The Mirror Paradox: Exploring Special Relativity Theory

[Imaxx]

TL;DR Summary

A question about a paradoxical situation related to the special theory of relativity.

Let's imagine that there are two people who has the same biological age. One is inside a spaceship, currently crossing a Quasar(A) in a constant velocity near the speed of light toward the Earth and the other is on Earth(B). In this situation, what will happen when A moves to the Earth and meets B? Let's say the Earth does not orbit or rotate in order to keep this situation only on the inertial coordinate system. The moment A arrives at the Earth after a long time, what will happen when they see each other?

According to the special theory of relativity, B will see a person younger than himself(A) in the spacecraft , and A will observe that a person younger than himself(B) is standing on the Earth. It sounds like a paradoxical situation, but according to the special theory of relativity that phenomenon is not a contradiction.

But let's go further and imagine that there is a mirror on Earth next to B. When A looks into the mirror, he will see his old face, and when B watches A through the mirror, he should see a face younger than what A saw.

Can light originated from the same object be simultaneously reflected as a different phenomena(light) in the mirror?

 

[PeroK]

Let's imagine that there are two people who has the same biological age. One is inside a spaceship, currently crossing a Quasar(A) in a constant velocity near the speed of light toward the Earth and the other is on Earth(B).

In which reference frame are these two people the same age?

Look up the relativity of simultaneity. Special Relativity is not just time dilation.

 

[Ibix]

According to the special theory of relativity, B will see a person younger than himself(A) in the spacecraft , and A will observe that a person younger than himself(B) is standing on the Earth

This is not correct. Exactly what they see will depend on factors you did not specify, but guessing that you intended them to be the same age at the start of the experiment using the Earth's simultaneity criterion, the traveller (A) will be younger when they meet up.

But let's go further and imagine that there is a mirror on Earth next to B. When A looks into the mirror, he will see his old face, and when B watches A through the mirror, he should see a face younger than what A saw.

This makes no sense to me. When are they looking into these mirrors? When they meet up? If so, see my previous paragraph.

Can light originated from the same object be simultaneously reflected as a different phenomena(light) in the mirror?

No.

 

[jbriggs444]

Let's imagine that there are two people who has the same biological age. One is inside a spaceship, currently crossing a Quasar(A) in a constant velocity near the speed of light toward the Earth and the other is on Earth(B). In this situation, what will happen when A moves to the Earth and meets B? Let's say the Earth does not orbit or rotate in order to keep this situation only on the inertial coordinate system. The moment A arrives at the Earth after a long time, what will happen when they see each other?

The highlighted words are key. You are making an assertion about a simultaneous state of affairs at two locations that are distant from one another.

That means that you need to account for the relativity of simultaneity. Those two things are simultaneous only for a particular frame of reference. We infer that you mean the Earth frame. We infer that because you said that person A is "in a constant velocity near the speed of light". If we were using person A's rest frame then person A would not be moving near the speed of light. Person A would be stationary.

According to the special theory of relativity, B will see a person younger than himself(A) in the spacecraft

No. B is moving. If we shift to B's rest frame, we mess with simultaneity. In this frame, the two twins did not start the scenario simultaneously. From B's point of view, A got a head start, is much older and is most of the way to B already.

When they meet, A will be older than B.

and A will observe that a person younger than himself(B) is standing on the Earth.

When they meet, A will be older than B.

Two explanations. One physical result.

 

[Ibix]

It's also worth noting that, as far as I know, there are no quasars closer than a couple of billion light years, far enough away that we can't use special relativity to describe this experiment since there's significant curvature on that kind of scale. I think that only makes a quantitative difference at the level of these answers which are all non-quantitative, but it may matter depending on how the conversation goes.

It would probably be easier to replace "quasar" with "another star a few light years away".

 

[Nugatory]

A question about a paradoxical situation related to the special theory of relativity.

There are two things that you should do before you try to take on this paradox:
1) Understand how time dilation is symmetrical: When A and B are moving relative to one another, A will find that B's clock is running slower than their own; B will find that A's clock is the one that is running slow; but after you allow for the relativity of simultaneity there is no contradiction.
2) Understand how the problem you have posed is different from the classic twin paradox (and how to resolve that paradox without blaming the acceleration). Again, the relativity of simultaneity will be crucial.

(Of course if this is the first time that you've considered the relativity of simultaneity, understanding that will be your first step. Google for "Einstein train simultaneity" and look at some of the many threads on the subject that we have here)

 

[Imaxx]

This is not correct. Exactly what they see will depend on factors you did not specify, but guessing that you intended them to be the same age at the start of the experiment using the Earth's simultaneity criterion, the traveller will be younger when they meet up.

