What Causes Time Dilation in the Twin Paradox?

[Mohammed_I]

I can't fully understand why a person who makes a journey into space in a high-speed rocket will return home to find his age less than an identical twin who stayed on Earth. It makes since for the twin who stayed on earth, but for the twin who traveled into space, he sees himself at rest and sees Earth moving at constant velocity. so he should feel that time on Earth is moving slower than his time, this way he will find his twin younger than him not older. Please tell me what I am missing to help me understand this thought experiment.

Thank you.

 

[gendou2]

The trick is that one twin (the one in the rocket ship) undergoes increased acceleration (much more than 9.8 m/s), while the other does not.

http://en.wikipedia.org/wiki/Twin_paradox

 

[Fredrik]

That only proves that the two twins aren't in identical situations. It implies that something can be different, but it doesn't explain why something is different.

The part that's hard to get is that there's no way for the astronaut twin to turn his ship around without having his brother's age "jump ahead" by a large amount. You need to understand simultaneity in SR before you can understand that.

Check out http://web.comhem.se/~u87325397/Twins.PNG .

I'm calling the twin on Earth "A" and the twin in the rocket "B".
Blue lines: Events that are simultaneous in the rocket's frame when it's moving away from Earth.
Red lines: Events that are simultaneous in the rocket's frame when it's moving back towards Earth.
Cyan (light blue) lines: Events that are simultaneous in Earth's frame.
Dotted lines: World lines of light rays.
Vertical line in the upper half: The world line of the position (in Earth's frame) where the rocket turns around.
Green curves in the lower half: Curves of constant -t^2+x^2. Points on the two world lines that touch the same green curve have experienced the same time since the rocket left Earth.
Green curves in the upper half: Curves of constant -(t-20)^2+(x-16)^2. Points on the two world lines that touch the same green curve have experienced the same time since the rocket turned around.

 

[Mohammed_I]

Thank you very much Fredrik, this helped me a lot. Thank you too Gendou

 

[neopolitan]

Check out http://web.comhem.se/~u87325397/Twins.PNG .

I wrote a longer post, but it got lost.

Fundamentally, I don't buy the discontinuity at the turn around point. "A" appears to age 7 or so years on "B"'s outward leg, and around 33 years on the return leg.

You can think about it with "A" sending out constant updates by radio signal, "I have aged 1 day" - once a day, every day and working out when and where these messages and "B" will be collocated. You should be able to see that "B" will receive 7 years worth of messages on the way out and 33 years worth on the way home. There will be a few messages during the turn around period, but not 25.6 years worth. (How many depends entirely on the rates of acceleration and thus how quickly "B" turns around.)

cheers,

neopolitan

 

[Dale]

You can think about it with "A" sending out constant updates by radio signal, "I have aged 1 day" - once a day, every day and working out when and where these messages and "B" will be collocated. You should be able to see that "B" will receive 7 years worth of messages on the way out and 33 years worth on the way home. There will be a few messages during the turn around period, but not 25.6 years worth. (How many depends entirely on the rates of acceleration and thus how quickly "B" turns around.)

You are correct, what you are describing is essentially the relativistic Doppler effect for a signal frequency of 12 uHz, but presumably you would want to correct for the light travel time. So you would get an "I have aged 1 day" signal, note how far you are from Earth at that time (in your frame), and calculate when the signal was sent to determine when he had aged. You wouldn't assume that he had aged when you received the signal, but rather when he sent the signal.

 

[neopolitan]

You are correct, what you are describing is essentially the relativistic Doppler effect for a signal frequency of 12 uHz, but presumably you would want to correct for the light travel time. So you would get an "I have aged 1 day" signal, note how far you are from Earth at that time (in your frame), and calculate when the signal was sent to determine when he had aged. You wouldn't assume that he had aged when you received the signal, but rather when he sent the signal.

Yep, that is why I only talk about receiving the signals. But the signals are just reification of the _information_ about the aging of "A" according to "B". It is the only valid way of thinking about it, of putting labels on a chart like the one that Fredrik put together, imho.

cheers,

neopolitan

 

[granpa]

Yep, that is why I only talk about receiving the signals. But the signals are just reification of the _information_ about the aging of "A" according to "B". It is the only valid way of thinking about it, of putting labels on a chart like the one that Fredrik put together, imho.

cheers,

neopolitan

its absolutely true that there is a jump at the turn around point. in fact if he accelerates decelerates reaccelerates again and again then his calculation of the twins age will jump back and forth again and again (the twin will appear to move backward in time). now I don't buy that this is a 'real' effect but it certainly happens. you forget that to calculate the twins age he must take into account the speed of light. but he always measures the speed of light to be c relative to himself in spite of the fact that he just changed velocity. obviously that much have a drastic effect on his calculation.

whether such relativistic effects are 'real' or not is another discussion entirely

 

[Fredrik]

Fundamentally, I don't buy the discontinuity at the turn around point. "A" appears to age 7 or so years on "B"'s outward leg, and around 33 years on the return leg.

It's 7.2 during the outbound trip, 7.2 during the return trip, and 25.6 during an instantaneous turnaround. Nothing important changes if you make the turnaround phase last longer.

You can think about it with "A" sending out constant updates by radio signal, "I have aged 1 day" - once a day, every day and working out when and where these messages and "B" will be collocated. You should be able to see that "B" will receive 7 years worth of messages on the way out and 33 years worth on the way home. There will be a few messages during the turn around period, but not 25.6 years worth. (How many depends entirely on the rates of acceleration and thus how quickly "B" turns around.)

Yep, that is why I only talk about receiving the signals. But the signals are just reification of the _information_ about the aging of "A" according to "B". It is the only valid way of thinking about it, of putting labels on a chart like the one that Fredrik put together, imho.

You might as well say that the definition of simultaneity in SR that's accepted by everyone isn't valid, because that's what I'm using.

 

[neopolitan]

You might as well say that the definition of simultaneity in SR that's accepted by everyone isn't valid, because that's what I'm using.

But are you using precisely what is "accepted by everyone"? And if you are using that, is that definition of simultaneity supposed to be a helpful device or a proclamation on how things actually are?

Ponder this ... in your example, "B" travels for 12 years. "A" ages 25.6 years in the turnaround period. How does the universe know to age "A" 25.6 years, and not 51.2 years or 102.4 years (which apply if "B" plans to travel for 24 or 48 years?

