I Can anyone clarify the relativistic twin paradox for me?

[Curious Kev]

TL;DR Summary

explanation of the relativistic time twin paradox

Can anyone clarify the relativistic twin paradox for me?

Here's my understanding of it. Twins on earth synchronize their clocks. One twin stays on earth and the other accelerates away in a space ship to 0.8c, say. After a time lapse of one year, say, on earth, the accelerated twin need not expend energy to reverse direction but uses a large star's gravitational field to swing him round to return back to earth without any speed alteration. As this twin approaches earth he decelerates and they compare their clocks. However ... since either twin can say that the other traveled at 0.8c in his frame, the clocks remain synchronized, even though only one of the twins has expended energy accelerating! Any comments?

 

[Orodruin]

Any comments?

Yes. It is wrong.

The travelling twin is younger when they meet up again.

Acceleration (or more accurately, proper acceleration) plays very little role in the twin paradox. Its only role is to turn the travelling twin atound. The real resolution of the paradox lies in properly applying spacetime geometry.

 

[Ibix]

However ... since either twin can say that the other traveled at 0.8c in his frame, the clocks remain synchronized, even though only one of the twins has expended energy accelerating!

No, the one who sligshotted round the star will be younger. He's taken a shorter-time route through spacetime.

 

[phinds]

You need to get clear on the terms "proper time", "time dilation", and "differential aging". I suggest looking up all three and study them 'til you understand them. Then the twin "paradox" will clarify itself to not being a paradox at all.

 

[DaveC426913]

...the other accelerates away in a space ship to 0.8c, say.

... uses a large star's gravitational field to swing him round to return back to earth...

A tiny nitpick. You cannot travel at .8c and use a star to reverse your direction. There was a discussion about this in the sci-fi forum as to what mass you'd need to deflect - let alone reverse - your trajectory.

This is moot, really - you are certainly allowed to handwave the required mechanism for the sake of the thought experiment - just know that, practically, it is not trivial.

 

[Nugatory]

However ... since either twin can say that the other traveled at 0.8c in his frame, the clocks remain synchronized, even though only one of the twins has expended energy accelerating! Any comments?

There is a sticky at the top this subforum: "When Discussing the Twin Paradox: Read This First". https://www.physicsforums.com/threads/when-discussing-the-twin-paradox-read-this-first.1048697/

At every stage of the journey earth twin and traveler are moving relative to one another. Thus we might argue that the travelling twin is at rest while earth twin is moving, apply the time dilation formula, and conclude that time is passing more slowly for the earth twin; or we might argue that it is earth twin who is at rest and traveler who is moving and arrive the opposite conclusion. This apparent contradiction is why we call it a "paradox"; but it comes about from not using the time dilation formula correctly.

There are easier ways of analyzing this problem than using the time dilation formula (@Ibix gave the best in #3 above; this answer is elaborated in many older threads here) but if you are going to use the time dilation formula you must also allow for the relativity of simultaneity. This is explained in the "Time Gap" section of the FAQ I linked above, and you will also find it helpful to draw a Minkowski diagram identifying the points on earth twin's worldline that happen "at the same time" that traveler turns around. Crucially, those are two different points according to whether we're using the frame in which traveler is at rest on the outbound leg or the frame in which traveler is at rest on the inbound leg.

 

[Sagittarius A-Star]

However ... since either twin can say that the other traveled at 0.8c in his frame, the clocks remain synchronized, even though only one of the twins has expended energy accelerating!

No. Consider a black clock on earth and a blue clock passing the earth with $0.8 c$ towards the star.

Near the star, the blue clock passes a red clock, which moves with $0.8 c$ in opposite direction towards the earth. When the red clock passed the blue clock, it showed coincidently the same time.

In the rest frame of the blue clock, the black clock on earth is moving.
In the rest frame of the red clock, the black clock on earth is moving, too.

But the black clock on earth experiences still more proper time between the two earth-meeting events than the blue and red clock together, see the simultaneity planes in the following diagram.