This makes no sense to me. When are they looking into these mirrors? When they meet up? If so, see my previous paragraph.

No.

Forgive me for my lack of understanding in physics and English, but why will the traveler be younger when they meet? Isn't this situation symmetrical to each other?

 

[Ibix]

Forgive me for my lack of understanding in physics and English, but why will the traveler be younger when they meet? Isn't this situation symmetrical to each other?

It depends on initial conditions that you didn't specify. Namely, how you determined that the two people were "the same age". You can do that in various different ways, using the simultaneity definitions of different frames. Usually people imply that they've adopted the simultaneity of Earth's rest frame, in which case the initial conditions are not symmetric: one of your people is moving, one is not. You can adopt a different simultaneity criterion, for example the one associated with the frame where both people are moving towards each other. In this case the situation is symmetrical, and the people's ages are the same when they meet up (presumably these are different people from the first two, at least one of whom was born at a different time).

You need to look up and understand the relativity of simultaneity. Basically, your experimental design can be interpreted in different ways because you haven't specified what you mean by "same age" for people not in the same place. Depending on how you interpret that, you get different experiments and different outcomes.

 

[Nugatory]

Forgive me for my lack of understanding in physics and English, but why will the traveler be younger when they meet? Isn't this situation symmetrical to each other?

There is an asymmetry in the way that you specified the initial ages.

STOP RIGHT NOW and learn about the relativity of simultaneity. It is the basis for just about every apparent inconsistency and so-called "paradox" in special relativity.

There is no question about the age of the traveller as he passes the quasar (call that $A_T$) - we could place a camera on the quasar to make the necessary observation as the traveller moves past, and that camera will show what it shows.

However you are also specifying that the Earth person is "the same biological age". That wording is a convenient shorthand for "The age of the Earth person is $A_T$ at the same time that the traveller passes by the quasar and is recorded by a camera there to be of age $A_T$”. And because of the relativity of simultaneity, the meaning of "at the same time" depends on which frame we're using to analyze the problem.

If they are both the same age using the frame in which the Earth is at rest (the events "earth person is of age $A_T$" and "traveller is of age $A_T$" are simultaneous), those two events will not be simultaneous using the frame in which the traveller is at rest and the Earth is moving. Using that frame, the event "earth person is of age $A_T$" occurs long before the event "traveller is of age $A_T$" so the Earth person is older at the start of the experiment.

Earth person explains the fact that traveller is younger by saying that they started out the same age but time dilation means that less time passed for traveller. Traveller explains the same facts by saying that Earth person started out older so of course is older at the the meeting.

 

[Ibix]

STOP RIGHT NOW and learn about the relativity of simultaneity.

Just wanted to repeat that bit. Not understanding the relativity of simultaneity is the root cause of almost every problem people have with relativity.

 

[Dale]

Isn't this situation symmetrical to each other?

No. In the Earth’s frame they are initially the same age. In the Traveler’s frame they are initially different ages. The situation is therefore not symmetric.

This is due to the relativity of simultaneity as mentioned by several others.

 

[Vanadium 50]

STOP RIGHT NOW and learn about the relativity of simultaneity.

This is good advice.

That's Step 1. Step Zero is to decide if you are asking us how relativity works (good) or telling us that you have discovered a mistake in it (bad). With your user name, I suspect it's the latter. That would not be good.

 

[Orodruin]

It’s still in my signature, right? I’ve been gone so long I don’t remember (and on mobile so I don't see signatures ...)

 

[PeterDonis]

It’s still in my signature, right?

Yes, indeed.

 

[Fervent Freyja]

but why will the traveler be younger when they meet?

It’s already been shown that there’s a possibility that a space traveler could arrive younger than their twin on earth, and that space-flight conditions may be protective against normal aging on earth.

Twin studies show telomeres lengthened during spaceflight: https://science.sciencemag.org/content/364/6436/eaau8650

“Telomeres are repetitive features of chromosomal termini essential for maintaining genomic integrity; they protect physical DNA ends from degradation and prevent them from triggering inappropriate DNA damage responses (DDRs). Telomere length shortens with cell division and thus with age, as well as with a variety of lifestyle factors, such as stress, and environmental exposures, including air pollution and radiation.“

 

[Orodruin]

space-flight conditions may be protective against normal aging on earth.

