The issue of "distance" is subtle, and I frankly don't have a good handle on how to resolve it. Suffice to say that you need to find some way to define a co-ordinate system before you you can speak of what happens at a given distance.

But you can do this. Suppose you take an accelerometer and use it to measure the gravitational force experienced at your location by virtue of your acceleration. Suppose you also take two more accelerometers. Hold one at a fixed distance "up", and dangle the other at a fixed distance "below" your location. You will find that in fact, the one "below" you shows a stronger force (corresponding to a greater proper acceleration) and the one held "above" you shows a weaker force, corresponding to a smaller proper acceleration. It follows, I am pretty sure, that you will find that the (apparent) acceleration of objects which are "falling" from your perspective show them falling deeper into a stronger apparent gravitational field. There is also a horizon behind you beyond which you cannot see, as light from there can never reach you (as long as you continue with the same constant proper acceleration).

This has been covered in other threads, but I don't have a good reference quick to hand. If I find one, I'll add a link.

Cheers -- sylas

 

[michelcolman]

Second... surprisingly, the force does drop off with distance, in one quite objective sense.

That's interesting. But anyway, it does not drop off nearly as quickly as gravity does. The important thing is that the potential keeps increasing with distance without limit, while for a gravitational field, it approaches a limit rather quickly.

 

[sylas]

That's interesting. But anyway, it does not drop off nearly as quickly as gravity does. The important thing is that the potential keeps increasing with distance without limit, while for a gravitational field, it approaches a limit rather quickly.

In both cases, the force experienced increases without limit as you proceed slowly "downwards".. though the relation between acceleration and distance is not the same, I think.

On a planet, you will hit the surface, but other than that the gravitation forces are the same as for a black hole with the same mass; and the force required to hold in place diverges to infinite at the black hole horizon.

In a spaceship with constant acceleration, the force diverges to infinite at a distance c²/a from points on the ship where the acceleration is a, as measured in an inertial frame in which the ship instantaneously at rest.

Cheers -- sylas

 

[Dale]

That's interesting. But anyway, it does not drop off nearly as quickly as gravity does. The important thing is that the potential keeps increasing with distance without limit, while for a gravitational field, it approaches a limit rather quickly.

You are thinking specifically about a Schwarzschild spacetime. What you say is correct for the Schwarzschild spacetime, but does not apply for other spacetimes such as the FLRW spacetime.

 

[michelcolman]

In both cases, the force experienced increases without limit as you proceed slowly "downwards".. though the relation between acceleration and distance is not the same, I think.

What I meant, was a limit when going upwards.

For a gravitational field, you only need the amount of energy corresponding to escape velocity to get away from the object. So the time dilation between the surface and infinity is that corresponding to that energy. The gravitational potential at large distance simply does not change much anymore. A clock at a million light years away will not go perceptibly faster than one at 10 light years.

In an accellerational field (not sure if that's the right term), the potential keeps increasing with distance without limit. No matter how much energy you spend to get ahead of the accellerating rocket, if it's a finite amount of energy, the continuously accellerating rocket will catch up with you at some point. Since time dilation is determined by the difference in potential in the field, this means there's no limit to how fast clocks far ahead of you will be spinning.

I thought that was a pretty significant difference between gravitation and accelleration.

 

[phyti]

My favorite scenario is one that eliminates the acceleration problem thusly: A ship at .866c travels past the earth. At the moment of passage, the ship's stopwatch "high fives" a stopwatch on Earth and this action resets both stopwatches to zero. A year later by ship time the ship "high fives" a second ship also going at .866c but at a heading back to Earth. This contact causes the returning ship to set it's stopwatch to the timeout time on the first ship setting both stopwatches to the same time which would in fact be 1 year ship time.

The second ship arrives at the Earth one year after the ship to ship contact and a photographer takes a picture of the stopwatch on the ship and finds that it has timed out 2 years. The Earth stopwatch on the other hand has timed out exactly 4 years according to the accepted understanding of relativity.

Now according to wiki, if instead of my description one uses a reversal of a single ship the paradox is resolved because the ship is in a gravitational state upon reversal and this causes the Earth time to jump ahead about 3 years in the time it takes for the ship to turn around and get back to speed explaining the 4 year age of the Earth in a 2 year ship time. I can accept this.

But using my scenario where there is no acceleration, this argument cannot be used. Still, I will take a leap of faith here and figure that at the time of ship to ship clock reading, and the shift over to the reference frame of the second ship, the Earth instantaneously jumps into the future by about 3 years. (it's not 2 years like you might think because of the time dilation of Earth time as seen by the ship according to the simultaneity graph in wiki just before and just after the ship to ship contact) Is everyone in agreement that this is what would happen? I can accept this as well as there seems to be no alternative way out of this if I'm to hold true to relativity.

-Switching between 2 frames is physically possible, instantaneous reversal is not. The jump in time doesn't happen, it's the result of using the Einstein clock synch method. In both cases, the switch or reversal happens in zero time, therefore nothing can change, i.e., neither clocks nor distances have changed. The time jump is a mathematical manipulation via the clock synch method, i.e., the clock synchronization of the inbound frame is opposite that of the outbound frame. Setting either clock does not alter the Earth clock. If the acceleration was included in the reversal of one ship the 'jump' would be eliminated

Now here is my problem. A person on Earth at some early time in this 4 year (earth time) time span and after 1 additional year for the light to reach this person, he sees the two ships a light year away making ship to ship contact. And in order to match reality, he will see something much much stranger. For about the next 3 years, he will see that both ships have not moved one single inch, they are frozen in time litereally waiting for the 3 additional year passage of time to pass on Earth and resolve the paradox. After the 3 years of looking, suddently the observer on Earth will once again see the ships traveling on their merry way, one of which will reach the Earth in about a half year time and of course when this happens, the person will have aged 4 years.