Suppose, just suppose, that "B" travels to the turnaround point at a velocity of 0.8c, the heads off at 0.8c on the way back, but half-way home suffers a breakdown and has to limp the rest of the way at 0.001c. We'd have to assume that A and B are long lived, of course, but your figures will no longer work out. The instantaneous aging you posit at the turnaround point is an implication that the universe somehow sees the future and applies the right instantaneous aging to "A" (based on "B"'s future speeds).

I don't think that is part of the standard understanding of simultaneity.

cheers,

neopolitan

 

[neopolitan]

its absolutely true that there is a jump at the turn around point. in fact if he accelerates decelerates reaccelerates again and again then his calculation of the twins age will jump back and forth again and again (the twin will appear to move backward in time). now I don't buy that this is a 'real' effect but it certainly happens. you forget that to calculate the twins age he must take into account the speed of light. but he always measures the speed of light to be c relative to himself in spite of the fact that he just changed velocity. obviously that much have a drastic effect on his calculation.

whether such relativistic effects are 'real' or not is another discussion entirely

Try thinking about it with "B" calculating "A"'s age purely on the basis of the number of reports of one day's aging. You will see that the discontinuities just don't happen.

I will recant of course, if you can show how 25.6 years worth of daily reports will be received by "B" instantaneously, or over a relatively short period, if "B" chooses to decelerate and accelerate back homewards. (I would then also like to see how the other discontinuities work, for example if "B" decelerates on the way home and "A" moves backward in time - are the daily reports rescinded and reissued? and how?)

cheers,

neopolitan

 

[Fredrik]

But are you using precisely what is "accepted by everyone"? And if you are using that, is that definition of simultaneity supposed to be a helpful device or a proclamation on how things actually are?

That definition of simultaneity tells you how to associate an inertial frame with a physical observer moving with constant velocity. There's nothing controversial about it.

Any coordinate system is a "helpful device" because you can use it to describe "how things actually are". The problem is of course that if you use a coordinate system to describe how things actually are, the description may sound very different from a description made using another coordinate system. (Take the pole-and-barn paradox for example: In the pole's frame, there's a time when the whole pole is inside the barn. In the barn's frame, there isn't). Because of these apparent contradictions, I can't encourage the use of expressions like "how things actually are". It's better to talk about e.g. "how things are in this particular frame".

Ponder this ... in your example, "B" travels for 12 years. "A" ages 25.6 years in the turnaround period. How does the universe know to age "A" 25.6 years, and not 51.2 years or 102.4 years (which apply if "B" plans to travel for 24 or 48 years?

The universe isn't doing anything, so it doesn't have to know anything. The reason it's 25.6 and not any other number is that B is 16 light-years from Earth and that the slope of a simultaneity line is always v when the slope of the world line is 1/v. You can verify for yourself that with v=0.8c, that works out to 25.6 years.

Suppose, just suppose, that "B" travels to the turnaround point at a velocity of 0.8c, the heads off at 0.8c on the way back, but half-way home suffers a breakdown and has to limp the rest of the way at 0.001c. We'd have to assume that A and B are long lived, of course, but your figures will no longer work out. The instantaneous aging you posit at the turnaround point is an implication that the universe somehow sees the future and applies the right instantaneous aging to "A" (based on "B"'s future speeds).

If B slows down to 0.001c on the way back, he will be in a frame where his "now" is simultaneous with an earlier event on Earth, so A will be younger than "before". That doesn't mean that the universe has "seen" or "done" something. It only means that B is using a different coordinate system.

I don't see the problem.

 

[neopolitan]

I don't see the problem.

Maybe you have the right idea about how simultaneity in SR is applied, but I really doubt it after long discussions elsewhere.

To bring in simultaneity, if you must have it, consider "A" as a simple transmitter, pulsing out "A has aged x days" repeatedly (where x(m) = x(m-1)+1). Consider "B" as a similar transmitter with a similar message, "B has aged z days" (similar values of z).

Using the relevant equations, "A" may consider the sending of one message to "B" and "B"'s sending of another message to be simultaneous, whereas "B" would not agree.

Once "B" reaches the turn around point, there will a number of signals from "A" still on the way. Since the speed was 0.8c, for a period of 12 years, then in _classical_ terms there would be about 9.6 years worth of signals inbound (and last signal from "A" should be "A has aged 876 days"). In SR terms there will be 12.8 years worth of signals still on the way and the last message will be "A has aged 2629 days". (I am using your figure of 7.2 years btw, I haven't checked its validity. In any event, the actual figures are not important, it is the whole concept that matters).

"B" then turns around and goes back through the incoming barrage of signals. Over a period of 12 years ship time, in _classical_ terms, "B" would experience 12 years of signals, plus the 9.6 years worth which were inbound at the turnaround point.

In SR terms, "B" will travel through 32.8 years worth of signals before getting home and asking what the hell is wrong with "A"'s transmission schedule.

What won't happen is the sudden delivery of 25.6 years worth of signals at the turn around point or the double delivery of signals (which would be the implication of "B" ending up in 'a frame where his "now" is simultaneous with an earlier event on Earth').

The only way you could organise double delivery of signals is by traveling faster than the speed of light (and it would involved some doubling back too, unless "A" started sending out pulses before "B" left Earth).

Your mathematics might be spot on, Fredrik, but the application of them is questionable.

cheers,

neopolitan

 

[Jonathan Scott]

What won't happen is the sudden delivery of 25.6 years worth of signals at the turn around point or the double delivery of signals (which would be the implication of "B" ending up in 'a frame where his "now" is simultaneous with an earlier event on Earth').

A stream of signals arriving does not define simultaneity. You also have to calculate how far they traveled and hence when they originated. It is that calculation which has an abrupt discontinuity at the turn-around point.

 

[neopolitan]

A stream of signals arriving does not define simultaneity. You also have to calculate how far they traveled and hence when they originated. It is that calculation which has an abrupt discontinuity at the turn-around point.

But you agree that "B" will receive a rational stream of signals from "A"?

Additionally, why exactly is there an abrupt discontinuity at the turn around? When pointed in one direction and getting "A has aged x days", B will calculate that A is a certain distance away (with the assumption that the message traveled c). Once turned around and getting "A has aged x+1 days" and being in approximately the same position, why does B make a calculation that A must have aged 25.6 years over the past day?

