Understanding the Twin Paradox: Exploring Contradictions in Special Relativity

[Al68]

Hello Al68.

Quote:-

--consider that the Earth accelerated relative to the ship, and have the Earth twin age less,---

The Earth twin would age less if the Earth accelerated and the ship did not, but in these scenarios it is not usually the Earth that accelerates because this is not a very realisteic option and so would not help to pose the twins "paradox". As normally stated the twins "paradox" is meant to show a real difference in age from a realistic scenerio and present this as a paradox, which we all know can be resolved with SR. If we made the Earth in effect the "traveller" then we use an impractical and unrealistic, though not impossible scenario which lessens the effect of the "paradox" as usually given by making it wholly improbable anyway to the learner of SR at who it is aimed as an example for study.

Quote:--

--SR only cares about coordinate acceleration and not "proper" acceleration.---

The whole point here is that one accelerates and the other does not. I don't know how you define proper acceleration and coordinate acceleration, but what we care about in this scenario is absolute acceleration as detected by an accelerometer. Only one, the ship or the Earth experiences this in our present example. As normally proposed it is the ship which experiences the acceleration. If the Earth experiences it and the ship does not then,as you say, the result is reversed and the Earth twin ages less.

I don't think you are missing anything you may just have the wrong idea about acceleration. Acceleration is absolute and physical and coordinate independent.

Matheinste.

By coordinate acceleration I mean change in relative velocity. By proper acceleration I mean as measured by an accelerometer, like you're talking about. But the biggest thing I see that "causes" the ship's twin to age less in the twins paradox is the simple fact that he didn't travel as far relative to Earth as the Earth twin did relative to the ship, each as measured in his own frame. Simple common sense tells me that at 0.8c, a shorter trip equals less elapsed time (t=d/v in any frame). The resolution's conclusion just follows this stipulation. It doesn't resolve the big picture "clock paradox" for scenarios which may be different. Some of the resolutions say that acceleration is the key to the problem, but they claim this as an axiom without showing why this is true. After all, the Earth does accelerate (change velocity) relative to the ship. By definition coordinate acceleration equals change in relative velocity per unit time.

Just as a side note, most on this board are probably aware that Einstein believed the "clock paradox" was unresolvable in SR. He was fully aware of the (now) common resolutions and rejected them. Any thoughts on why he believed this?

Thanks,
Al

 

[Al68]

I don't get why you think you can replace one problem with another that doesn't look anything like the original.

The original problem specifices three events and three frames. If you change any of that, it's a different problem.

Well, because I'm interested in the big picture "clock paradox", not just a single scenario whose conclusion only applies to narrowly defined conditions.

And I'm curious why Einstein thought it was unresolvable in SR, even after being fully aware of the now common resolutions. Since the reason probably isn't that Einstein "didn't understand SR", "didn't understand how simultaneity works", etc.

I'm sure that when Einstein presented the "clock paradox" and said that the ship's twin should be able to claim that the Earth twin aged less, he obviously didn't mean by using the same specified three events and three frames.

Thanks,
Al

 

[matheinste]

Hello Al68.

Quote:-

---Some of the resolutions say that acceleration is the key to the problem, but they claim this as an axiom without showing why this is true----

To start with this point. To get back to the smaller picture let us talk about the role of acceleration in the standard twin paradox. This has been explained very well in many threads in this forum. I may be able to rephrase a few things but better people than me seem not have been able to make it clear to you so I don’t hold out much hope but I will try.

Take a ship and the Earth at rest relative to each other. Now in all that follows the words ship and Earth wherever they appear in the text can be interchanged. Because they are not in relative motion with regards each other let us for our purposes consider them at rest. In a standard spacetime diagram we can represent their common worldline as a vertical line, corresponding to the time axis.

Let the ship moves off in any direction, for our convenience we will show this displacement as being along the horizontal axis in our spacetime diagram. It does not matter whether to the left or to the right. The worldline of the ship will therefore be a straight line let us say to the right, and its angle upwards to the horizontal will be proportional to its velocity which may be anything less than c.
After some time the ship halts and heads back to the left at some speed making again an tilted upward straight line this time to the right to the left until it reaches the Earth again. We will assume that the initial acceleration, the deceleration at the turnaround point, the acceleration back towards Earth and the final deceleration at Earth to be very high for a very short period, this of course being only a thought experiment. We do this to make the time spent in the acceleration phases as short as possible, in theory infinitesimally small. It has in fact no effect on the outcome.