-You didn't reason about this long enough. If the ships remain moving at constant speed, how can they stop? If you want a reciprocal pov for the 'time jump', it happened in zero time, therefore it will be zero time for everybody.

post #9
I'm going to stay logical here and assume that there is no "real" sudden jump in simultaneity and that a sudden fast forward in Earth time from the ship is not something the ship would see. But as far as I can see, there is only one thing happening here that can be taken as the gospel truth and that is that in my 2 ship scenario, the traveling stopwatch will show less time elapsing than the stopwatch on earth. In addition, what the ship sees is one of two things in order for logic to prevail. The first is that the ship sees the clock on Earth moving slowly, then at some point (the turnaround or meeting point) must see it speed up rapidly, then slow down again on the way home, or secondly it must see the clock on Earth always moving faster which defies SR. There can be no third alternative. This implies an asymmetry between the ship and the Earth either during the whole trip or only during the turnaround time. Since according to SR there is complete symmetry during the trip out, and again during the trip back, this means that at the instant of signal transfer between the two ships, something profound must happen such as a jump into the future. Now I for one cannot accept this as it defies all logic, so the only other answer is that there is something different about the ship and the Earth that makes it non-symmetrical during the entire trip. And if this is true then SR is not correct in it's prediction of symmetry during the time the ship travels out and again during the time the second ship travels home.

-The symmetry results from the clock synch method, and SR is symmetrical by design.
We know the Earth clock runs at a constant rate so it's the rate of receiving signals that changes with the ships change in speed, i.e., doppler effects.
Two points to add here.
In the dynamic universe, it's a fair assumption that all objects are in motion, thus all lose time (all objects have frequencies, molecular, atomic, etc), so your clock comparisons are relative ones.
The time lost by fast moving clocks is cumulative and not recoverable. There is no speed by which a clock gains time!
For single paths, the rates of clocks can be compared but this is insufficient to determine which one is the slowest (unless you can determine absolute speed in space, which is still debated). When one of two identical clocks returns to the other, i.e., a closed path, then any asymmetry is revealed because the number of signals is conserved. In the twin scenario, the twin who returns is younger. If the Earth twin left in a ship to catch up to the other, he would have experienced the same acceleration, but would be the younger. In general you don't need to exchange signals, just a reuniting of the clocks for comparison.

 

[ghwellsjr]

-Switching between 2 frames is physically possible,...

Frames are not physical, they're in our minds, they are coordinate systems to enable us to specify and analyze scenarios like the ones in Buckethead's original post. If you start out using one FoR and half way though the scenario you switch to another one, of course you can get time jumps or any other nonsense you want.

instantaneous reversal is not. The jump in time doesn't happen, it's the result of using the Einstein clock synch method. In both cases, the switch or reversal happens in zero time, therefore nothing can change, i.e., neither clocks nor distances have changed. The time jump is a mathematical manipulation via the clock synch method, i.e., the clock synchronization of the inbound frame is opposite that of the outbound frame. Setting either clock does not alter the Earth clock.

Agreed, jumps in time don't happen but it has nothing to do with Einstein clock synch method. You could call it a mathematical manipulation of switching between frames of reference, nothing more.

If the acceleration was included in the reversal of one ship the 'jump' would be eliminated

The 'jump' has nothing to do with any acceleration or lack thereof.

Buckethead was talking about two different scenarios: one from the wikipage that had only one ship that accelerated and reversed direction but not in zero time and the other one that he devised that included two ships, neither one of which ever accelerated.

-You didn't reason about this long enough. If the ships remain moving at constant speed, how can they stop? If you want a reciprocal pov for the 'time jump', it happened in zero time, therefore it will be zero time for everybody.

He wasn't talking about ships that actually stopped, he was talking about the observer on Earth seeing the ships stopped, frozen in time, or some such nonsense.

-The symmetry results from the clock synch method, and SR is symmetrical by design.

The symmetry has nothing to do with any clocks being synchronized. I don't know why you are bringing this up. Clock synchronization concerns multiple clocks at rest with one another but physically separated from one another. There are no two clocks in this scenario that are at rest with one another.

We know the Earth clock runs at a constant rate

Yes, the Earth clock runs at a constant rate but so do the two ship's clocks in Buckethead's scenario. Why do you single out the Earth clock as if it were different somehow?

so it's the rate of receiving signals that changes with the ships change in speed, i.e., doppler effects.

You're getting close, it has only to do with the doppler effects, but the Earth does not receive a change in the rate of those signals when the ships "change speed" but some time later. However the ships receive a change in the rate of the signals from Earth at the moment when they "change speed". It's this difference between the times at which the Earth versus the ships detect the change in speed that accounts for the difference in accumulated time between the Earth's and the ship's clocks.

Two points to add here.
In the dynamic universe, it's a fair assumption that all objects are in motion, thus all lose time (all objects have frequencies, molecular, atomic, etc), so your clock comparisons are relative ones.
The time lost by fast moving clocks is cumulative and not recoverable. There is no speed by which a clock gains time!
For single paths, the rates of clocks can be compared but this is insufficient to determine which one is the slowest (unless you can determine absolute speed in space, which is still debated). When one of two identical clocks returns to the other, i.e., a closed path, then any asymmetry is revealed because the number of signals is conserved. In the twin scenario, the twin who returns is younger. If the Earth twin left in a ship to catch up to the other, he would have experienced the same acceleration, but would be the younger. In general you don't need to exchange signals, just a reuniting of the clocks for comparison.

Don't know what this is all about--too complicated for my simple mind--and although you don't need to exchange signals, that's the whole point of the Twin Paradox, if they aren't observing each other's clocks as running slower than their own, then why should it be surprising when one of the clocks ends up with a lower time on it?