My thinking is that B should make a calculation like this:

To the best of my knowledge we (A and B) have been separating at a constant 0.8c for a period of 12 years, so that our current separation is 9.6 light years. Any signal from A will have taken 9.6 years to get to my current position. Therefore, A is currently 9.6 years older than as indicated in the latest message I have received.

Righto, I will turn around now. Done.

Ok, now I am pretty much in the same spot as I was yesterday, with a separation of 9.6 light years from A. It is a day or so later. Again, any signal from A will have taken 9.6 years to get here, and A is currently 9.6 years older than indicated in the latest message

No discontinuity.

Is there any other sensible way that B should calculate how great the separation between the two of them is?

cheers,

neopolitan

PS Thinking about it some more ...

"A" will appear to have been sending out signals at a greater rate than "B" might have otherwise suspected, so that by the turnaround point "B" will already have received 2.4/0.6 years of signals. If "B" is clever, then an assumption will be made to the effect that "A" has and will continue to transmit signals at the same rate, then "B" should expect that there are 9.6/0.6 years worth of signals on the way at the turnaround point.

9.6/0.6 + 2.4/0.6 = 16 + 4 = 20 years

On the way back "B" will expect to intercept a total of 36 years worth of messages.

Still no discontinuity :)

 

[Jonathan Scott]

But you agree that "B" will receive a rational stream of signals from "A"?

Additionally, why exactly is there an abrupt discontinuity at the turn around? When pointed in one direction and getting "A has aged x days", B will calculate that A is a certain distance away (with the assumption that the message traveled c). Once turned around and getting "A has aged x+1 days" and being in approximately the same position, why does B make a calculation that A must have aged 25.6 years over the past day?

The change of B's frame of reference at the turn around makes all the difference to the time and space separation.

You can illustrate the ordinary 2D rotation equivalent by using "forwards" and "sideways" instead of "time" and "space" when moving in a plane. If two objects start from the same spot and move off in different directions at an acute angle relative to one another, then they are both moving less in each others' "forward" direction than the other. However, if one then turns a corner towards where the other one is going, the other one is suddenly much further "forward".

 

[granpa]

why exactly is there an abrupt discontinuity at the turn around?

I already told you the reason why in post 8. he has to take the speed of light into account. the speed of light remains the same even when he changes velocity. can't you see that thet wall change his calculation?

 

[Fredrik]

Neopolitan, you need to make an effort to understand simultaneity. Suppose that a light signal is emitted in the positive x direction at (t₁,0), gets reflected at some point along the x-axis and returns at (t₂,0), then the reflection event must be simultaneous with (t₁+(t₂-t₁)/2,0). That's how simultaneity works. Pick t₁=-T and t₂=T for simplicity. The emission event is (-T,0) and the return event is (T,0). The reflection event must be simultaneous with (0,0), and since the speed of light is 1, the spatial coordinate must be T, so the reflection event is (0,T).

Now draw the world line of someone who's moving with speed v in the positive x direction, and imagine that this person is doing the same thing we just did to find out which events are simultaneous. Note that the slope (\Delta t/\Delta x) of the world line is 1/v. You will find that to this person, two events are simultaneous if and only if they are on a line with slope v.

That's what I used to draw the diagram. Before the rocket turns around, B considers events on a blue line simultaneous. After the turnaround, he considers events on a red line simultaneous.

 

[Fredrik]

To bring in simultaneity, if you must have it, consider "A" as a simple transmitter, pulsing out "A has aged x days" repeatedly (where x(m) = x(m-1)+1). Consider "B" as a similar transmitter with a similar message, "B has aged z days" (similar values of z).

This isn't the way to "bring in simultaneity". See my previous post.

Once "B" reaches the turn around point, there will a number of signals from "A" still on the way. Since the speed was 0.8c, for a period of 12 years, then in _classical_ terms there would be about 9.6 years worth of signals inbound (and last signal from "A" should be "A has aged 876 days").

You need to specify a frame if you're going to refer to a certain time. Since you're saying "12 years" instead of "20 years", I assume that we're considering B's frame, and since you're talking about things that happened before the turnaround, I assume that we're talking about the first of B's frames, the one that's moving away from Earth.

The 9.6 and 876 figures are wrong. When (in B's frame) the rocket turns around, only 7.2 years have passed on Earth. The last message that arrived was sent by A when he had aged 4.0 years. (The turnaround event is (20,16) in A's frame, so a light signal that arrives at that event must have been emitted at (4,0)).

In any event, the actual figures are not important, it is the whole concept that matters.

I don't think it matters. It doesn't have anything to do with simultaneity.

"B" then turns around and goes back through the incoming barrage of signals. Over a period of 12 years ship time, in _classical_ terms, "B" would experience 12 years of signals, plus the 9.6 years worth which were inbound at the turnaround point.

He's going to experience 36 years of signals, since he only received 4 years of signals before he turned around. (I see now that you realized that yourself).

What won't happen is the sudden delivery of 25.6 years worth of signals at the turn around point or the double delivery of signals (which would be the implication of "B" ending up in 'a frame where his "now" is simultaneous with an earlier event on Earth').

The only way you could organise double delivery of signals is by traveling faster than the speed of light (and it would involved some doubling back too, unless "A" started sending out pulses before "B" left Earth).

Huh. You lost me here. First of all, he's only ending up in 'a frame where his "now" is simultaneous with an earlier event on Earth' if he slows down on the way back (which is the scenario you were considering at the time). The turnaround puts him in a frame where his "now" is simultaneous with a much later time on Earth. 25.6 years later to be exact.

I have no idea why you're talking about double delivery of signals and stuff like that.

Additionally, why exactly is there an abrupt discontinuity at the turn around? When pointed in one direction and getting "A has aged x days", B will calculate that A is a certain distance away (with the assumption that the message traveled c). Once turned around and getting "A has aged x+1 days" and being in approximately the same position, why does B make a calculation that A must have aged 25.6 years over the past day?

Simultaneity. (There's obviously no discontinuity in the incoming messages, but the simultaneity lines get tilted the other way when the rocket turns around).

 

[neopolitan]

The change of B's frame of reference at the turn around makes all the difference to the time and space separation.

You can illustrate the ordinary 2D rotation equivalent by using "forwards" and "sideways" instead of "time" and "space" when moving in a plane. If two objects start from the same spot and move off in different directions at an acute angle relative to one another, then they are both moving less in each others' "forward" direction than the other. However, if one then turns a corner towards where the other one is going, the other one is suddenly much further "forward".

