Why Does the Ship's Twin Observe Earth's Clock Jump in the Twin Paradox?

[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.

 

[Fredrik]

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.

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?

I like talking about these things too, but I would like to point out that ideas like "Mach's principle" or "ideal clocks" have no place in a discussion about the twin paradox. The twin paradox is the (false) claim that special relativity predicts two contradictory things about the twins' ages when they meet again. Special relativity is just Minkowski space, and the twins are just three straight lines. If you find a way to eliminate the contradiction that involves the properties of clocks (ideal or not), or some "principle" that isn't a part of SR, then you haven't solved the problem. You still wouldn't have any idea if there really is a contradiction in SR, or what SR really says! If you want to really solve the problem, you have to do it using the properties of Minkowski space, and nothing else.

 

[DrGreg]

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?

Despite what others have said, it is possible to set up a coordinate system for an accelerating observer. But that coordinate system is not an inertial frame so it doesn't behave like inertial frames do.

Warning: the following requires a knowledge of hyperbolic functions and calculus.

If (t, x) are the inertial coordinates of an inertial observer I (ignore y and z as being constant), consider a new coordinate system (T, X) defined by

x + \frac{c^2}{g} - x_0 = \left( X + \frac{c^2}{g} \right) \cosh \frac {g(T-T_0)}{c}
t - t_0 = \left( X + \frac{c^2}{g} \right) \sinh \frac {g(T-T_0)}{c}​

These are Rindler coordinates. T is the proper time of an accelerating observer A located at X = 0, and with a constant proper acceleration of g. Two events with the same value of T are simultaneous in A's co-moving inertial frame and X measures distance from A in the co-moving inertial frame. The velocity of A relative to I is

\frac{dx}{dt} = c \tanh \frac {g(T-T_0)}{c}​

with Lorentz factor

\gamma = \cosh \frac {g(T-T_0)}{c}​

The coordinate acceleration of A relative to I is

\frac{d^2x}{dt^2} = g \, sech^3 \frac {g(T-T_0)}{c} = \frac {g}{\gamma^3}​

Choose values of T₀, t₀ and x₀ to synchronise clocks and distances between the A and I frames in the way you want. When T = T_0, A is stationary relative to I at t = t_0, x = x_0. (So I is the co-moving inertial frame at that moment.)

References

Rindler, W. (2006 2nd ed), Relativity: Special, General and Cosmological, Oxford University Press, Oxford, ISBN 978-0-19-856732-5, Section 3.8, pp.71-73 and Section 12.4, pp.267-272.

Gibbs, P. and Koks, D. (2006), http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html , Usenet Physics FAQ, accessed 19 June 2008.

Anonymous, (undated), "Born rigidity, Acceleration, and Inertia", MathPages, accessed 19 June 2008.

 

[matheinste]

Hello Fredrik.

Quote:-

---we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration.---

I only included this comment because some people might think that a clock in the mechanical sense might be affected by acceleration via its mechanism. To make a point an ordinary clockwork clock may well be affected either slowing down or speeding up due to acceleration. Also people may not know whether or not atomic clocks are directly affected by acceleration by way of the physics involved. The use of an ideal clock is just to remove any possible objections of this sort from a situation which for some reason many find confusing.

Of course Mach's views on inertia are irelevant to a universe peopled by earth, space stations, twins and the like, i.e. matter.

Matheinste.

 

[Fredrik]

Despite what others have said, it is possible to set up a coordinate system for an accelerating observer.

I didn't mean that it's impossible to define a coordinate system that takes the accelerated observer's world line to be its time axis. (I said something to that effect in another thread, and you were right to correct me then). What I meant is that it doesn't make much sense to think such coordinates as representing the accelerating observer's point of view. I'm sure there are lots of ways to slice up space-time into a one-parameter family of space-like hypersurfaces that we can (if we want to) think of as representing space at different times. Why should the choice defined by Rindler coordinates be the "correct" choice?

 

[yuiop]

Hello Fredrik.