By the way, I want to emphasize that during the entire scenario that Buckethead described, the Earth observer see the ship's clocks as always running at half the speed of his own clock and the ships see Earth's clock as always running at half the speed of their own clocks. That's the whole point of the paradox, each sees the other's clocks as always running slower than their own and yet the ship's clock ends up with one half of the total time accumulated compared to Earth's clock.

It's easy to understand the resolution of the Twin Paradox from the frame of reference in which the Earth is stationary because the ship's clocks actually run at one half the speed, but the ships still see the Earth clock as running slow.

You can analyze this from any other frame of reference and the description that I gave in post #2 will apply unchanged because relativistic doppler and relative speed are invariant under any inertial frame of reference transformation, but you can't switch from one frame to another half way through the scenario. Whoever came up with that idea?

One other point that may be confusing is how the relativistic doppler relates to the time dilation factor. They are not the same thing. The time dilation factor cannot be directly observed, it can only be indirectly observed via the relativistic doppler which is the ratio of the distant observed clock speed to the local clock speed. This ratio is a combination of the normal kind of doppler caused by relative speed between the source and the receiver and the time dilation and length contraction. This results in a ratio which is lower than expected when traveling away from the source and higher than expected when traveling toward the source. If they are at rest with one another, the ratio is 1. The relativistic doppler formula relates the doppler ratio to the relative speed and the Lorentz factor relates the relative speed to the time dilation. So just measuring the doppler ratio is how an observer determines both the speed (delayed in time) of the source and the time dilation of the source and it's the time dilations that remain constant for both legs of the trip for both the Earth and the ships.

EDIT: I'm not sure I have the definition of the doppler ratio correct, in may be the reciprocal but it doesn't matter to the point I am making.

 

[John232]

Clock would read a slower time? Slower in comparison to what? Satellites will run either faster or slower than a clock on the surface, depending on the height of the orbit. A low orbit clock runs slower. A high orbit clock runs faster. The surface clock is not in an orbit, of course.

You can't say a clock runs "slower" unless you indicate what it is is compared with. In some cases (like clocks in orbit or on the surface) you can get an unambiguous answer as to which is running faster and which is running slower. In other cases (like clocks moving past one another in unaccelerated motions in flat space) you get different answers from the perspective of each clock.

I think we are diverging from the original topic, however. The main point for the original topic is that the time dilations follow from velocities, not from accelerations.

Cheers -- sylas

I think we got on this because I thought every twin paradox was solved with the right answer from determining what object did the acceleration.

Say, one clock was made out of satallites. One satallite would send a message to the other back and forth creating ticks to form a clock. The gravitational field of the Earth would slightly bend the transmission from each satallite makeing it take longer for the clock to tick than another satallite network that was in space not in orbit around a planet.

Then it would be the same as a satallite network that was traveling together accelerating to create the same curvature that it would have if the network was stationed around Earth.

These two clocks made of satallites should read the same amount of time since they would have expereinced the same amount of curvature in the paths of their transmissions to each other. Or, should they read differently because of the velocity required for one network to obtain the velocity to give the same amount of curvature that would mimic the network being around planet Earth?

 

[Al68]

If the acceleration was included in the reversal of one ship the 'jump' would be eliminated

No, it wouldn't. We'd have the same "jump", and for the exact same reason. The "jump" isn't an actual jump in observed (or actual) time, it's a jump in the "corrected" Earth time after correcting it for the signal delay in two different frames with two different velocities relative to earth.

 

[sylas]

I think we got on this because I thought every twin paradox was solved with the right answer from determining what object did the acceleration.

No... as I explained earlier back in msg #13. The solution is simply that one twin remained in one inertial frame and the other didn't.

The relevance of acceleration is only that it is a way to change velocity, and hence give an observer a different inertial frame. You can see that it is not the acceleration which is critical by the alternatives I gave in msg #13, where the same frame shift is achieved by a teleportation device, or by a twist in spacetime. You can also express the twin "paradox" using two coasting space ships that pass by one another, and synchronize as they pass. No acceleration is involved.

The two passing ships have different frames, and THAT is the key to the solution.

These two clocks made of satallites should read the same amount of time since they would have expereinced the same amount of curvature in the paths of their transmissions to each other. Or, should they read differently because of the velocity required for one network to obtain the velocity to give the same amount of curvature that would mimic the network being around planet Earth?

Your problem with this example is that satellites in different orbits can remain synchronized, but an accelerating twin cannot remain synchronized with an inertial twin. The satellites are a different matter entirely from the twin paradox, and solved using gravitational dilation AND velocity based dilation. The time dilations involved during the acceleration up into orbit are entirely and exclusively solved the same way... gravity, and velocity. Not acceleration.

Added in edit. Think of THIS example. Two twins are out in space. One remains at rest (inertial). The other accelerates around the inertial twin, in circular motions. Not an orbit, as the gravity involved is negligible. Since the twins can remain at a fixed separation distance, they can remain synchronized, and each one agrees that one moving in circles has slower clocks. The rate at which the clock slows depends ONLY on the velocity of the circular motion; and this can be achieved with any acceleration magnitude you like.

Cheers -- sylas

 

[ghwellsjr]

Think of THIS example. Two twins are out in space. One remains at rest (inertial). The other accelerates around the inertial twin, in circular motions. Not an orbit, as the gravity involved is negligible. Since the twins can remain at a fixed separation distance, they can remain synchronized, and each one agrees that one moving in circles has slower clocks.

Huh? Two clocks "remain synchronized" but one of them is slower? How does that work?

 

[sylas]

Huh? Two clocks "remain synchronized" but one of them is slower? How does that work?

Thanks for picking that up. Bad phrasing on my part, sorry!