"B" may have to a new basis of determining, in his new frame, which distant events about which he receives information in the future are simultaneous with which local events (which may be in his past), but he won't reassess events which were simultaneous with events in his previous frame. All such events are in the absolute past and cannot be resurrected.

I do not think that a range of spatially colocated events at "A" (not colocated in time) can be simultaneous for "B", ever. That would violate causality. (For example, "B" cannot receive two events which one which is dependant on the other with a delay. Imagine an egg dropping to the floor under the influence of gravity, "B" cannot receive the event "egg whole" and "egg broken" together since there is a time component to the fall between.)

Your "turning the corner" simile only works if "B" changes speed, which he doesn't really ... except during the actual turnaround. "B" changes velocity, speed is the same, only the direction changes.

cheers,

neopolitan

 

[matheinste]

Hello neopolitan.

Surely we are talking about events in B's frame(s) being simultaneous with events in A's frame and not about colocated events in space being simultaneous.I don't see how anything happening in the twins scenario violates causality. Don't forget that B is only in one frame at a time and there is no suggestion of events in one of these frames being simultaneous (for B)with events in the other. A's world of course carries on as normal and even to B it is seen as a continuous forward time progression albeit very much speeded up at turnaround.

Matheinste.

 

[neopolitan]

I can't fully understand why a person who makes a journey into space in a high-speed rocket will return home to find his age less than an identical twin who stayed on Earth. It makes since for the twin who stayed on earth, but for the twin who traveled into space, he sees himself at rest and sees Earth moving at constant velocity. so he should feel that time on Earth is moving slower than his time, this way he will find his twin younger than him not older. Please tell me what I am missing to help me understand this thought experiment.

Thank you.

I wonder if anyone else is still thinking about the OP's original question?

I have kept it in mind and attempted not to get tangled inextricably in simultaneity issues.

My point, in my first post in this topic is that there is no real discontinuity.

If there is a discontinuity it is based on the conceit that there is no change to "B"'s frame. However, in the real universe, "B" will undergo observable accelerations and will know that velocity changes have occured.

If "B" ignores that, and only if "B" ignores that, then the purely mathematical discontinuity will arise. That discontinuity will be balanced by an equal and opposite mathematical discontinuity from "A"'s point of view (as calculated by "B").

In reality, there is no discontinuity, as can be understood by considering the flows of information possible between "A" and "B".

Fredrik, I'll highlight something for your edification. Please observe the words which are bold and underlined.

Once "B" reaches the turn around point, there will a number of signals from "A" still on the way. Since the speed was 0.8c, for a period of 12 years, then in _classical_ terms there would be about 9.6 years worth of signals inbound (and last signal from "A" should be "A has aged 876 days")

Of course 9.6 years and 876 days are wrong, classical physics doesn't apply.

Matheinste,

I think that fundamentally we agree, given your words:

[QUOTE+matheinste]A's world of course carries on as normal and even to B it is seen as a continuous forward time progression albeit very much speeded up at turnaround.[/QUOTE]

There is no real discontinuity as presented in Fredrik's diagram (only a mathematical discontinuity created under the conditions given above).

I do disagree that the progression is particularly sped up at the turnaround, since there is a limitation to the transmission of information from "A" to "B" (and indeed from "B" to "A"). "B" will receive more information from "A" on the way back home than on the outward leg. There will be no special bulk delivery of signals from "A" during the turnaround.

If you don't understand this, then I implore you to read the preceding posts to understand what is actually being discussed.

cheers,

neopolitan

 

[Jonathan Scott]

"B" may have to a new basis of determining, in his new frame, which distant events about which he receives information in the future are simultaneous with which local events (which may be in his past), but he won't reassess events which were simultaneous with events in his previous frame. All such events are in the absolute past and cannot be resurrected.

I do not think that a range of spatially colocated events at "A" (not colocated in time) can be simultaneous for "B", ever. That would violate causality. (For example, "B" cannot receive two events which one which is dependant on the other with a delay. Imagine an egg dropping to the floor under the influence of gravity, "B" cannot receive the event "egg whole" and "egg broken" together since there is a time component to the fall between.)

That's not the way relativity and Lorentz transformations work.

There's no problem with causality. Once B has changed velocity, this changes the relationship between space and time in such a way that light in the new frame is still seen to move at c. This affects calculated times and distances. Spacelike and timelike separations remain spacelike and timelike so there is no effect on causality. You can calculate this and see it on a space-time diagram. The change in velocity doesn't cause any sort of discontinuity of what B actually sees (although the rate at which information is received effectively becomes Doppler shifted). Anything that has been literally "seen" is in the timelike past. Changing direction affects the calculation of exactly how far it was in the timelike past and where it happened, relative to B's new frame of reference.

Your "turning the corner" simile only works if "B" changes speed, which he doesn't really ... except during the actual turnaround. "B" changes velocity, speed is the same, only the direction changes.

In the 2D analogy, turning the corner DOES have an effect on how far forward something is. If A and B are both moving at the same speed from the same point in different directions, but A then turns in the same direction as B, then before turning A will consider that it is "ahead" of B in whatever direction A is going, but after turning, it will definitely be "behind" B in the direction that A and B are going. (Because of the different signature, this is the reverse of the time/space case, where the one which turns has aged more).

 

[neopolitan]

I have no idea why you're talking about double delivery of signals and stuff like that.

Then you don't understand what has been written before. Think about it, consider hypersurfaces of simultaneity, which are equivalent to signals sent out at the speed of light.

A distant event which is simultaneous with a local event in my frame is one which arrives after a period of time defined by x/c where x is the separation between me and the event.

This is why, when "B" turns around and observes the last message from "A" implying an ageing of 4 years, "B" can calculate that that latest signal was sent 9.6 years ago in the "B-frame" and that that equates to 16 years ago in the "A-frame" and that that means that a signal sent simultaneously with "B" turning around will indicate that "A" has aged 20 years.

The double delivery comes in when you say that "B" slowing down will make "B" simultaneous with earlier events in the "A" frame. If one moment "B" is simultaneous with an event 25 years after departure (in the "A-frame") and the next is simultaneous with an event 22 years after departure, then "B" should rightly expect to receive two lots of signals between 22 years and 25 years. But I am saying that this does not happen.

Part of the problem, I guess, is trying to use information that you haven't got yet. Only once a signal has been received from "A" can "B" work out what event that signals transmission was simultaneous with.