Via:
https://en.wikipedia.org/wiki/Twin_paradox#Relativity_of_simultaneity

 

[Halc]

You cannot travel at .8c and use a star to reverse your direction. There was a discussion about this in the sci-fi forum as to what mass you'd need to deflect - let alone reverse - your trajectory.

Why not? All you need to do is get close enough to its photon sphere. OK, i don't think even a neutron star can be sufficiently dense to let you get that close, so is the restriction that it is still something commonly referred to as a 'star'?

Of course the tidal forces will kill you (unless the 'star' is super massive), but hey, you can take it if you managed the typical acceleration we tend to put the poor guy through at all the events of interest. Turns out you don't need fuel to match Earth velocity at the end. Just need a big ball pit.

 

[PeterDonis]

either twin can say that the other traveled at 0.8c in his frame

No, they can't, because only the stay-at-home twin's frame is an inertial frame in the required sense. The traveling twin's frame is not; even though he is in free fall the whole time, the spacetime geometry he travels through is not flat, so his "rest frame" doesn't work like an inertial frame in flat spacetime in SR. In particular, he cannot assign a well-defined "speed" to the stay-at-home twin that stays the same the whole time.

 

[Dale]

You cannot travel at .8c and use a star to reverse your direction.

You could if the star was also going about 0.8 c.

Edit: I guess actually you would need a line of stars each going a slightly different velocity to do it.

 

[DaveC426913]

Why not?

Because I was the one who posited the 'relativistic 180' in the sci fi writing forum and was informed it was effectively impossible.

As soon as I find the reference I'll post it. It was in one of StratoIncendus' threads about his genship mutiny story.

Still, it's moot, as I said. This thread is a thought experiment, not a story plot line.

 

[Orodruin]

Because I was the one who posited the 'relativistic 180' in the sci fi writing forum and was informed it was effectively impossible.

As soon as I find the reference I'll post it. It was in one of StratoIncendus' threads about his genship mutiny story.

Still, it's moot, as I said. This thread is a thought experiment, not a story plot line.

A star would be impossible. A black hole would make it possible.

 

[Curious Kev]

No, they can't, because only the stay-at-home twin's frame is an inertial frame in the required sense. The traveling twin's frame is not; even though he is in free fall the whole time, the spacetime geometry he travels through is not flat, so his "rest frame" doesn't work like an inertial frame in flat spacetime in SR. In particular, he cannot assign a well-defined "speed" to the stay-at-home twin that stays the same the whole time.

Do you mean that the loss of time in the one traveling away from the earth is due to the time spent accelerating? Because he need not be accelerating for the whole journey. What if the acceleration is turbocharged for just one minute? Would the loss of time occur only in that minute? Furthermore, isn't the stay-at-home clock accelerating in the traveling frame? That means they're doing the same thing. Thanks for your interest.

 

[PeterDonis]

Do you mean that the loss of time in the one traveling away from the earth is due to the time spent accelerating? Because he need not be accelerating for the whole journey. What if the acceleration is turbocharged for just one minute? Would the loss of time occur only in that minute? Furthermore, isn't the stay-at-home clock accelerating in the traveling frame? That means they're doing the same thing. Thanks for your interest.

Read the Insights article that @Nugatory linked to in post #6. There's a reason why it has the title it has.

 

[Dale]

Do you mean that the loss of time in the one traveling away from the earth is due to the time spent accelerating? Because he need not be accelerating for the whole journey. What if the acceleration is turbocharged for just one minute? Would the loss of time occur only in that minute? Furthermore, isn't the stay-at-home clock accelerating in the traveling frame? That means they're doing the same thing. Thanks for your interest.

The loss of time is not due to the acceleration except in the sense that the acceleration breaks the symmetry and clearly identifies which twin is which.

Think of it this way. Say you have a piece of paper with two points on it. Draw a straight line between the points. Now draw a non-straight line between the points, say one with two straight segments and a sharp bend.