What I mean is that the two clocks remain at a fixed distance from each other, so that observers at the clocks can agree without ambiguity as to which clock is running slower, and by how much.

This principle is used in the GPS system, for example, to run the clocks on the satellite slightly slower, so that they match clocks on Earth.

You can't do this with two inertial clocks with some velocity relative to each other. The distance is constantly changing. Each observer receives signals from a receding clock at a slower rate, and from an approaching clock at a faster rate; so you can't slow down or speed up either clock to get a match in the rate for each observer.

There's probably a different word to use for this; but what I mean is that by slowing down one of the clocks you can get a match in the ticking rate as observed by either observer.

Cheers -- sylas

 

[phyti]

refer to drawing:
For simplicity the times are referenced as A or B followed by the year.
Twin B leaves twin A moving at .8c, reverses direction at B12, and returns.
Fig. 1 shows A's view of B's trip. The axis of simultaneity (gray) for B is (A7.2, B12) outbound, and (B12, A32.8) inbound. The instantaneous jump from A7.2 to A32.8 is due to excluding any period of acceleration for B to transfer from the outbound to the inbound frame of reference. The ratio of B-time to A-time is 24/40 = .60.

Fig. 2 is B's view using the Einstein simultaneity convention. The discontinuous motion of A at A4 reflects the switching of frames without acceleration. The extreme distortion of times and locations, using this convention, is noted with A4 simultaneous with B-36, 36 years before they parted! The ratio of A-time to B-time is 4/6.7 = 36/60 = .60 for both path segments.

At this point the slow clock rate is reciprocal.

Fig. 3 is B's view using a horizontal axis of simultaneity, i.e. a translation of positions, and A moving at .8c. The initial conditions place A in the 'chosen' static frame and B moving therefore light speed relative to B is c-v and c+v (magenta). The ratio of A-time to B-time is 4/2.4 = 36/21.6 = 1.67, in agreement with the result for the closed path in fig. 1. The time dilation is now asymmetrical as calculated by both A and B. The extreme space and time shifts are also removed.

Fig. 4 shows a more realistic case with a short period of acceleration for B transitioning between frames. B would explain the curved portion of A's motion as resulting from an equivalent g-field during his acceleration. This also provides an asymmetrical view with 40 A events to 24 B events.

The axis of simultaneity for B determines where B locates the A events, and that axis is determined by the clock synchronization. The simultaneity definition is the source of the distorted coordinates, where unequal path lengths are defined as equal, for the purpose of preserving constant light speed.

Absolute vs. relative speeds.
The question: How much time is required for a car moving at 60 mph, to overtake a car moving at 50 mph with a 1 mile lead?
The answer: distance/(v1-v2) = 1/(60-50) = 1/10 = .1 hr = 6 min.
It's the relative or closing speed that determines the answer. Neither car would expect the other to approach at 60 mph. If the lead car used 60 for the chase car rate, the initial separation would have been 60*.1 hr = 6 miles, not 1 mile. The absolute car speed is relative to the ground. The relative car speed is relative to the other car. They are two different types of relations. If light replaces the chase car, its speed c is relative to space, defined as an invisible but fixed frame of reference, and its relative speed as c-v, with v the speed of the object being chased. The fact that relative light speed is different from c, doesn't contradict its absolute speed, no more than the 10 mph closing speed alters the 60 mph chase car speed.

This quote clarifies the nature of the clock synch method and its purpose.
(M is the midpoint of the distance AB)

Relativity - The Special and the General Theory, Albert Einstein 1961, page 27:
"That light requires the same time to traverse the path AM as for the path BM is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity."

https://www.physicsforums.com/attachments/30035

 

[phyti]

A good example of this is a spaceship under a continuous constant acceleration (measuring acceleration at a given point on board gives a fixed acceleration) and which remains a fixed length (as measured by the people on board).

Using special relativity, you can calculate some consequences that may seem surprising at first glance.

Clocks at the front and the back of the spaceship are running at different rates. The clock at the front runs faster than the one at the rear. Also, the acceleration experienced at the front and back is different. There is a greater acceleration experienced at the rear than at the front. Any given point on the ship will have a fixed acceleration; but different points on board will show different fixed values, if the length remains constant for those on board.

If the acceleration decreases from back to front, i.e., the direction of motion, then the ship contracts. If the direction is reversed using a propulsion unit on the front end, the ship will contract more. By reversing directions enough times, the ship could be reduced to any desired length, and effectively disappear to outside observers. This would be a great contribution to stealth technology.
What do you think?

 

[sylas]

If the acceleration decreases from back to front, i.e., the direction of motion, then the ship contracts. If the direction is reversed using a propulsion unit on the front end, the ship will contract more. By reversing directions enough times, the ship could be reduced to any desired length, and effectively disappear to outside observers. This would be a great contribution to stealth technology.
What do you think?

Length contraction, like time dilation, depends on velocity, not acceleration.

The differing proper acceleration of the front and back of a spaceship is a normal consequence of the fact that the length as measured on board remains constant. During the acceleration phase of the ship this may be an added wrinkle for describing the matter from the perspective of an inertial observer.

Any paradox (nearly always) arises by failing to consider the relativity of simultaneity. Length contraction goes hand in glove with differing perspectives of what is simultaneous for the front and rear.

If the ship stops accelerating at some point, then from the perspective of those on board, the front and the rear stop accelerating at the same time. For an external observer, the rear (which had a greater acceleration) will stop accelerating before the front, and all parts of the ship end up at the same velocity, and with the corresponding length contraction.

When it comes to applications of stealth technology, you can, of course, get length contraction by virtue of high velocity. The issue of length is confounded by that of energy; people will notice side effects from something passing through the air above at velocities approaching that of the speed of light. Also, if you play tricks with simultaneity to achieve a lasting length contraction somehow, you are actually squashing your stealth vehicle. You can achieve the same contractions more easily with a large pile driver; though there may be negative consequences for the pilot and instrumentation on board.