That has nothing to do with differential aging in relativity though.

 

[Fervent Freyja]

That has nothing to do with differential aging in relativity though.

I couldn’t help but point it out though.

It’s a valid question: would the results be relativistic or biological? How could it even be distinguished once the space-traveler arrives back on earth?

 

[mfb]

It’s a valid question: would the results be relativistic or biological? How could it even be distinguished once the space-traveler arrives back on earth?

By a clock in the spacecraft . Clocks don't have telomeres and they are more precise anyway. Please keep this discussion about physics.

 

[Fervent Freyja]

By a clock in the spacecraft . Clocks don't have telomeres and they are more precise anyway. Please keep this discussion about physics.

Yes, Sir!

 

[Ibix]

It’s still in my signature, right? I’ve been gone so long I don’t remember (and on mobile so I don't see signatures ...)

Incidentally, at least on my phone, if I go to a landscape view I get something a lot more like desktop, including signatures. Landscape is not a convenient way to do anything except check sigs...

 

[Ian J Miller]

Am I allowed to alter the problem slightly? Suppose for practical purposes we have a common reference frame. Suppose this has been well thought out in advance, and ship A passes a star at, say 0.6c, and as it does so, it receives a message from a planet near the star that sets the clock. The ship now travels to Earth where a similar clock has had its time set. Each clock records the set time and the then current time. The common reference frame is set by a pulsar, the frequency of which is gradually decaying and hence acts as a third clock. By being well thought out in advance, the parties can agree at what time A should pass the star, in terms of the pulsar's clock. It won't be exact, but one should get near enough to get the relativistic effect to be significantly larger than any other defect. Now, when A passes B they exchange current time and set times on each clock. Now, A should think B's clock has dilated, and vice versa for B hence they cannot agree on which clock has the greatest elapsed time difference- is that not right?

 

[PeterDonis]

Suppose for practical purposes we have a common reference frame.

What objects are at rest in that frame? It seems like you intend Earth and the other star and the pulsar to all be at rest in that frame, but you should explicitly specify that (or whatever you intend).

Also, if you are trying to say that everybody will use this common reference frame even if they are not at rest in it, then you can no longer appeal to the symmetry of the situation as you do when you say:

A should think B's clock has dilated, and vice versa for B

If A is using the "common reference frame" instead of his own rest frame, A will not think B's clock has dilated, because A knows that B is at rest in the common reference frame while A himself is not.

The common reference frame is set by a pulsar, the frequency of which is gradually decaying and hence acts as a third clock.

How does this set a "common reference frame"? You can't just wave your hands and say so. You have to explain how.

the parties can agree at what time A should pass the star, in terms of the pulsar's clock

It depends on what you mean by this. If you mean, what frequency A sees the pulsar to be pulsing, in light signals it is receiving from the pulsar at the instant A passes the star, yes, everyone will agree on that. But that might not be the same thing as what you mean by "what time A passes the star, in terms of the pulsar's clock".

It won't be exact, but one should get near enough to get the relativistic effect to be significantly larger than any other defect.

I have no idea what you mean by this.

when A passes B

You mean "B" as in the B who, in the OP, is on Earth the whole time?

they exchange current time

Meaning, they each tell each other what their own clock reads?

and set times on each clock

I don't see why this needs to be included, since resetting the clocks at this point has no effect on anything else you're describing.

they cannot agree on which clock has the greatest elapsed time difference

Time difference from what to what? A and B weren't co-located before, so they have no previous direct comparison of their clocks. We have two different events for A--passing the star and passing Earth/B--but we only have one event for B, so we have no elapsed time for B to compare to at this point. You need to specify how B is determining his elapsed time.

To fill in all the blanks above the way I think you intend to fill them--though, as noted, you should not leave this to guesswork but should explicitly specify all these things:

I assume that the "common reference frame", as noted above, is a frame in which the Earth, the other star, and the pulsar, are all at rest.

I assume that the starting point for B's elapsed time is the event on Earth that happens at the same time, according to the common reference frame, as A passing the star.

Given those assumptions, B's elapsed time when A and B pass each other will be greater than A's, and both observers will agree on that.

B will explain his greater elapsed time in terms of time dilation: A's clock runs slower than his because A is moving in the common reference frame, while B is at rest.

A will explain B's greater elapsed time in terms of time dilation plus B's clock being set wrong: A will say that B set his clock to start his elapsed time, not at the same time A passed the star, but long before that--so long that even the fact that B's clock runs slow compared to A's, in A's rest frame, was not enough to compensate for B setting his elapsed time to start way in advance of A passing the star.