The bent path is longer than the straight path. Would you say that the extra length is in the bend or something similar? I wouldn’t. The bend clearly distinguishes the straight from the non-straight path, but the extra length is along the whole path, not just the bend.

 

[Orodruin]

Think of it this way. Say you have a piece of paper with two points on it. Draw a straight line between the points. Now draw a non-straight line between the points, say one with two straight segments and a sharp bend.

The bent path is longer than the straight path. Would you say that the extra length is in the bend or something similar? I wouldn’t. The bend clearly distinguishes the straight from the non-straight path, but the extra length is along the whole path, not just the bend.

Obligatory promotion of self-written material: https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

 

[jbriggs444]

The bent path is longer than the straight path. Would you say that the extra length is in the bend or something similar?

What I am about to write has already been mentioned in this thread. Not much new to see here.

Imagine a bug crawling down the bent path. The bug looks to the left (or right as appropriate) at the other path. The bug's line of sight is at 90 degrees to his direction of travel. He is gratified to see that his twin crawling down the straight path has failed to keep up. The twin bug is behind our bug's 90 degree line of sight.

Our bug reaches the bend in the path. The twin bug is well behind his line of sight now. But as our bug turns the corner, his 90 degree line of sight sweeps backward along the straight path. Now the bug on the straight path is ahead of the new line of sight.

The bug on the straight path stays ahead and finishes the course ahead of the bug on the bent path.

The situation in the twin's paradox is the same except that it is hyperbolic geometry trigonometry. Instead of sweeping backward at turnaround, the line hyper-plane of simultaneity sweeps forward along the stay at home twin's world line when the travelling twin makes the turn on his bent path.

If one were to adopt a particular choice of non-inertial coordinates during turnaround, one might claim that the clock of the stay at home twin is dramatically accelerated during the travelling twin's turnaround. However, this speed up should probably not be regarded as a "real" effect. It is just the cosmetic result of a set of coordinate assignments.

Edit: Thank you, @robphy for the terminology correction.

 

[robphy]

hyperbolic geometry

Hyperbolic trigonometry (since a hyperbola is used to measure rapidities) in [flat] Minkowski spacetime,
analogous to circular trigonometry (since a circle is used to measure angles) in [flat] Euclidean geometry.

(Hyperbolic geometry is a curved riemannian geometry. Similarly spherical/elliptical geometry is a curved riemannian geometry .)

 

[DaveC426913]

Imagine a bug crawling down the bent path. The bug looks to the left (or right as appropriate) at the other path. The bug's line of sight is at 90 degrees to his direction of travel. He is gratified to see that his twin crawling down the straight path has failed to keep up. The twin bug is behind our bug's 90 degree line of sight.

Our bug reaches the bend in the path. The twin bug is well behind his line of sight now. But as our bug turns the corner, his 90 degree line of sight sweeps backward along the straight path. Now the bug on the straight path is ahead of the new line of sight.

The bug on the straight path stays ahead and finishes the course ahead of the bug on the bent path.

It sounds like it might be a Minkowski diagram, but this is what I get:
Does this display what you're describing?

Up to mark 5, blue observes he's ahead of green, but as he turns (between 5 and 6) blue sees green pull into the lead, and ultimately win.

(Diagram not to scale. Blue will never see green move backward.)

 

[phinds]

Furthermore, isn't the stay-at-home clock accelerating in the traveling frame? That means they're doing the same thing.

No, they are not doing the same thing. Speed is relative, acceleration is not. The stay at home is not accelerating.

 

[Herman Trivilino]

The situation in the twin's paradox is the same except that it is hyperbolic geometry trigonometry.

Just to pick nits here, I'd say it's hyperbolic space.

 

[robphy]

Just to pick nits here, I'd say it's hyperbolic space.

In the case of ordinary geometry, would you use spherical (or circular or elliptic) space?
...since a circle plays the role of "the circle in that geometry".

https://en.wikipedia.org/wiki/Hyperbolic_space says

Riemannian manifold of constant sectional curvature equal to −1

which is an agreement with what I said above...a curved riemannian geometry.