What I can say, with great certainty is that there is no _real_ discontinuity.

cheers,

neopolitan

 

[matheinste]

If you don't understand this, then I implore you to read the preceding posts to understand what is actually being discussed.
olitan.

Quote:-

--If you don't understand this, then I implore you to read the preceding posts to understand what is actually being discussed.---

I have read them. all this has been discussed time and time again and the answer is well known and accepted. Your view is incoreect.

Matheinste.

 

[Fredrik]

Think about it, consider hypersurfaces of simultaneity, which are equivalent to signals sent out at the speed of light.

A distant event which is simultaneous with a local event in my frame is one which arrives after a period of time defined by x/c where x is the separation between me and the event.

What you're defining here is just the future light cone of the local event. The apex of the light cone is not simultaneous with any other event on the light cone.

This is why, when "B" turns around and observes the last message from "A" implying an ageing of 4 years, "B" can calculate that that latest signal was sent 9.6 years ago in the "B-frame" and that that equates to 16 years ago in the "A-frame" and that that means that a signal sent simultaneously with "B" turning around will indicate that "A" has aged 20 years.

Oddly enough, you're getting this part exactly right. It's as if you're using the correct definition of simultaneity for this part, even though you just defined it incorrectly.

The double delivery comes in when you say that "B" slowing down will make "B" simultaneous with earlier events in the "A" frame. If one moment "B" is simultaneous with an event 25 years after departure (in the "A-frame") and the next is simultaneous with an event 22 years after departure, then "B" should rightly expect to receive two lots of signals between 22 years and 25 years.

This is just wrong. The deliveries obviously only depend on B's location in spacetime, but simultaneity only depends on velocity. So a change of simultaneity isn't going to make anything too weird happen to the deliveries.

 

[neopolitan]

If you don't understand this, then I implore you to read the preceding posts to understand what is actually being discussed.
olitan.

Quote:-

--If you don't understand this, then I implore you to read the preceding posts to understand what is actually being discussed.---

I have read them. all this has been discussed time and time again and the answer is well known and accepted. Your view is incoreect.

Matheinste.

What, pray tell, is my view that is incorrect? I am just asking for a quick summary of what you think my view is, as expounded in this thread. Of course, it would help if you highlight areas which, when taken in context, indicate a huge misunderstanding.

I ask this because people often jump in and attack what they think is a misunderstanding, without trying to gain an understanding themselves.

Note also, that I have accepted the mathematical discontinuity which results from "B" changing frame. What I argue is that this is not any real discontinuity and is based on the rather unlikely scenario that "B" does not accept that there actually has been a change of frame.

This might derive from my being an engineer by training rather than a physicist. We use the calculations as tools, rather than laws. If I were designing a spaceship for "B", complete with a clock that shows "B" the time elapsed for "A", I would not plan on having that clock jump from 7.2 years to 32.8 years at turnaround. Nor would I have a clock which just quickly scrolled forwards to 32.8 years.

If you were designing such a clock, how would it operate?

cheers,

neopolitan

 

[granpa]

if you want to know the 'real' time on Earth then just put a string of synchronized clocks between Earth and the turn around point. all he has to do is look out the window and read the clock. each clock he observes will seem to him to tick at a rate of 1/gamma. but the total elapsed time as told by the nearest clock will be gamma times the clock on board his ship

when he stops at the turn oround point all the clocks will ogain be synchronized and ticking at the normal rate.

if all you want his clock to do is tell the 'real' time on Earth then by all means design it however you want. but if as an engineer you want his clock to be useful to him in timing events on board his ship then you should reconsider.

 

[Fredrik]

if you want to know the 'real' time on Earth then just put a string of synchronized clocks between Earth and the turn around point.

You are probably already aware of this, but the term "real" is very inappropriate. What these clocks would tell you is just what event on Earth your "now" is simultaneous with in A's frame. But A's frame is obviously not any more real than B's.

 

[granpa]

You are probably already aware of this, but the term "real" is very inappropriate. What these clocks would tell you is just what event on Earth your "now" is simultaneous with in A's frame. But A's frame is obviously not any more real than B's.

thats why it was in quotes

 

[atyy]

You are probably already aware of this, but the term "real" is very inappropriate. What these clocks would tell you is just what event on Earth your "now" is simultaneous with in A's frame. But A's frame is obviously not any more real than B's.

Within the framework of SR in which inertial frames are special, wouldn't A's frame be more "real" in the sense that it is a single inertial frame?

 

[matheinste]

Hello neopolitan.

Quote:-

---If I were designing a spaceship for "B", complete with a clock that shows "B" the time elapsed for "A", I would not plan on having that clock jump from 7.2 years to 32.8 years at turnaround. Nor would I have a clock which just quickly scrolled forwards to 32.8 years.----

But this is what happens from the point of view of B.When A and B are reunited there is an age difference, it is not just an optical illusion caused by light travel times.

The turnaround can take as long as you like if you want to limit the rate of advance of A's time as seen by B.

Matheinste

 

[neopolitan]

What you're defining here is just the future light cone of the local event. The apex of the light cone is not simultaneous with any other event on the light cone.

No, I was talking about the future light cone of the distant event. The information about an event, ie a signal, takes a period of x/c to reach a distant observer (and that has to be x in the observers frame). An event which occurred a period of x/c before this signal is received is by definition simultaneous with the event which transmitted the signal - in terms of the receiver's frame.

Oddly enough, you're getting this part exactly right. It's as if you're using the correct definition of simultaneity for this part, even though you just defined it incorrectly.

Yeah, odd isn't it.

This is just wrong. The deliveries obviously only depend on B's location in spacetime, but simultaneity only depends on velocity. So a change of simultaneity isn't going to make anything too weird happen to the deliveries.

This is pretty much my point. You won't get weird deliveries. The simultaneity confusion arises because you insist on trying to determine the time a signal was emitted in terms of a frame which was not valid when the signal was emitted. In the real world, you just won't do that.

Try putting it this way, call the midpoint of the turnaround an event. At this event, "A" and "B" have zero separation speed. There is a world-line joining these events (two, if you want to do one for "A" and one for "B"). There is causality linking all "B" events post this event and all "A" events post this event.

"B" could choose to stay in place until all signals which were en route catch up. Let's say 9.6 years. During this time, 16 years worth of signals would be received. Then "B" could head off again, same velocity and intercept 20 years of signals during 12 years of shipboard time (plus the 9.6 years worth of signals which were on the way).