Quote:-

---we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration.---

Yes.. A grandfather clock with a pendulum would certainly be affected by acceleration, but it also far from an ideal clock.

 

[DrGreg]

I didn't mean that it's impossible to define a coordinate system that takes the accelerated observer's world line to be its time axis. (I said something to that effect in another thread, and you were right to correct me then). What I meant is that it doesn't make much sense to think such coordinates as representing the accelerating observer's point of view. I'm sure there are lots of ways to slice up space-time into a one-parameter family of space-like hypersurfaces that we can (if we want to) think of as representing space at different times. Why should the choice defined by Rindler coordinates be the "correct" choice?

You are right that there are other choices of accelerated coordinate system. And it is debatable as to exactly what the accelerated observer's "point of view" is. Nevertheless it is conventional to consider the co-moving inertial frame to represent the "instantaneous" view, and Rindler coordinates are the only coordinates (I think) that are compatible with this view in the sense that:

- the observer is at fixed spatial coordinates X = Y = Z = 0
- at X = 0 (but not at other positions), T is the proper time of the observer
- every surface of constant T coincides with the plane of simultaneity of the corresponding co-moving inertial frame
- within each such simultaneity plane, the Rindler spatial coordinates X, Y, Z coincide with the co-moving inertial frame's spatial coordinates

That, in my view, makes Rindler coordinates a more "natural" choice than any others. Of course all "points of view" are a mathematical construct, even in inertial frames. They don't reflect what you see with your eyes; the frame point of view is something you have to calculate retrospectively from observations made after the events being measured, and it depends on what conventions you choose to adopt to perform the calculation.

And I think Rindler coordinates would answer the question put in post #65: they give us a way of seamlessly (up to continuous first derivative) interpolating between the two points of view of inertial motion before and after acceleration. The attached left-hand diagram illustrates the accelerated twin's point of view in the Twins Paradox. (The right-hand diagram shows the inertial twin's point of view.)

 

[phyti]

Is this pretty much a consensus view?

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

Don't know about consensus view, but do know time dilation is an experimentally verified fact, and it explains the differences in observer perceptions.
If a and b are two cities 200 miles apart, and you fly between them at 100 mph, you arrive in 2 hr. IF you fly between them at 200 mph, you arrive in 1 hr. The distance between them did not change, you got there quicker! You can't give an unqualified statement such as 'the space contracted' without explaining how. This is a popular misconception, because SR does not state it. The transformation rules apply to the varied observations/perceptions of different frames so as to preserve the one set of actual physical events. SR is like an accounting method that reconciles the perceptions, but is does not alter the actual events.
There is one event, but many perceptions.
Mach:
If the mass of the universe is on average, uniformly distributed (including the lumps), and considering the vast distances involved, the net gravitational effect is zero. Any inertial effects are the result of local mass, (within the solar system). Two space ships would still resist acceleration because of the ships mass.
Consider, if all matter had local effects, it would be impossible to conduct an isolated experiment, and you would get random variations from distant events.

I would like to clarify... that isolated experiments have a purpose as control elements, but
in hypothetical scenarios, this does not represent a real world situation.

Sorry I didn't respond sooner.

 

[Al68]

I like talking about these things too, but I would like to point out that ideas like "Mach's principle" or "ideal clocks" have no place in a discussion about the twin paradox. The twin paradox is the (false) claim that special relativity predicts two contradictory things about the twins' ages when they meet again.

Well, you're right, most of my questions were about situations very different from the twins paradox. I referenced it just because everyone is familiar with it. Maybe I should have used a different title for the topic.

And yes, Mach's principle is a little off track, but interesting. Einstein was the one who coined the phrase "Mach's principle" while discussing why inertial frames are different from non-inertial frames, and the seemingly circular logic of saying that Newton's laws "hold good" in inertial frames, and we know a frame is inertial (a priori) because Newton's laws "hold good". He considered this a "defect" of SR. I wouldn't call it a defect, just an unanswered question.

DrGreg, I think I should be careful what I ask for. It's been a couple of years since college. I think I'll have to take some time to understand your post.

Thanks,
Al