However, the space under discussion is
https://simple.wikipedia.org/wiki/Minkowski_spacetime

has a metric signature of (-+++), and describes a flat surface when no mass is present.

which is a flat pseudo-riemannan geometry (and is not a hyperbolic space).

 

[Orodruin]

I agree with @robphy Minkowski space is not hyperbolic space. Hyperbolic trigonometry is useful in both.

 

[Curious Kev]

Forgive the length of this but I think it's a clear argument.

Suppose the history of acceleration is unknown. Two spaceships A and B pass each other in opposite directions at 0.8c in field free space. They each then start to send the other a signal at one second intervals. That way, the frequency of pulses allows a comparison of clock speeds. Their clock rates should be the same. But they find that they are not.

We now learn that about a week ago, they were stationary relative to each other (though separated in space) and synchronised their clocks. (Maybe they were coincident and accelerated in opposite directions for several days before decelerating). Then B ignited its extremely powerful thrusters for about a minute (they're robot pilots so no one dies) and set off at 0.8c relative to A and (as stated) eventually passes A. So the length of B's second is 5/3 times A's second as the comparison of pulse frequencies testifies. When a time of one week has passed on A's spacecraft then 3/5 weeks has passed on B's.

Now here's the point of all this. This is not about theory of observation it's about reality. B's clock has accelerated and decelerated for two minutes in total which is negligible compared with one week. No one can sensibly refer to non-inertial frames and claim that the time loss of 2/5 weeks occurred during these two minutes of being a non-inertial frame. It occurred for B while being an inertial frame. Furthermore, while B is accelerating then A is too (in the sense of B measuring a change in velocity in A). One could define acceleration as that which is measure by an accelerometer but what is really meant by this is that one type of acceleration (B) involves a gain or loss of energy while the other (A) does not.

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock. And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration.

 

[Sagittarius A-Star]

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock. And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration.

Acceleration does not change physically a clock. This clock hypothesis was experimentally verified in particle accelerators.
Source:
https://en.wikipedia.org/wiki/Time_dilation#Clock_hypothesis

The reciprocal time dilation while inertial movement in flat spacetime comes from the relativity of simultaneity.
Source:
https://en.wikipedia.org/wiki/Twin_...psed_times:_how_to_calculate_it_from_the_ship

 

[PeroK]

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock. And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration.

That's a clear example of just how wrong someone can be.

 

[PeroK]

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock. And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration.

If that were true, in order to manufacture an atomic clock, you would need to know the acceleration history of all the atoms and protons and electrons that made up the clock. Every atomic clock would run at a slightly different rate. The GPS system relies on clocks being extremely accurate. Your anti-clock hypothesis would make the GPS system unworkable, due to atomic clocks being increasingly inherently different.

You might look up the Hafele-Keating experiment. They put atomic clocks on commercial airliners to measure differential ageing associated with different global journeys. The clocks in question confirmed the theory of relativity and not you anti-clock hypothesis.

 

[Curious Kev]

But with two atoms that were stationary relative to each other, their acceleration history would be the same!

 

[jbriggs444]

But with two atoms that were stationary relative to each other, their acceleration history would be the same!

Not if they were in motion relative to each other prior to that. You are postulating that clock rate now depends on acceleration history in the past.

 

[PeroK]

But with two atoms that were stationary relative to each other, their acceleration history would be the same!

Not if they were created in different supernova explosions.

 

[Curious Kev]

I meant to say that their acceleration history in terms of energy per unit mass gained or lost would cancel out if they were stationary relative to each other. Of course, tracing a complete acceleration history is not possible with an unlimited time regression. But this equality seems reasonable to me.

 

[PeroK]

I meant to say that their acceleration history in terms of energy per unit mass gained or lost would cancel out if they were stationary relative to each other.