Or, "B" could choose not to wait and head off instead. In this case, "B" would intercept 16 years worth of signals where were already on the way, plus the 20 years worth of signals which "A" sends during the 12 shipboard years it takes "B" to get back.

If "B" chooses to slow down on the way back, it will have two effects, one is that "A"'s apparent exuberance with signally will die down and the other is that "B" will intercept signals from "A" at a reduced rate.

"B" should then designate a new event, the slowing down event and consider what will happen with the flow of signals before and after this event. Again, "B" could chose to stop, gain zero separation rate with "A", receive all signals which were en route before stopping, and then keep going at a slower rate ... or "B" could chose to just slow down, receive those en route signals and then the ones sent after the event which simultaneous with the slow down (simultaneous according to "B", of course).



If you ignore these events, you will calculate discontinuities. This may be the accepted thing to do, but it's not characteristic of the real universe.

cheers,

neopolitan

 

[neopolitan]

Hello neopolitan.

Quote:-

---If I were designing a spaceship for "B", complete with a clock that shows "B" the time elapsed for "A", I would not plan on having that clock jump from 7.2 years to 32.8 years at turnaround. Nor would I have a clock which just quickly scrolled forwards to 32.8 years.----

But this is what happens from the point of view of B.When A and B are reunited there is an age difference, it is not just an optical illusion caused by light travel times.

The turnaround can take as long as you like if you want to limit the rate of advance of A's time as seen by B.

Matheinste

So if you were designing this clock, what would it do?

I would like to clarify.

cheers,

neopolitan

 

[neopolitan]

if all you want his clock to do is tell the 'real' time on Earth then by all means design it however you want. but if as an engineer you want his clock to be useful to him in timing events on board his ship then you should reconsider.

I had to laugh.

The clock in question has one specific function, which I stated. In the modern world we are overrun by ordinary clocks, I have about five in front of me (computer, alarm, wall clock, time on mobile, time on printer, oh, and one on the wireless phone unit, so it is six). I didn't even consider that designing a special clock to give time elapsed for "A" might cause "B" undue hardship :)

cheers,

neopolitan

 

[atyy]

If you ignore these events, you will calculate discontinuities. This may be the accepted thing to do, but it's not characteristic of the real universe.

Is your point that everyone ages according to his own proper time, and that B can calculate the proper time of A?

 

[matheinste]

Hello neopolitan.

Quote:-

---So if you were designing this clock, what would it do?---

Why would i need such a clock. Why not just calculate A's time. Should you wish for a display of the results of this time calculation then that is easy enough.

Matheinste.

 

[Fredrik]

The information about an event, ie a signal, takes a period of x/c to reach a distant observer (and that has to be x in the observers frame). An event which occurred a period of x/c before this signal is received is by definition simultaneous with the event which transmitted the signal - in terms of the receiver's frame.

(Units such that c=1). So if a signal is emitted at (0,0) and received at (x,x), the event that occurred a time x before the event (x,x) (at spatial coordinates x I assume) has coordinates (0,x) and is simultaneous with the emission (0,0). At least we agree about something. We have found that a simultaneity line is a line of constant time coordinate.

So far this definition of simultaneity agrees with the one I posted in #18, but have you really thought about what it means? Do you agree or disagree with my claim that the blue lines are simultaneity lines of the rocket when it's moving away from Earth? What about the simultaneity lines when the rocket has turned around?

This is pretty much my point. You won't get weird deliveries.

But your claim was of the form "if [something that's 100% true] then we get weird deliveries", so your point must have been that the true statement was actually false. (Reductio ad absurdum). But your claim was false. You were wrong to think that the assumption implies something obviously incorrect ("weird deliveries").

In the real world, you just won't do that.

This isn't a valid argument.

Try putting it this way, call the midpoint of the turnaround an event. At this event, "A" and "B" have zero separation speed. There is a world-line joining these events (two, if you want to do one for "A" and one for "B"). There is causality linking all "B" events post this event and all "A" events post this event.

I haven't been able to decrypt what you're saying here. What two events? I assume that one is (20,16) in A's frame, but what's the other one? Also, a world line is a curve that represents the motion of an object. You want to imagine something moving from (20,16) to some other event? Why?

(lots of sentences containing the word "signals")

You're much too focused on those signals. I don't see how they are relevant at all. You seem to think that they somehow forbid us from using the inertial frame associated with B's return trip, but you haven't given us a reason for that.

If you ignore these events, you will calculate discontinuities. This may be the accepted thing to do, but it's not characteristic of the real universe.

The way you talk about these things is pretty strange to me. How is it not a characteristic of the real universe that two different global coordinate systems disagree about stuff? And what "discontinuity" are you talking about? What function is supposed to be discontinous? All we have here are (at least) two inertial frames that describe things differently.

 

[Al68]

This might derive from my being an engineer by training rather than a physicist. We use the calculations as tools, rather than laws. If I were designing a spaceship for "B", complete with a clock that shows "B" the time elapsed for "A", I would not plan on having that clock jump from 7.2 years to 32.8 years at turnaround. Nor would I have a clock which just quickly scrolled forwards to 32.8 years.

If you were designing such a clock, how would it operate?

It would run slower than a "standard" clock (by the appropriate factor dependent on relative velocity) for all inertial motion. When it senses the acceleration (turnaround), it would run faster than "normal" by a factor proportional to the rate of acceleration and proportional to its distance from "A"s clock.

While such a clock would not be "needed", it is interesting to note how it would have to work. It would have to be rigged so that acceleration would cause it to speed up. Great question, neopolitan. Why am I the only person to even attempt to answer it?

Al

 

[matheinste]

Hello Al68.

This is purely a reply to your remark in #40.

To disagree with a point of view is absolutely acceptable as is my right to put forward my view, which on this subject is very much mainstream. I know you do not mean to criticize religion as such ( but it would not bother me if you do ) but the implication is that answers i have given to the best of my admittedly limited ability are given as a matter of unthinking, blind faith, in the manner of a crusade. My answers and those of many others on this question may in your eyes be incorrect but do not deserve to be called religious with all the underlying disparaging connotations of the word. The word itself is of course not offensive but its implications in your remark are.

Matheinste

 

[Al68]

Hello Al68.

This is purely a reply to your remark in #40.