Cheers -- sylas

 

[Buckethead]

I'm slowly (painfully slowly) seeing the light here I think. BTW, thanks to everyone for taking the time to help me get through this. I'm a logic junkie and getting this right helps me to sleep at night.

One thing I'm glad to weed out is that acceleration has absolutely nothing to do with the twin paradox. I wish that I hadn't read so many other twin paradox threads in this forum that insisted that it did and that everything works out if you just remember one is accelerating and the other one is not. ACCELERATION HAS NOTHING TO DO WITH IT. Don't anyone try and change my mind on this or I will surely go out of my mind.

OK, now that I have that straight I would like to ask if the following conclusions are correct. These assume the scenario of 2 ships passing each other as in my opening post.

1. As ship 1 passes Earth on the way out, it sees the clock on Earth moving slower and the Earth sees the clock on the ship moving slower and each will continue to measure the others clock moving slower by the available Earth-ship communications as the two separate.

2. The observations made in the above conclusions are not "real" events since the idea of "simultaneity" has no real meaning at this time. In other words we are truly only talking about "observations". Another way to say this might be, that both clocks really are moving more slowly with respect to the other, but it's irrelevant since no real syncing can be done until the test is over at which time only one will have aged more slowly (the ship in this case).

3. At the point in time when the two ships pass each other to sync clocks, the encoded time of the Earth clock from ship 1 and ship 2 is the same (since they are getting the same time code message) but because ship 2 is going in the opposite direction Earth appears to be much further away, so ship 2 calculates that the actual time of Earth is much later than the calculated time of ship 1.

4. Ship 2 will continue to notice the Earth clock is moving slower but because of the calculated later time in step 3, when the ship arrives at Earth the Earth time will actually be later.

Did I get all that right so far?

One additional question. I had learned that if you travel to a star 10 l.y. away near the speed of light that you will observe that you arrive in less than 10 years. Is that true or not? I thought it was, but if the observed distance to a star you are traveling to seems to move further away the faster you go, it appears to all cancel out and you can never really get very far no matter how fast you go.

 

[yuiop]

2. The observations made in the above conclusions are not "real" events since the idea of "simultaneity" has no real meaning at this time. In other words we are truly only talking about "observations". Another way to say this might be, that both clocks really are moving more slowly with respect to the other, but it's irrelevant since no real syncing can be done until the test is over at which time only one will have aged more slowly (the ship in this case).

In Einstein's SR, both clocks are "really" running slower than each other. In Lorentz Ether Theory, one observer's clock is "really" running slower than the other. The slow clock (and length contracted rulers) of this observer causes this observer to incorrectly measure the clock of the other observer to be running slower than his own. The end result is that it is impossible to determine which observer has absolute motion and which observer really has the slower running clock. The mathematical predictions of SR and LET are identical and the differences are purely philosophical. There is no way in either theory to determine which inertial clock is really slower and there is also no way to determine if the philosophy of SR or LET is the correct philosophy. Nature has the last laugh because some things about how nature works are truly un-knowable to the human mind.

One additional question. I had learned that if you travel to a star 10 l.y. away near the speed of light that you will observe that you arrive in less than 10 years. Is that true or not?

It's true. If you travel at 0.8c it will take you 7.5 years by your own clock.

I thought it was, but if the observed distance to a star you are traveling to seems to move further away the faster you go, it appears to all cancel out and you can never really get very far no matter how fast you go.

You have this the wrong way round. The observed distance to the star according to the traveller is 6 lightyears (when traveling a 0.8c). The traveller concludes he has traveled 6 lightyears in 7.5 years which equate to a velocity of 6/7.5= 0.8c. The Eath to star distance is length contracted from the travellers point of view. Taking both length contraction and time dilation into account, you can in principle travel a distance of a million light years in a year of your own biological time or more extremely, anywhere in the universe in a second of your own time (without exceeding the speed of light).

 

[sylas]

ACCELERATION HAS NOTHING TO DO WITH IT. Don't anyone try and change my mind on this or I will surely go out of my mind.

Shrug. We all lose our minds and some point on the way to understanding relativity, or it feels like it. It doesn't last.

Acceleration does have something to do with it, because acceleration is what causes the change in frame. Acceleration is not directly a cause of time dilation, but in so far as accelerating is how you change frames, you can't say it has NOTHING to do with it. You just have to follow the nature of the association -- which means focusing on relative velocities and how they change.

1. As ship 1 passes Earth on the way out, it sees the clock on Earth moving slower and the Earth sees the clock on the ship moving slower and each will continue to measure the others clock moving slower by the available Earth-ship communications as the two separate.

Keep in mind the difference between "seeing" the clock, where you have to consider the change in how long it takes for light to reach you as well as dilation effects. The dilation effect refers not to what you SEE the other clock doing, but to what you infer the other clock is doing "at the same time" as your clock.

Hence, a clock moving at 60% light speed relative to you is running 1.25 times more slowly, no matter its direction. But what you SEE of the other clock is something else again.

If the other clock is moving tangentially to you, then the distance to the clock remains unchanged, and you see the ticking proceeding 1.25 times more slowly.

If the other clock is moving away from you, you see the ticking twice as slowly, because the signals are taking longer and longer to reach you.

If the other clock is moving towards you, you see it ticking twice as fast! It is still running more slowly by the same dilation factor of 1.25, but because the signals take less and less time to reach you, you actually see them speeded up by a factor of two, rather than slowed down.

In each case, the dilation factor remains the same, and depends only on velocity.