All we can tell you is that there is no experimental evidence to support an "acceleration history matters" theory. And, indeed, there is a considerable amount of evidence that the clock hypothesis holds. I.e. that acceleration does not fundamentally affect the running time of a clock.

 

[Curious Kev]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

 

[jbriggs444]

I meant to say that their acceleration history in terms of energy per unit mass gained or lost would cancel out if they were stationary relative to each other.

So you are saying that acceleration history only matters in terms of relativistic mass gain or loss resulting from said acceleration?

You accelerate a twin, he gains mass and his clocks run slow as a result?

And you seem to accept implicitly that synchronization is absolute, discarding the relativity of simultaneity.

The result is the Lorentz Ether Theory. That theory is deprecated because it makes all of the same predictions as special relativity while carrying around useless baggage.

 

[PeroK]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

The explanation for differential ageing is the geometry of spacetime. There is no structural change to a particle when accelerated, for example.

What we are doing here is dissecting your alternative theory, based on acceleration history and physical structural change. Technically, this is bending the rules of the forum.

The global positioning system comes as close as experimentally possible in proving the special theory of relativity and the affect of a gravitational field on the geometry of spacetime.

 

[Dale]

Forgive the length of this but I think it's a clear argument.

You don’t need to be presenting clear arguments. You need to be asking clear questions that will help you learn how the universe actually works.

One big problem is that your description is not at all clear as you assume it is. In many cases what you state is unclear because you state a relative quantity without stating what frame it is relative to.

For example, in relativity there are two kinds of time: proper time and coordinate time. Coordinate time is relative meaning it is the time in a specific reference frame. It is defined in the entire reference frame. In contrast, proper time is invariant, it is the time that is physically read on a single clock and is only defined where the clock is located.

Because proper times are only defined where the clocks are located, proper times can only be compared when the clocks are at the same location. When comparing distant clocks, one of the times is a coordinate time, not a proper time. The comparison is relative to the frame whose coordinate time is used.

Similarly with acceleration. Proper acceleration is the physical acceleration measured by an accelerometer and it is invariant. Coordinate acceleration is the second derivative of position in a given frame. It is relative to the given frame.

Energy is also a relative quantity. Different frames will disagree on the energy, so the frame must be specified.

With this understanding, let’s look back and see exactly how unclear your argument is:

the frequency of pulses allows a comparison of clock speeds

In which frame? The frequency comparison only gives the Doppler shift, not a comparison of clock speeds without specifying the frame.

So the length of B's second is 5/3 times A's second

In which frame?

When a time of one week has passed on A's spacecraft then 3/5 weeks has passed on B's.

In which frame? Which is the coordinate time and which is the proper time?

It occurred for B while being an inertial frame.

B’s frame is non inertial. The frame includes time.

while B is accelerating then A is too (in the sense of B measuring a change in velocity in A)

Which of these is proper acceleration and which is coordinate acceleration? For the coordinate acceleration, relative to which frame is the coordinate acceleration measured?

One could define acceleration as that which is measure by an accelerometer but what is really meant by this is that one type of acceleration (B) involves a gain or loss of energy while the other (A) does not.

This is simply wrong. Proper acceleration is absolute, gain or loss of energy is relative.

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock.

Gainnor loss of energy is a relative quantity, so relative to which frame is which clock physically changing?

And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration

Since frames include time, it makes no sense to speak of reverting to be an inertial frame.

 

[Sagittarius A-Star]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

As I wrote in posting #7, you can replace the travelling clock by 2 clocks. One is moving inertially out-bound, the other inertially in-bound. Together they count a round-trip proper time, that leads also to the twin paradox in combination with the earth-clock. But none of the 2 travelling clocks is or was accelerated.

 

[Nugatory]

No one can sensibly refer to non-inertial frames and claim that the time loss of 2/5 weeks occurred during these two minutes of being a non-inertial frame

Part of the problem here may be that you seem to be misunderstanding what a frame, whether inertial or not, is. A frame is a convention for assigning coordinates (time and space) to events so there as no such thing as "two minutes of being a non-inertial frame" - what we're doing is using one convention for a while, then using another. It is not at all surprising to find that when the traveller changes the convention they use to assign times to the earth clock, the time they've assigned to the earth clock changes. There is no physical change associated with either clock.