To disagree with a point of view is absolutely acceptable as is my right to put forward my view, which on this subject is very much mainstream. I know you do not mean to criticize religion as such ( but it would not bother me if you do ) but the implication is that answers i have given to the best of my admittedly limited ability are given as a matter of unthinking, blind faith, in the manner of a crusade. My answers and those of many others on this question may in your eyes be incorrect but do not deserve to be called religious with all the underlying disparaging connotations of the word. The word itself is of course not offensive but its implications in your remark are.

Matheinste

Matheinste, my reference to religion was not to imply unthinking or blind faith, just to point out that in science, unlike religion, "well known and accepted" does not equal "correct and not subject to revision". Of course, I realized my remark was out of line when I reread it after posting. That's why I deleted it right away. My apologies.

Al

P.S. I don't believe your answer on this question is incorrect, just incomplete.

 

[phyti]

I can't fully understand why a person who makes a journey into space in a high-speed rocket will return home to find his age less than an identical twin who stayed on Earth. It makes since for the twin who stayed on earth, but for the twin who traveled into space, he sees himself at rest and sees Earth moving at constant velocity. so he should feel that time on Earth is moving slower than his time, this way he will find his twin younger than him not older. Please tell me what I am missing to help me understand this thought experiment.
Thank you.

You are missing the fact that the traveling twin, his clock, and everything in his space can/ship is functioning at a slower rate, therefore he ages more slowly. Referring to the drawing, he records 8 Earth years during his 7, while moving at .6c.

Jumping time is a misconception due to the instantaneous reversal of direction. In reality, it would be a gradual process and the Earth signal frequency would increase accordingly.

 

[neopolitan]

I haven't been able to decrypt what you're saying here. What two events? I assume that one is (20,16) in A's frame, but what's the other one? Also, a world line is a curve that represents the motion of an object. You want to imagine something moving from (20,16) to some other event? Why?

The two events are "A" and "B" at the time when "B" turns around. Since there is no separation speed there should be no simultaneity disagreement, both should (with the application of SR calculations) agree that these events are (t=20,x=0), "A"'s position according to "A" and (12,0), "B"'s position according to "B". (Couplets are: (20,0), (20,16) and either (12,-9.6), (12,0) or (12,0),(12,9.6) depending on "B"'s choice of origin since both choices have some merit even if the former is standard.)

The something moving from one event to another event is the information which I have discussed previously, a signal. A signal moving from "A" to "B". My contention is that even if "B" undergoes a change of frame, the calculations which "B" uses should not be used in such a way to indicate that this signal sent by "A" at (20,0) was simultaneous with any event at "B" earlier than (12,0).

That is a consequence of the implication in your diagram (http://web.comhem.se/~u87325397/Twins.PNG" ) that according to "B", "A" suddenly ages 25.6 years.

With the information that "B" has to hand, there is no need to make such a ridiculous claim - even if it may be standard simultaneity fare.

You're much too focused on those signals. I don't see how they are relevant at all. You seem to think that they somehow forbid us from using the inertial frame associated with B's return trip, but you haven't given us a reason for that.

The signals are an attempt to get you to understand that it is unreasonable and unnecessary to state that "A" suddenly ages 25.6 years. The signals are also representative of the information flow from "A" to "B". There is real information about "A" which is accessible to "B", but it is speed limited so "B" will never get it instantaneously. The best "B" can do is use the information received to make projections which are valid for the prevailing frame.

The way you talk about these things is pretty strange to me. How is it not a characteristic of the real universe that two different global coordinate systems disagree about stuff? And what "discontinuity" are you talking about? What function is supposed to be discontinous? All we have here are (at least) two inertial frames that describe things differently.

This sudden ageing of 25.6 years is the discontinuity that I am referring to.

The 25.6 years is based on realigning the frames with the end result, so that "A" is a nice 40 years old when "B" gets there.

However, it is not real. The clock I discussed with Matheinste won't suddenly scroll forward from 7.2 to 32.8 years. And here is why not ... the 32.8 year figure is based on "A" not moving at all during the 20 years. That means that the clock would have to somehow predict the future.

This is totally separate from the issue that the calculation behind the 32.8 years is based on a combination of situations, the bastard son of two frames, and that the calculation totally ignores how information flows in the universe.

"B" should, at the turnaround, make a projection that "A" has aged a total of 20 years. Not 32.8 years.

cheers,

neopolitan

PS Perhaps you might like to create a chart which maps the "A" events which are, according to "B", simultaneous with "B" events. Make all the events ageing events, ie '"B" has aged x days, this is simultaneous, according to "B" with "A" having aged y days' and plot y against x.

In my version, there will be a straight line (with a little bump in the middle if I am going to be pedantic), since "B" effectively maintains the same speed the whole time (0.8c) and I will not be ignoring the information that "B" receives.

In your version, there will be three straight lines - (0,0) to (12,7.2), (12,7.2) to (12,32.8) and (12,32.8) to (24,40).

Which sounds more representative of a realworld situation?

PPS phyti has approximately the right sort of diagram. His figures are for a shorter trip and show the situation in a different way, but at each end of phyti's diagonal lines are the simultaneous events which I suggest you chart, Fredrik.

 

[Al68]

"B" should, at the turnaround, make a projection that "A" has aged a total of 20 years. Not 32.8 years.

I think everyone agrees with that, assuming that you mean when the ship comes to momentary rest with earth. That's when Earth's clock will read 20 yrs. The 32.8 yrs is what the Earth clock will read according to "B" after the acceleration and the ship is headed back. The Earth clock readings of 7.2, 20, and 32.8 yrs are only a "jump" in time seen by the ship if we consider the turnaround instantaneous. We could just as easily say Earth's clock jumps from 7.2 to 20 during an instantaneous deceleration, then from 20 to 32.8 during an instantaneous acceleration. And we could break it down into segments as small as we like, and for each segment during the acceleration, Earth's clock will advance more than the ship's clock according to "B" (or co-moving inertial observers).

Al

 

[Fredrik]

The two events are "A" and "B" at the time when "B" turns around.

That's what I thought you meant, but you said weird things like that there's a world line that connects them.

...that according to "B", "A" suddenly ages 25.6 years.

With the information that "B" has to hand, there is no need to make such a ridiculous claim - even if it may be standard simultaneity fare.

You seem to be missing the point of the twin paradox. The false conclusion that there is a paradox was found by considering both twins' points of view, and doing it incorrectly. You can't solve that problem by saying that B's point of view is unnecessary. A resolution must explain why B is younger when they meet again, even though B can say (correctly) that his brother is aging at a slower rate at any point during the trip except the turnaround

This sudden ageing of 25.6 years is the discontinuity that I am referring to.