2. The observations made in the above conclusions are not "real" events since the idea of "simultaneity" has no real meaning at this time. In other words we are truly only talking about "observations". Another way to say this might be, that both clocks really are moving more slowly with respect to the other, but it's irrelevant since no real syncing can be done until the test is over at which time only one will have aged more slowly (the ship in this case).

No. Simultaneous DOES have a meaning. The point to grasp is that the meaning is relative. That is, events that are simultaneous in one frame may not be simultaneous in another.

You CAN draw conclusions about simultaneity, and they are meaningful.

3. At the point in time when the two ships pass each other to sync clocks, the encoded time of the Earth clock from ship 1 and ship 2 is the same (since they are getting the same time code message) but because ship 2 is going in the opposite direction Earth appears to be much further away, so ship 2 calculates that the actual time of Earth is much later than the calculated time of ship 1.

Yes.

4. Ship 2 will continue to notice the Earth clock is moving slower but because of the calculated later time in step 3, when the ship arrives at Earth the Earth time will actually be later.

Both ships with infer that the clock on Earth is moving more slowly. But the one approaching will SEE it sped up. Both ships will agree on what time Earth clocks will show when the approaching ship gets to Earth. Of course, the (x,t) location of this arrival event will differ for the inertial frame of the different ships.

Did I get all that right so far?

Point 3 was the important one, I think.

One additional question. I had learned that if you travel to a star 10 l.y. away near the speed of light that you will observe that you arrive in less than 10 years. Is that true or not? I thought it was, but if the observed distance to a star you are traveling to seems to move further away the faster you go, it appears to all cancel out and you can never really get very far no matter how fast you go.

Got that one backwards. As you move faster, the distance between Earth and the star CONTRACTS, not lengthens.

Eg. If a ship approaches a star 10ly away (from Earth's perspective) at 60% light speed, then it will take about 16 years and eight months (16.666 years) for the ship to get there (though of course it takes 10 years more to see the arrival).

From the point of view of the ship, the distance is contracted to 8 light years, and at 60% light speed (the speed the star is approaching) it will take 13 years and 4 months. (13.3333 years.)

The effects of relativity, if you could travel at sufficiently high relative velocities, allow you to travel all over the galaxy in as little experienced time as you like. From your own perspective as the traveler, all the distances between stars would be reduced.

Cheers -- sylas

 

[ghwellsjr]

You are still mixed up on several points but you are making progress. Let me comment:

1) The easiest way for each ship and Earth to communicate their time to the others is through a clock that emits a bright flash periodically, say once an hour. Each observer has two counters, one to count its own outgoing flashes and one to count the other observer's incoming flashes. When they are at their closest approach, they each reset all their counters to zero. Then as they move apart, they will each observe that the incoming flashes are coming in at a slower rate than their outgoing flashes. They can each calculate the ratio of the rate of incoming flashes to outgoing flashes and it will be a number less than one and they both will get the same ratio. From this ratio, they each can determine the relative speed between them and from that, they can each determine the time dilation factor. Look up relativistic doppler for more information.

2) You should not consider the measurements to be not real, even though you are right that simultaneity is not an issue here but that's because simultaneity is only a concern when you are comparing results between two different frames of reference and we are not defining any frame of reference in this explanation. Later on, if you want to, you can revisit this scenario from different frames of reference and you will discover that what I am describing here is the same no matter which frame of reference you use. Just remember, what each observer measures and observes will be the same however you analyze the situation.

3) Ship 1 communicates the value on its counters to ship 2 which then sets its counters accordingly. The ships at this point cannot tell by observing the flashes how far away the Earth is. Ship 1 can calculate how far it has traveled by simply multiplying the number of outgoing flashes by the distance traveled per flash which is .866 light hours. Ship 2 can do the same calculation (because the value in the outgoing counter from ship 1 has been communicated to it) and will arrive at the same distance. There is no meaning to your statement that the Earth appears further away or "the actual time of Earth is much later than the calculated time of ship 1". These kinds of conclusions would be frame dependent and not invariant. We aren't concerned about a frame in this analysis.

4) As ship 2 takes over the role of counting incoming flashes to outgoing flashes and calculating the ratio of their rates, it immediately sees the ratio as much larger, in fact, it is the reciprocal of what ship 1 saw. But using the Relativistic Doppler formula, it calculates exactly the same relative speed between itself and Earth and therefore, exactly the same time dilation as ship 1 saw. We should mention that at the point of switch over, ship 1 shuts off its flashes and ship 2 turns on its flashes, we don't want the Earth observer to later on get confused seeing flashes from two different ships at the same time. And be aware that the observer on Earth is completely unaware of this "turn-around" event happening and keeps measuring the same low Relativistic Doppler rate as before for a very long time, but they both continue to observe the same time dilation throughout this entire scenario.

Now here is the key to the different aging: from the moment of "turn-around" to the end of the scenario, the ships have spent an equal amount of time counting incoming flashes from Earth, half of them at a low rate (ship 1) and half of them at a high rate (ship 2). But the Earth doesn't see the transition from low rate to high rate until much, much later because it has to wait for all those flashes that were in transit from the ships' "turn-around" event to Earth to finally get back to Earth. When they do see the "turn-around" event, long after it happened, they will start counting the high rate for a relatively short period of time and this results in a much lower count on Earth's incoming counter than on the ship's incoming counter. Remember, counters on clocks keep track of accumulated time.

I explained all this, by the way, in post #2. Also, this is a description of what actually happens and has nothing to do with the Theory of Special Relativity or any other theory. As I said earlier, once you understand what is actually happening, you can go ahead, pick a frame of reference and "explain" it again using Special Relativity. A good frame of reference to start with would be the one in which Earth is at rest. Then you can do it again with the frame of rest for ship 1 and again for the rest frame of ship 2 and then a fourth frame could be the "average" between ship 1 and Earth where they are each traveling at the same speed in the opposite direction. Doing this explanation in many different frames will give you great insight into how Special Relativity works but it is never necessary to "solve" any problem in more than one frame because they all give the same result.