In spacetime the greater the spacetime distance traversed, and on a spacetime diagram the longer the line(s) representing the path taken, the SHORTER the proper time experienced by anything traveling along this path. So the earth, traveling in a straight line, travels the shortest distance and therefore its clocks record the LONGEST POSSIBLE time. This is contrary to common sense but is accounted for by the time dimension being involved in the calculation. I would show the mathematics but I am not yet proficient in latex. Remember spactime distance is not the same thing as spatial distance. The actual speeds and distances are immaterial, the principle is simply that the longest spacetime diagram path has the shortest proper time, that is shows less ageing. Proper time is of course what the ship and Earth experience themselves.

Anyhow to make the path deviate from the vertical axis of the spacetime diagram requires a change of velocity, that is an acceleration. Every time the spacetime path changes direction, as it must in this scenario, an acceleration is involved. It is the change in path that causes the difference in proper times and for this change in path an acceleration is needed. The acceleration does not cause the clock differences but is needed to alter the spacetime path.

The point of turnaround of the ship can be anywhere but in the scenario we have chosen it is somewhere to the right on the spacetime diagram to the right of the vertical worldline of the earth.

There have been authors who have claimed that GR is necessary to resolve the paradox but it is generally accepted that GR is not needed. As to Einstein, what can I say, I don’t know the facts.

This has all been posted before and anyway it is the best I can do and only hope it is accurate. If anybody spots any mistakes would they please point them out and we can put them right rather than let Al think we are in disagreement over the point.

Matheinste.

 

[Al68]

Hello Al68.

Quote:-

---Some of the resolutions say that acceleration is the key to the problem, but they claim this as an axiom without showing why this is true----

To start with this point. To get back to the smaller picture let us talk about the role of acceleration in the standard twin paradox. This has been explained very well in many threads in this forum. I may be able to rephrase a few things but better people than me seem not have been able to make it clear to you so I don’t hold out much hope but I will try.
..
Let the ship moves off in any direction, for our convenience we will show this displacement as being along the horizontal axis in our spacetime diagram. It does not matter whether to the left or to the right. The worldline of the ship will therefore be a straight line let us say to the right, and its angle upwards to the horizontal will be proportional to its velocity which may be anything less than c.

Thanks for your reply. Your post is very clear, and everything you said is clear to me. It's what everyone is omitting that I'm asking about. Why must we attribute the deviation from the initial common worldline to the observer who underwent proper acceleration? Why can't we just as rightly (according to SR) attribute this deviation to the other observer and draw the Earth's worldline to the right when it's velocity relative to the ship changes (coordinate acceleration)?

Why can't we just randomly choose which observer to draw vertically on the spacetime diagram? I know that sounds like a silly question, but that very question is part of the reason Einstein pursued GR.

Maybe the answer is so intuitively obvious that nobody considers it necessary to mention, but when I ignore my intuition and use only Einstein's 1905 SR paper, I can't find the answer. (And neither could he).

I understand what happens after we decide who's worldline to draw vertically, that's just simple math. But it's that decision that seems, for lack of a better word, Newtonian.

I mentioned Mach's principle in an earlier post, but nothing came of it. I realize that when the ship fires it's thrusters, Earth's velocity changes relative only to the ship, while the ship's velocity changes relative to earth, and relative to every other single body in the universe. This is obviously not considered important by anyone, since nobody brought it up. Is it relevant?

There have been authors who have claimed that GR is necessary to resolve the paradox but it is generally accepted that GR is not needed. As to Einstein, what can I say, I don’t know the facts.

Well, he was one of those authors, although most consider his GR resolution to be flawed.

Thanks,
Al

 

[Mentz114]

Al68,

Why can't we just randomly choose which observer to draw vertically on the spacetime diagram? I know that sounds like a silly question, but that very question is part of the reason Einstein pursued GR.

Because one of them has a curved worldline which cannot be transformed into a straight vertical line by Lorentz transformation.

It sort of follows that to handle force-free acceleration, you need to curve the axes.