 

[ersmith]

Forgive the length of this but I think it's a clear argument.

Suppose the history of acceleration is unknown. Two spaceships A and B pass each other in opposite directions at 0.8c in field free space. They each then start to send the other a signal at one second intervals. That way, the frequency of pulses allows a comparison of clock speeds. Their clock rates should be the same. But they find that they are not.

Indeed. After correcting for light travel time, each finds the other's clock to be ticking more slowly. The geometric reason for this is explained in earlier messages, and in particular in Orodruin's insight at https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock. And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration.

Let's consider an analogy. Suppose car A and car B are in New York and have identical odometers, both set to 0. Car A travels to Boston in as straight a line as possible. Car B also goes to Boston, but takes a windy path. Would you explain the difference in their final odometer readings by attempting to construct a theory of how turning the steering wheel in the car imparts energy to the odometer, causing it to tick faster? This *might* be possible in some sense, but it would require a very complicated theory to cover all possible cases. The far simpler explanation is just that the odometer measures distance traveled, and car B traveled a longer distance.

Clocks are odometers in spacetime, and time has the opposite sign in the metric to space (so a straight line is the *longest* path in spacetime, rather than shortest). Acceleration corresponds to turning or bending the clock's worldline. Otherwise the analogy is quite exact.

 

[Nugatory]

No one can sensibly refer to non-inertial frames and claim that the time loss of 2/5 weeks occurred during these two minutes of being a non-inertial frame

You may want to go back and review post #17 by @jbriggs444 in his thread. Keep it in mind as you study the "time gap" section of the twin paradox FAQ that we've already referred you to.

 

[Dale]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

I agree. There isn’t an explanation in terms of structural change.

I do not agree that that means there isn’t a reason for time dilation. There are good reasons, but they are geometrical, not structural or mechanical.

 

[PeterDonis]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

That's right, because one clock does not "run slower than another" in any invariant sense. As far as invariants are concerned, all clocks tick at exactly the same rate: one second per second of arc length along their worldlines. There is no "structural change" anywhere.

To put this another way: "time dilation" is not the result of doing anything to a clock. It's a result of looking at the clock in a different frame. It's the hyperbolic geometry analogue of the apparent size of an object changing when you view it from a different angle: you changing your viewing angle doesn't do anything to the object itself. There is no "structural change" involved.

 

[robphy]

Here is a visualization of mine that might help.

It displays the proper time along piecewise-inertial worldlines
using a ticking light clock, which provides a relativity-friendly mechanism for the wristwatch's behavior.

It uses a "circular light clock" because it was based on generalizing the Michelson-Morley apparatus.

From the relativity-principle and the speed-of-light principle,
the visualization displays the resulting effects of
time-dilation, length-contraction, and the relativity-of-simultaneity

VisualizingProperTime - the Clock Effect / Twin Paradox with Circular Light Clocks (alert: it has sound effects)


The spacetime-volume enclosed in the causal-diamonds
(the intersection of the future light-cone of one tick with the past-light-cone of the next tick)
are equal. (The volume is related to the square-interval between the tick-events.)

(In the (1+1) case, the equality of areas can be visualized on "rotated graph paper".
See https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/)

(To see that the causal diamonds along different piecewise-inertial worldlines are related by a boost.
visit
robphy-CircularLightClocks-VisualizingProperTimeInSpecialRelativity
https://www.geogebra.org/m/pr63mk3j .)

The ticking light-clock mechanism is purely kinematic.
As others have said, it's really about the spacetime geometry.
(Between two timelike-related events, A and Z,
the arc-length [using the spacetime metric] along a worldline from A to Z
depends on the worldline.)

 

[Orodruin]

It's the hyperbolic geometry analogue

Minkowski geometry.