 

[D H]

You should not consider the measurements to be not real

Your use of a double negative threw me for a loop. I initially read that statement as "You should not consider the measurements to be real", a statement with which I was about to take exception until I re-read it.

Those measurements, IMO, are very real. Measurements are what distinguish philosophical ramblings from physics.

 

[Buckethead]

Got that one backwards. As you move faster, the distance between Earth and the star CONTRACTS, not lengthens.

But in an earlier post you said the returning ship would see the Earth as being 4 times as far away as the leaving ship. Doesn't this mean the returning ship would see the Earth as being twice as far as a ship in the same location that is not moving relative to the Earth?

 

[Mike_Fontenot]

But in an earlier post you said the returning ship would see the Earth as being 4 times as far away as the leaving ship. Doesn't this mean the returning ship would see the Earth as being twice as far as a ship in the same location that is not moving relative to the Earth?

It's the usual problem of what is meant by "sees" ... that term gets used with different meanings, by different people, in special relativity.

The image currently being SEEN by an approaching ship IS smaller than the image currently being SEEN by a stationary ship at that same location. But the approaching ship is seeing the Earth NOT as it currently IS, but as it WAS at an earlier time, when it was farther away. When those on the approaching ship correct for that delay in the propagation of light, they will conclude that the Earth is closer than what those on the stationary ship conclude.

Mike Fontenot

 

[Buckethead]

It's the usual problem of what is meant by "sees" ... that term gets used with different meanings, by different people, in special relativity.

The image currently being SEEN by an approaching ship IS smaller than the image currently being SEEN by a stationary ship at that same location. But the approaching ship is seeing the Earth NOT as it currently IS, but as it WAS at an earlier time, when it was farther away. When those on the approaching ship correct for that delay in the propagation of light, they will conclude that the Earth is closer than what those on the stationary ship conclude.

Mike Fontenot

But there is only one true way to determine the DIFFERENCE in distance between what a stationary ship and the approaching ship measures as the distance to Earth and that is to listen to an encoded time message from Earth. If it says "good morning travelers, the time now is 7:00AM" , then they can both conclude that they are the same distance from Earth. And this of course will be what they both hear since they are in the same location. So it may appear smaller (as I said earlier, because its an illusion) and it can also appear to be closer because their time has slowed down and they will reach Earth in less time than they thought, but both of these are an illusion because in fact the moving ship and the stationary ship and the departing ship are all the same distance from the Earth as indicated from the encoded message from Earth. So it appears I am still missing something.

 

[michelcolman]

But in an earlier post you said the returning ship would see the Earth as being 4 times as far away as the leaving ship. Doesn't this mean the returning ship would see the Earth as being twice as far as a ship in the same location that is not moving relative to the Earth?

When you are traveling towards or away from the earth, the Earth will "be" closer to you than if you would be standing still at exactly the same location. You could measure this distance by multiplying your travel time with your speed, so it's a "real" distance. Also, experiments sending a laser beam back and forth would be perfectly consistent with this distance. Of course people on Earth would not agree, and neither would people at the same location as you but that are not moving relative to earth. For you, however, this shorter distance will be perfectly real.

The visual effect, however, is a bit different.

If you are moving towards the earth, its physical size will appear smaller because your field of view widens. Basically, you are receiving the same light as a stationary observer, but the rays seem to come from smaller angles. In fact, if you go fast enough, you will even see objects that are at an angle behind you but that appear to be in front of you! So, when accellerating to really high speeds, you will get the visual impression that you are going backwards. Everything in front of you will also become brighter and blueshifted.

When moving in the opposite direction, away from Earth and looking backwards, the Earth will look bigger but fainter and redshifted. While accellerating, the objects behind you (that you are accellerating away from) will actually appear to be coming closer to you. So in this case too, the accelleration will create the illusion of going backwards.

You can see these effects in action on:
http://www.anu.edu.au/Physics/Searle/

 

[michelcolman]

But there is only one true way to determine the DIFFERENCE in distance between what a stationary ship and the approaching ship measures as the distance to Earth and that is to listen to an encoded time message from Earth. If it says "good morning travelers, the time now is 7:00AM" , then they can both conclude that they are the same distance from Earth. And this of course will be what they both hear since they are in the same location. So it may appear smaller (as I said earlier, because its an illusion) and it can also appear to be closer because their time has slowed down and they will reach Earth in less time than they thought, but both of these are an illusion because in fact the moving ship and the stationary ship and the departing ship are all the same distance from the Earth as indicated from the encoded message from Earth. So it appears I am still missing something.

What's missing is how long it took for the message to reach you. One person may say that Earth's message was sent an hour ago, while the other will say that the same message was sent two hours ago. That means they will disagree on the current time on earth, and the current distance from earth.

If you are moving away from earth, you will say that the Earth is moving away from you and therefore it is now further away from you than it was when the message was sent. Somebody receiving the same message but going towards earth, will say that the Earth is coming towards him and therefore it is now closer than it was when the message was sent. That also means current time at Earth is now later, since the message had to travel a longer distance so it must have been sent longer ago. Both will be right, from their point of view! The speed of light relative to themselves is indisputably c, and therefore their conclusions are the only possible explanation.

 

[Buckethead]

When you are traveling towards or away from the earth, the Earth will "be" closer to you than if you would be standing still at exactly the same location. You could measure this distance by multiplying your travel time with your speed, so it's a "real" distance. Also, experiments sending a laser beam back and forth would be perfectly consistent with this distance. Of course people on Earth would not agree, and neither would people at the same location as you but that are not moving relative to earth. For you, however, this shorter distance will be perfectly real.