I know. I didn't ask because I needed to hear that part of the answer. I asked what function is supposed to be discontinuous to get you to think about how you can't even specify that function without explicitly mentioning the switch from one inertial frame to another.

The clock I discussed with Matheinste won't suddenly scroll forward from 7.2 to 32.8 years. And here is why not ... the 32.8 year figure is based on "A" not moving at all during the 20 years. That means that the clock would have to somehow predict the future.

No, it just has to keep track of the rocket's position and velocity at all times. The rest is a straightforward calculation. And yes, if you slow down on the way back, it will go backwards. If you programmed it right, that is.

...the calculation totally ignores how information flows in the universe.

It should ignore that. Think about what a coordinate system is. It's just a function that assigns numbers (coordinates) to events.

"B" should, at the turnaround, make a projection that "A" has aged a total of 20 years. Not 32.8 years.

Half way through the turnaround, yes. (When A and B have the same velocity). But not when the turnaround is complete.

PS Perhaps you might like to create a chart which maps the "A" events which are, according to "B", simultaneous with "B" events. Make all the events ageing events, ie '"B" has aged x days, this is simultaneous, according to "B" with "A" having aged y days' and plot y against x.

Check out for example the second text box from the bottom in my spacetime diagram. It states both what B is experiencing at the event where the blue line intersects B's world line, and what A is experiencing at the event where the blue line intersects A's world line. (Reminder: the blue line is a simultaneity line of B).

...at each end of phyti's diagonal lines are the simultaneous events which I suggest you chart, Fredrik.

Simultaneous? They aren't simultaneous unless the signals are transmitted by tachyons moving at 1.25c.

This discussion doesn't seem to be going anywhere, so I might withdraw from it after this.

 

[Al68]

The clock I discussed with Matheinste won't suddenly scroll forward from 7.2 to 32.8 years.

No, but if it works the way I described earlier, the time (according to the "normal" clock "B") it would take for this special clock to advance from 7.2 to 32.8 yrs would be inversely proportional to the rate of acceleration. Since the rate of acceleration can't realistically equal infinity, the special clock won't "jump" time, some time will elapse on the "B" clock while the special clock advances from 7.2 to 32.8 yrs smoothly (assuming the rate of acceleration is constant). And this special clock would not need to predict the future, if the ship stops accelerating at any time, the clock would sense the lack of acceleration and resume running slow by the gamma factor compared to clock "B".

Al

 

[yogi]

I have a different take in the twins thing - first of all, solve the problem by reducing it to two one way trips - and double the result to get the total age difference - so on the outward bound one way trip use two clocks, one on earth, one at the destination - they are synchronized in the Earth frame and always read the same - the distance to the traget is vt where t is the lapsed time in the earth-target frame, and ct is the temporal distance the Earth frame has moved during the one way trip - so the space time path followed by the traveler is a composite of the the space and temporal increments - during this interval, the travelers clock logs a time t' and the temporal distance for the traveler is ct'

so (ct)^2 = (ct')^2 + (vt)^2 from which you get Gamma and the amount of time that difference between the two frames, that is t' = Gamma(t)

You don't have to send signals, you don't need to get involved with turn around accelerations, and it doesn't make any difference whether the earth-target frame is moving or the traveler - all you need is Minkowski orthogonality of space and time and realize that each frame has taken a different space time path so since one frame has moved only in time and the other frame has experienced both a space and time increment - the intervals will be equal, but the components of the interval in each case will be different

 

[neopolitan]

Fredrik and Al,

Perhaps we can break the deadlock if you explicity state the assumptions of your conclusion(s).

Here are the assumptions of my conclusion, as far as I can tell:

• "B" keeps track of signals from "A".
  
• "B" is aware of accelerations undergone.
  
• "B" assumes under the conditions of the experiment that "A" does not undergo any significant accelerations.
  
• "B" is aware of relativity and uses that knowledge appropriately in calculations.
  
• "B" calculates "A"'s age on an ongoing basis.
  
• Most significantly: "B" does not at any time assume that the prevailing inertial frame is eternally valid (due to awareness of accelerations).

I do think that the last is important, and is possibly the sticking point.

Please present your assumptions, then have a go at mine, if you so wish.

cheers,

neopolitan

PS as yogi points out, signals are not really strictly necessary, assumption 4 would eliminate the need for assumption 1. But I keep the signals to force some real universe thinking in the example.

 

[Al68]

I asked what function is supposed to be discontinuous to get you to think about how you can't even specify that function without explicitly mentioning the switch from one inertial frame to another.

I think he might be referring to a function describing clock "A" in B's frame, which would have a discontinuity if we assume the rate of acceleration to be infinity, and the elapsed time on clock "B" to be zero during the turnaround. The Earth's clock would read 7.2 then 32.8 yrs, but not values in between. But this is just an artifact of treating the turnaround as instantaneous. If we say the turnaround is just "near" instantaneous, then the Earth's clock will advance from 7.2 to 32.8 very quickly according to clock "B", but it will read all times in between with no discontinuity.

Al

 

[Al68]

Fredrik and Al,

Perhaps we can break the deadlock if you explicity state the assumptions of your conclusion(s).

Here are the assumptions of my conclusion, as far as I can tell:

• "B" keeps track of signals from "A".
  
• "B" is aware of accelerations undergone.
  
• "B" assumes under the conditions of the experiment that "A" does not undergo any significant accelerations.
  
• "B" is aware of relativity and uses that knowledge appropriately in calculations.
  
• "B" calculates "A"'s age on an ongoing basis.
  
• Most significantly: "B" does not at any time assume that the prevailing inertial frame is eternally valid (due to awareness of accelerations).

I do think that the last is important, and is possibly the sticking point.

Please present your assumptions, then have a go at mine, if you so wish.

cheers,

neopolitan

Hi neopolitan,

I actually see what you're saying, I think. I answered your question about how a clock would work that is supposed to show Earth time on the ship. And I agree with you that there is a discontinuity associated with instantaneous turnaround, but it's not real because instantaneous turnaround is not possible. Such a clock would simply run fast relative to clock "B" during any acceleration (toward earth). And it would always show the time on Earth simultaneous with any moment on the ship (or a co-moving inertial frame).

I would use your assumptions, although #1 wouldn't be necessary.

Al