M

 

[Al68]

Al68,

Why can't we just randomly choose which observer to draw vertically on the spacetime diagram? I know that sounds like a silly question, but that very question is part of the reason Einstein pursued GR.

Because one of them has a curved worldline which cannot be transformed into a straight vertical line by Lorentz transformation.

I'm asking about the decision that led to us to consider that one of them has a curved worldline, not how we treat the problem afterward.

I don't know how else to put it.

Thanks,
Al

 

[matheinste]

Hello again Al68.

Quote:-

----Why must we attribute the deviation from the initial common worldline to the observer who underwent proper acceleration? Why can't we just as rightly (according to SR) attribute this deviation to the other observer and draw the Earth's worldline to the right when it's velocity relative to the ship changes (coordinate acceleration)?

Why can't we just randomly choose which observer to draw vertically on the spacetime diagram? I know that sounds like a silly question, but that very question is part of the reason Einstein pursued GR.----

A vertical worldline in a spacetime diagram represents a body in inertial motion, for our purposes this is the same as being at rest. If a body accelerates it undergoes a spatial displacement with changing time which would have to be represented by a sloping line to the left or right in the spacetime diagram.

I do not know for sure, this being a quick answer, but I suppose with some corresponding alterations to the slope of the spatial displacement axis, that is horizontal axis, and a corresponding alteration to the other objects worldline, some sort of modified spacetime diagram may be possible in which an accelerated observer may be given a vertical worldline. I would be interested in any answers to this which may be forthcoming. But why complicate things. The outcome remains the same as far as time differentials are concerned.

As for Mach’s principle, I think there would be no difference I the posing of or resolution of the paradox if nothing else existed in the universe. Acceleration would still be absolute.

I wrote this while before Mentz114 posted his last reply. He says all that needs to be said on that point and i suspect that what he has said about the Lorentz transformation answers my question about a modified spacetime diagram. Not possible ??

Mateinste.

 

[matheinste]

Hello Al68

In answer to the curved worldline—

A spacetime diagram plots distance or displacement against time. For constant velocity this plot or graph is a straight line. For accelerated motion it is not.

Matheinste.

 

[Fredrik]

Well, because I'm interested in the big picture "clock paradox", not just a single scenario whose conclusion only applies to narrowly defined conditions.

It's instructive to consider a scenario defined by three time-like straight lines chosen at random (except that no two of them are parallel), where we imagine physical observers traveling on those world lines and comparing their clocks at the events where two lines meet. This scenario contains everything that's relevant from the standard twin "paradox". (Note that there's no acceleration).

And I'm curious why Einstein thought it was unresolvable in SR, even after being fully aware of the now common resolutions. Since the reason probably isn't that Einstein "didn't understand SR", "didn't understand how simultaneity works", etc.

A person who understands SR would never believe that this is unresolvable in SR, and I do believe that Einstein understood SR.

I'm sure that when Einstein presented the "clock paradox" and said that the ship's twin should be able to claim that the Earth twin aged less, he obviously didn't mean by using the same specified three events and three frames.

I don't know how he presented it (or even that he did), so I can only assume that if he said anything like that, he must have meant that a very naive application of the time dilation formulas which completely ignores any other effects due to relativity of simultaneity leads to the result that the Earth twin aged less (and also that the Earth twin aged more).

 

[Fredrik]

Why must we attribute the deviation from the initial common worldline to the observer who underwent proper acceleration? Why can't we just as rightly (according to SR) attribute this deviation to the other observer and draw the Earth's worldline to the right when it's velocity relative to the ship changes (coordinate acceleration)?

Why can't we just randomly choose which observer to draw vertically on the spacetime diagram? I know that sounds like a silly question, but that very question is part of the reason Einstein pursued GR.

I think you're missing an important thing here. We're not trying to find the best possible theory to handle this scenario. We're just trying to find out what special relativity says about it. And it's clear from any formulation of special relativity that straight lines have a very special significance.

Maybe the answer is so intuitively obvious that nobody considers it necessary to mention,

It's not obvious that there are no better theories, but we're not looking for a better theory. The twin paradox is about finding the mistake in an incorrect application of the rules of SR.