In other words, as is almost always the case when people get SR problems wrong, you are leaving out relativity of simultaneity: what happens "at the same time" as A passing the star is different in the two frames (B's rest frame, which is what you are calling the "common reference frame", and A's rest frame).

 

[Ian J Miller]

I apologise in that I expressed my question of whether one can avoid the simultaneity issue through a third reference clock poorly. (Or maybe I have missed something, which is why I am asking.) I shall try again. The question is, what do A and B see if they can coordinate to set their initial times on their clocks to some third clock. Assume as the pulsar slows, the parties agree that when it reaches precisely frequency x in a frame of reference defined as stationary, i.e. as seen by the star and by Earth, they set their clocks at time t zero. Traveller A in advance sets his ship into motion, and times his acceleration such that he reaches velocity 0.6c as he passes a given star precisely when pulsar reaches frequency x in the agreed fram of reference and then maintains constant velocity until he reaches Earth, with his clock running in the ship's frame of reference. Observer B sets his clock to time zero when the pulsar reaches frequency x, and makes any correction for the different distances, knowing that c is constant, and his clock runs constantly in his frame of reference. But when the ship passes Earth they exchange information and see what each other's clock says. What do they see, and what do they each see if they each take a rapid measurement of the pulsar's new frequency? Since B and the pulsar are essentially in the same frame of reference B should see the pulsar giving the expected frequency, but what does A see?

 

[PeterDonis]

whether one can avoid the simultaneity issue

One can't.

Assume as the pulsar slows, the parties agree that when it reaches precisely frequency x in a frame of reference defined as stationary

Since A is not at rest in the "stationary" frame, A cannot directly observe the pulsar's frequency in this frame. He will directly observe some different frequency. However, if he knows his velocity relative to the "stationary" frame (which he could measure by measuring his velocity relative to something at rest in that frame), he can adjust the pulsar frequency he directly observes using the relativistic Doppler shift formula, to obtain the pulsar's frequency in the "stationary" frame. So if this is what you intended, then yes, A can use this method to adjust his motion so he is passing the given star at 0.6c at the right time, and set his clock to zero at that time.

What do they see, and what do they each see if they each take a rapid measurement of the pulsar's new frequency?

A and B both see that B's clock has more elapsed time than A's clock. B and A explain this exactly the way I described in post #22.

A sees the same relativistic Doppler shifted pulsar frequency that he saw when he passed the given star and set his clock to zero, since his speed relative to the stationary frame (and therefore relative to the pulsar) has remained the same.

 

[mfb]

I apologise in that I expressed my question of whether one can avoid the simultaneity issue through a third reference clock poorly.

You cannot. You can have the clocks at the two stars synchronized as seen by these stars, or synchronized as seen by the pulsar, or synchronized as seen by the spacecraft , but you can never have them synchronized both for the stars and the spacecraft , or more generally for multiple observers that move (in the connecting direction) relative to each other.

 

[Dale]

@Ian J Miller Wouldn’t you be better off simplifying the scenario? The more you add complexities the less you will gain from an answer.

It is far better to consider the heart of your question and produce the simplest scenario capturing the essence.

whether one can avoid the simultaneity issue through a third reference clock

One cannot. The only way to avoid the simultaneity issue is to have the important simultaneous events happen at the same place.

 

[Ian J Miller]

Thank you.

 

[Piet Venter]

You cannot. You can have the clocks at the two stars synchronized as seen by these stars, or synchronized as seen by the pulsar, or synchronized as seen by the spacecraft , but you can never have them synchronized both for the stars and the spacecraft , or more generally for multiple observers that move (in the connecting direction) relative to each other.

So according to you the GPS system can never work, because the GPS clocks can’t be synchronized to the earth clock?

 

[Dale]

So according to you the GPS system can never work, because the GPS clocks can’t be synchronized to the earth clock?

The GPS clocks are only synchronized in the earth centered inertial frame. @mfb is correct that “you can never have them synchronized both for the stars and the spacecraft , or more generally for multiple observers that move (in the connecting direction) relative to each other”. The GPS is not a counterexample. The GPS clocks are not synchronized in any inertial reference frame where the earth is moving.

 

[PeroK]

So according to you the GPS system can never work, because the GPS clocks can’t be synchronized to the earth clock?

Some time ago someone posted a document on the GPS synchronization process. It not only takes into account SR and GR but the travel time of light through the atmosphere. And, the latter is affected by atmospheric conditions like air pressure and rain.