The visual effect, however, is a bit different.

If you are moving towards the earth, its physical size will appear smaller because your field of view widens. Basically, you are receiving the same light as a stationary observer, but the rays seem to come from smaller angles. In fact, if you go fast enough, you will even see objects that are at an angle behind you but that appear to be in front of you! So, when accellerating to really high speeds, you will get the visual impression that you are going backwards. Everything in front of you will also become brighter and blueshifted.

When moving in the opposite direction, away from Earth and looking backwards, the Earth will look bigger but fainter and redshifted. While accellerating, the objects behind you (that you are accellerating away from) will actually appear to be coming closer to you. So in this case too, the accelleration will create the illusion of going backwards.

You can see these effects in action on:
http://www.anu.edu.au/Physics/Searle/

Thank you, this is what I thought which is why in a much earlier post I said that the Earth only looks further away, an illusion. And yes I can see how it will "actually" be closer because of the slowing of time for the returning traveller. These are all consistant with my understanding.

Somehow though, this still feels circular to me because you are saying the Earth is closer upon returning because all measurements indicate that it is closer (which is fine). But of course it is not closer than the departing ship when they are in the same location because of the encoded time message which is the same for both.

So far I am just seeing that everything is just an illusion except for the slowed time when the ship returns to Earth. The departing ship sees the Earth aging more slowly , but this is an illusion, the two ships see different distances to Earth and this is also an illusion, and the returning ship sees the Earth clock moving more slowly which is also an illusion, so the only thing that is not an illusion is the clock on the two ships is going slower than the clock on Earth. I don't think I'll ever understand this.

 

[Mike_Fontenot]

[...]
But there is only one true way to determine the DIFFERENCE in distance between what a stationary ship and the approaching ship measures as the distance to Earth and that is to listen to an encoded time message from Earth. If it says "good morning travelers, the time now is 7:00AM" , then they can both conclude that they are the same distance from Earth.
[...]

No, that's not correct. They DO each hear EXACTLY the same message, but when they each CORRECTLY allow for the transit time of that message, they get DIFFERENT answers. They are BOTH correct.

And there are easier ways for them to determine their current distance from the earth.

They can use the Lorentz equations.

They can use the length-contraction result.

They can use the time-dilation result, combined with their known velocity with respect to the earth.

All four methods yield exactly the same result.

Mike Fontenot

 

[Buckethead]

What's missing is how long it took for the message to reach you. One person may say that Earth's message was sent an hour ago, while the other will say that the same message was sent two hours ago. That means they will disagree on the current time on earth, and the current distance from earth.

If you are moving away from earth, you will say that the Earth is moving away from you and therefore it is now further away from you than it was when the message was sent. Somebody receiving the same message but going towards earth, will say that the Earth is coming towards him and therefore it is now closer than it was when the message was sent. That also means current time at Earth is now later, since the message had to travel a longer distance so it must have been sent longer ago. Both will be right, from their point of view! The speed of light relative to themselves is indisputably c, and therefore their conclusions are the only possible explanation.

OK, yes I can see this, but I think it's fair to say that when one ship says it was sent a different time than the other, they are simply using their instruments and calculators to say that and that in fact the message had to have been sent at the same time (it's only one message) and had to have been received at the same time (they are both there to receive it), and the message says the same time to both of them, so the only conclusion to be made from this is that THIS is the actual distance from Earth and it's one and the same distance for both ships. What I understand can't be determined is the actual distance in units as both ships will come up with a different number. So both numbers must be wrong since the two numbers MUST agree with each other since they are in the same location.

 

[Buckethead]

No, that's not correct. They DO each hear EXACTLY the same message, but when they each CORRECTLY allow for the transit time of that message, they get DIFFERENT answers. They are BOTH correct.

And there are easier ways for them to determine their current distance from the earth.

They can use the Lorentz equations.

They can use the length-contraction result.

They can use the time-dilation result, combined with their known velocity with respect to the earth.

All four methods yield exactly the same result.

Mike Fontenot

Agreed. But see post #67. You can't both be at different distances and at the same distance at the same time, this is a paradox

 

[Mike_Fontenot]

[...]
But see post #67. You can't both be at different distances and at the same distance at the same time, this is a paradox
[...]

They each agree how old the home twin was when she SENT the message ... her message tells them that.

And they each agree about how old THEY each were when they simultaneously RECEIVED her message.

But they disagree about how much the home twin aged during the message transit, and therefore they disagree about how old the home twin was when they RECEIVED her message.

There are no true paradoxes and/or inconsistencies in special relativity. But it is very easy to THINK you see an inconsistency, whenever you allow yourself to be even the slightest bit imprecise in your statements, or when you allow a subconsciousassumption to creep in, that is obviously true in Newtonian physics, but which is NOT true in special relativity. Everyone who has ever carried out any calculations in special relativity has been burned before (usually multiple times) because they haven't been sufficiently precise in their statements.

Mike Fontenot

 

[phyti]

They each agree how old the home twin was when she SENT the message ... her message tells them that.

And they each agree about how old THEY each were when they simultaneously RECEIVED her message.

But they disagree about how much the home twin aged during the message transit, and therefore they disagree about how old the home twin was when they RECEIVED her message.

There are no true paradoxes and/or inconsistencies in special relativity. But it is very easy to THINK you see an inconsistency, whenever you allow yourself to be even the slightest bit imprecise in your statements, or when you allow a subconsciousassumption to creep in, that is obviously true in Newtonian physics, but which is NOT true in special relativity. Everyone who has ever carried out any calculations in special relativity has been burned before (usually multiple times) because they haven't been sufficiently precise in their statements.

Mike Fontenot

But what if ships 1 and 2 do not synchronize their clocks upon leaving earth?
What would they conclude about the message?