I understand what happens after we decide who's worldline to draw vertically, that's just simple math. But it's that decision that seems, for lack of a better word, Newtonian.

"Minkowskian", or "special relativistic" would be pretty good ways to say it.

I mentioned Mach's principle in an earlier post, but nothing came of it. I realize that when the ship fires it's thrusters, Earth's velocity changes relative only to the ship, while the ship's velocity changes relative to earth, and relative to every other single body in the universe. This is obviously not considered important by anyone, since nobody brought it up. Is it relevant?

Mach's principle may be important to someone who knows SR and is trying to find a theory that includes gravity. Such a researcher might decide early on to only consider theories that satisfy some version of Mach's principle and throw away all other theories without further consideration. This is a lot like deciding to only consider theories where coordinate transformations between inertial frames with a common origin preserve the light-cone at the origin, when trying to find SR. (Only not as powerful, because light-cone preservation is almost the entire theory of SR).

 

[Mentz114]

Al68,

I'm asking about the decision that led to us to consider that one of them has a curved worldline, not how we treat the problem afterward.

I don't know how else to put it.

I thought I'd answered that. We didn't choose which one to treat inertially, and which one non-inertially.The twin who travels non-inertially nominates themselves. Acceleration is absolute.

M

[edit] This is what Matheinste and Fredrik are saying also, I think.

 

[Al68]

Hello Al68

In answer to the curved worldline—

A spacetime diagram plots distance or displacement against time. For constant velocity this plot or graph is a straight line. For accelerated motion it is not.

Matheinste.

Why not?

 

[Al68]

A person who understands SR would never believe that this is unresolvable in SR, and I do believe that Einstein understood SR.

Einstein did consider it unresolvable in SR. And said so, and tried to resolve it in GR.

 

[matheinste]

Herllo Al68.

The plot or graph of constant velocity against time is a straight line. For accelerated motion it is not.

You ask why not!

This is basic mathematics and physics of motion. If you do not know this you really should learn it as it is at a very basic level and if you do not understand this you have no chance of understanding anything in physics involving motion.

Matheinste.

 

[Al68]

Al68,


I thought I'd answered that. We didn't choose which one to treat inertially, and which one non-inertially.The twin who travels non-inertially nominates themselves. Acceleration is absolute.

M

[edit] This is what Matheinste and Fredrik are saying also, I think.

Sure proper acceleration is absolute, but coordinate acceleration is not.

And I know we didn't choose which twin travels non-inertially, but we did choose to treat the inertial frame differently.

 

[matheinste]

Hello Al68.

Quote

---And I know we didn't choose which twin travels non-inertially, but we did choose to treat the inertial frame differently. ----

Because it is different.

Matheinste.

 

[Mentz114]

Al68,

And I know we didn't choose which twin travels non-inertially, but we did choose to treat the inertial frame differently.

It is different.
M

 

[matheinste]

Hello Al68.

Regarding proper acceleration and coordinate acceleration, something i know little about, i believe that in flat spacetime, the sort of spacetime that SR deals with, they are the same and so coordinate acceleration is not relevant. A spacetime diagram's axes, that is coordinate system, are linear indicating a flat spacetime.

Matheinste.

 

[Fredrik]

The twin who travels non-inertially nominates themselves. Acceleration is absolute.

I agree, but I would like to add that you can see that without thinking in terms of acceleration. Consider my example with three non-parallel time-like lines, chosen at random. Note that all 3 observers in this case will agree which of the three events is the latest, and also which is the earliest. Therefore, they will also agree which of the three events corresponds to the turnaround event. There's no acceleration in this scenario, but it's still clear that the funny thing that happens with simultaneity at the "turnaround" event is what resolves the naive paradox.

Why not?

Because we're talking about special relativity, and that theory was constructed to satisfy the requirement that coordinate changes between inertial frames take straight lines to straight lines. This isn't mentioned explicitly, but Einstein's "postulates" don't make sense unless this is taken to be a part of what they mean.

In other words, the the world line of an inertial observer is straight by definition, in the theory that was used incorrectly to find the "paradox".

Einstein did consider it unresolvable in SR. And said so, and tried to resolve it in GR.