You should try to find that document and have a look through it.

 

[Demystifier]

Let's imagine that there are two people who has the same biological age.

That assumption is not as clear as you think it is. When I was 20, I had the same biological age as my dad when he was 20. But we didn't have the same biological age at the same time. Perhaps you wanted us to imagine two people who have the same biological age at the same time, but then you have to specify what do you mean by the "same time". The notion of "same time" is relative in the theory of relativity. As others told you, you will not get very far before you understand this central idea.

 

[Piet Venter]

The GPS clocks are only synchronized in the earth centered inertial frame. @mfb is correct that “you can never have them synchronized both for the stars and the spacecraft , or more generally for multiple observers that move (in the connecting direction) relative to each other”. The GPS is not a counterexample. The GPS clocks are not synchronized in any inertial reference frame where the earth is moving.

“The GPS clocks are only synchronized in the earth centered inertial frame.“

Perhaps you meant here the center of earth non rotating frame (which is approximately non inertial)? The surface of earth is of course not an inertial frame.

According to SR the earth clocks would be moving relative to the GPS satellite, hence the earth clocks must be unsynchronized due to relativity of simultaneity?

Yet the GPS system seems to work everywhere on earth, hence the predicted unsynchronization of earth clocks never happens?

 

[Ibix]

“The GPS clocks are only synchronized in the earth centered inertial frame.“

Perhaps you meant here the center of earth non rotating frame (which is approximately non inertial)? The surface of earth is of course not an inertial frame.

Yes.

According to SR the earth clocks would be moving relative to the GPS satellite, hence the earth clocks must be unsynchronized due to relativity of simultaneity?

Unsynchronized with respect to what?

Yet the GPS system seems to work everywhere on earth, hence the predicted unsynchronization of earth clocks never happens?

The GPS satellite clocks are deliberately set to tick the time of the Earth centered inertial frame, even though that is not their natural tick rate or simultaneity convention (there is no unique convention for orbiting bodies).

However, the "receiver" end of a GPS system has no need of high precision clocks, or consumer GPS units would have to be hundreds of thousands of pounds each. Relativistic effects on the receiver clocks are negligible compared to the inaccuracy of quartz clocks, and the GPS system is built to tolerate those much larger errors. So, while there are uncorrected relativistic issues with GPS receivers, they are completely negligible.

 

[PeroK]

Yet the GPS system seems to work everywhere on earth, hence the predicted unsynchronization of earth clocks never happens?

there's a nice story that the politicians and senior managers who commissioned the GPS system didn't believe in the theory of relativity, so they insisted that the system was first tried without the relativistic corrections. I.e. letting each clock run at its normal rate. Only when this failed did they allow the relativitistic corrections to be made. I.e. each satellite clock continually adjusts itself to take account of relativity.

I don't know whether the politicians and senior managers still doubt relativity. Quite what is the purpose of physicists lying about the laws of physics escapes me.

 

[Dale]

Perhaps you meant here the center of earth non rotating frame (which is approximately non inertial)? The surface of earth is of course not an inertial frame.

ECI is a pretty standard term. It is non-rotating and approximately inertial. The surface of the earth would be an earth-fixed frame, sometimes ECEF for earth centered earth fixed frame. It is non-inertial.

According to SR the earth clocks would be moving relative to the GPS satellite, hence the earth clocks must be unsynchronized due to relativity of simultaneity?

More explicitly and correctly, according to SR clocks synchronized in the momentarily co-moving inertial frame (MCIF) of any earth clock will be unsynchronized in the ECI frame because those clocks are moving relative to the ECI frame.

This is the relativity of simultaneity. Two inertial frames moving with respect to each other (like the ECI and the MCIF) disagree on simultaneity.

Note that the ECEF frame is a non-inertial frame, so that statement doesn't apply. The ECEF frame is designed to have the same simultaneity as the ECI frame. So clocks synchronized in the non-inertial ECEF frame will also be synchronized in the ECI frame but not in their own MCIFs.

Yet the GPS system seems to work everywhere on earth, hence the predicted unsynchronization of earth clocks never happens?

The GPS system doesn't interact with any system of clocks that are synchronized in the MCIF of any earth clocks. So why would that "unsynchronization" matter? The earth clocks in the GPS use the ECI time, not the synchronization of their MCIF. In fact, many of the earth clocks don't even measure their own time, they simply use multiple signals from the space clocks to calculate the ECI time at their location. In that sense, they aren't even actually clocks.