I think it's more likely that you have misunderstood what he said. SR is just the theory of Minkowski space, which is just $\mathbb R^4$ with some functions. Both the functions and $\mathbb R^4$ can be explicitly constructed from the axioms of set theory. Therefore, if SR really contains a paradox, all of mathematics falls with it. Maybe not all of it, but we definitely lose the integers, so bye bye 1+1=2.

I have explained this lots of times in this forum. I think the fact that almost no one understands this means that there's something very wrong with the way SR is presented in all the standard texts.

 

[Fredrik]

Regarding proper acceleration and coordinate acceleration, something i know little about, i believe that in flat spacetime, the sort of spacetime that SR deals with, they are the same and so coordinate acceleration is not relevant.

They are the same in any inertial frame on Minkowski space, but we can easily imagine a global coordinate system such that an accelerating object e.g. has x=0 at all times.

I learned recently that some authors actually consider such a coordinate system a part of GR instead of a part of SR. I find that quite bizarre.

 

[Fredrik]

Because it is different.

It is different.

Agreed. And to provide one more detail: It's different...in special relativity, which is the theory we're working with here.

 

[Al68]

Herllo Al68.

The plot or graph of constant velocity against time is a straight line. For accelerated motion it is not.

You ask why not!

This is basic mathematics and physics of motion. If you do not know this you really should learn it as it is at a very basic level and if you do not understand this you have no chance of understanding anything in physics involving motion.

Matheinste.

I should give up now, but I can't resist pointing out that the Earth's velocity relative to the ship is not constant. So the coordinate acceleration of the Earth relative to the ship is not zero.

Thanks,
Al

 

[matheinste]

Hello Al68

Quote:-

---So the coordinate acceleration of the Earth relative to the ship is not zero.---

In flat spacetime coordinate acceleration and proper acceleration are the same. We are dealing with flat spacetime. If the Earth's proper acceleration is zero then its coordinate acceleration is zero. This is the case if we choose the ship to be the traveller. In this case an accelerometer on the Earth will show no acceleration so its acceleration is zero.

Its my bedtime. goodnight.

Matheinste.

 

[Al68]

In flat spacetime coordinate acceleration and proper acceleration are the same.

Well, coordinate acceleration could be defined relative to a reference frame co-moving with the ship's clock. Proper acceleration cannot.

Thanks,
Al

 

[Fredrik]

I should give up now, but I can't resist pointing out that the Earth's velocity relative to the ship is not constant. So the coordinate acceleration of the Earth relative to the ship is not zero.

This is true (if you're talking about a coordinate system with the ship at x=0 both before and after the turnaround), but as I said in #54 (in a different way), Minkowski space was chosen as the space-time for SR because it makes it obvious that a coordinate transformation from one inertial frame to another takes straight lines to straight lines. I think you will find that your "ship frame" (which you still haven't defined fully) will violate this requirement, no matter how you finish its definition.

Why do I say that you haven't defined the ship's frame? Because its world line only defines the time axis. You haven't defined a way to assign time coordinates to events that aren't on the time axis.

 

[phyti]

Al68;

But the biggest thing I see that "causes" the ship's twin to
age less in the twins paradox is the simple fact that he didn't
travel as far relative to Earth as the Earth twin did relative
to the ship, each as measured in his own frame.
Simple common sense tells me that at 0.8c, a shorter trip
equals less elapsed time (t=d/v in any frame). The resolution's
conclusion just follows this stipulation.

The ship clock moving at .8c registers less time, but still
travels 16 lyrs. The ship twin assumes the distance is shorter,
as an explanation for his early arrival (6 yr instead of 10).
Again, this is not magic, his space journey does not alter the
known laws of physics, nor physical processes in the rest of
the universe.

It doesn't resolve the big picture "clock paradox" for
scenarios which may be different. Some of the resolutions say
that acceleration is the key to the problem, but they claim
this as an axiom without showing why this is true. After all,
the Earth does accelerate (change velocity) relative to the ship.

The Earth acceleration is perceived motion by the ship twin,
not a motion with a physical cause, therefore not symmetrical.
This is a key element in resolving the 'paradox issues'.
If the ship twin chooses to deny his own motion, and it's the
rest of the universe that starts moving, then a star 1000 lyr
distant would have had to begin moving 1000 yr ago to
accommodate his perception of the universe instantaneously
moving in the opposite direction!
This is nonsense and one reason why the motion is not symmetrical.
Another is conservation of energy. The amount of energy used to
move the ship would not be sufficient to move the rest of the
universe in the opposite direction with the same velocity! In
fact there is no available energy to move the universe.
If you perform these gedanken/thought experiments in isolation,
two bodies in space, a train and a station, an observer in a
moving box (with no windows), etc., you can invent all types of
paradoxes, because you don't have the additional information
that could resolve them.

matheinste;

The actual speeds and distances are immaterial, the principle
is simply that the longest spacetime diagram path has the
shortest proper time, that is shows less ageing. Proper time is
of course what the ship and Earth experience themselves.

The speed/velocity is material because the longest path was
achieved with greater speed, which is what slows the clock
rate. Examine the time dilation equation for 'v/c', the clock
rate is a function of object velocity to light velocity.

 

[matheinste]

Hello phyti

Quote:-

---The speed/velocity is material because the longest path was
achieved with greater speed, which is what slows the clock
rate. Examine the time dilation equation for 'v/c', the clock
rate is a function of object velocity to light velocity.----

I only said this to make the point that specific figures were not reacquired to show the principle of diffential time lapses. Any appropriate velocities and distances would do for the purpose of an example.

Matheinste.

 

[matheinste]

Hello phyti

This is a correction to my last post in which i quoted the wrong paragraph.

Quote:-

---The actual speeds and distances are immaterial, the principle
is simply that the longest spacetime diagram path has the
shortest proper time, that is shows less ageing. Proper time is
of course what the ship and Earth experience themselves.------

I only said this to make the point that specific figures were not reacquired to show the principle of diffential time lapses. Any appropriate velocities and distances would do for the purpose of an example.

Matheinste.

 

[phyti]

Matheinste;
I agree with your principle of longest path, least time.
I only mention the other factors to explain 'why' to those who might ask, to counter all the flim-flam, house of mirrors ideas that are still prevalent today, after 100 years.

 

[Al68]

Thanks everyone for the responses, some things are clearer.

There's still one thing that I don't have worked out. If we had real acceleration instead of instantaneous, it would be obvious that, from the ship's twin's view, the Earth and space station do not stay at rest with each other. So as the ship starts slowing down, the Earth and space station are getting farther apart. If the ship decelerates at 1 G proper acceleration, is it important that his coordinate velocity and acceleration relative to Earth will be different than relative to the space station?

I haven't seen this addressed in textbooks or resolutions on the net, since they all either use instantaneous acceleration, or just split the ship frames to keep the math simple. Is there a resolution on the net that shows the math for realistic acceleration?

Thanks,
Al

 

[matheinste]

Hello Al68.

Quote:-

---I haven't seen this addressed in textbooks or resolutions on the net, since they all either use instantaneous acceleration, or just split the ship frames to keep the math simple. Is there a resolution on the net that shows the math for realistic acceleration?-----

This is not really a direct answer to your question but just a few, hopefully relevant, remarks.----

Many people seem unsure whether or not the actual acceleration affects the clock rate of the accelerated twin. Having instantaneous arbitrarily high acceleration is an attempt to reduce, in the limit, the time spent in the acceleration phases to zero and so remove any possible effects this way. These accelerations are of course unrealistic in practice but the theory is of course not altered.

As far as I am aware acceleration has no direct effect on clock rates, we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration. This makes these instantaneous high accelerations unnecessary. We could use realistic acceleration rates where the acceleration phases occupy a considerable part of the journey time and integrate the instantaneous clock rates over these phases. This is because in accelerated motion, at any instant the clock rate is the same as that of an inertially moving clock with the same velocity at that instant and integration allows us to sum the accumulated time. We in effect do the same thing for constant velocity but it is a lot simpler. However the time periods involved then become rather long if you wish to see any marked age difference.

Matheinste.

 

[Al68]

Hello Al68.

Quote:-

---I haven't seen this addressed in textbooks or resolutions on the net, since they all either use instantaneous acceleration, or just split the ship frames to keep the math simple. Is there a resolution on the net that shows the math for realistic acceleration?-----

This is not really a direct answer to your question but just a few, hopefully relevant, remarks.----

Many people seem unsure whether or not the actual acceleration affects the clock rate of the accelerated twin. Having instantaneous arbitrarily high acceleration is an attempt to reduce, in the limit, the time spent in the acceleration phases to zero and so remove any possible effects this way. These accelerations are of course unrealistic in practice but the theory is of course not altered.

As far as I am aware acceleration has no direct effect on clock rates, we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration. This makes these instantaneous high accelerations unnecessary. We could use realistic acceleration rates where the acceleration phases occupy a considerable part of the journey time and integrate the instantaneous clock rates over these phases. This is because in accelerated motion, at any instant the clock rate is the same as that of an inertially moving clock with the same velocity at that instant and integration allows us to sum the accumulated time. We in effect do the same thing for constant velocity but it is a lot simpler. However the time periods involved then become rather long if you wish to see any marked age difference.

Matheinste.

I was really more interested in the coordinate position of Earth in the ship's frame during the acceleration, since the coordinate distance to Earth will "length expand" during the acceleration. And it seems like in effect, although the ship never exceeds c while moving inertially, it would exceed c relative to Earth during deceleration. And the ship's coordinate acceleration relative to Earth would not equal its coordinate acceleration relative to the space station. And it seems like earth, the space station, and the ship would not agree on the rate of deceleration.

Thanks,
Al

 

[Al68]

The ship clock moving at .8c registers less time, but still
travels 16 lyrs. The ship twin assumes the distance is shorter,
as an explanation for his early arrival (6 yr instead of 10).

Is this pretty much a consensus view?

The Earth acceleration is perceived motion by the ship twin,
not a motion with a physical cause, therefore not symmetrical.
This is a key element in resolving the 'paradox issues'.
If the ship twin chooses to deny his own motion, and it's the
rest of the universe that starts moving, then a star 1000 lyr
distant would have had to begin moving 1000 yr ago to
accommodate his perception of the universe instantaneously
moving in the opposite direction!
This is nonsense and one reason why the motion is not symmetrical.
Another is conservation of energy. The amount of energy used to
move the ship would not be sufficient to move the rest of the
universe in the opposite direction with the same velocity! In
fact there is no available energy to move the universe.
If you perform these gedanken/thought experiments in isolation,
two bodies in space, a train and a station, an observer in a
moving box (with no windows), etc., you can invent all types of
paradoxes, because you don't have the additional information
that could resolve them.

What do you think of Mach's principle that were it not for the mass in the rest of the universe, and an experiment like this were performed in isolation, the ship's twin would feel no acceleration, and inertia would not even exist?

Thanks,
Al

 

[matheinste]

Hello Al68.

What system of coordinates do you want to use and how would you depict the objects relative to it.

Matheinste.

 

[Fredrik]

There's still one thing that I don't have worked out. If we had real acceleration instead of instantaneous, it would be obvious that, from the ship's twin's view, the Earth and space station do not stay at rest with each other. So as the ship starts slowing down, the Earth and space station are getting farther apart. If the ship decelerates at 1 G proper acceleration, is it important that his coordinate velocity and acceleration relative to Earth will be different than relative to the space station?

No it isn't. The reason is that what you're describing isn't a coordinate system. You're describing a one-parameter family of coordinate systems (with proper time along the ship's world line being the parameter). It doesn't make sense to think of this infinite set of coordinate systems as the ship's point of view, not globally anyway. Each member of this set is a coordinate system that we can think of as the ship's point of view in an infinitesimally small region of space-time around the point on the ship's world line that's characterized by the same value of proper time as the coordinate system. We can not think of one of them, or all of them, as representing the ship's point of view in a region of space-time that includes both the Earth and the space station. (I assume "the space station" is the spatial location of the turnaround event in Earth's frame).

Is there a resolution on the net that shows the math for realistic acceleration?

If you mean, "describes things from the ship's point of view during realistic acceleration", the answer is no. There is no natural way to associate a coordinate system with the ship's world line. (This is not a problem that GR solves. Things are actually even worse in GR).

One thing we can do is calculate the age of either one of the twins at any event on his world line. It's just the integral of $\sqrt{dt^2-dx^2}$ along the world-line. This works no matter what the acceleration is.