Special relativity and expansion of the Universe, A paradox

[Rafeek AR]

Consider two bodies A &B are moving apart with a velocity V due to the expansion of space. According to an observer in A the body B is moving away and an observer in B feels the body A is moving apart. Can some one answer in which body the time dilates and why?. ( I am specifying once again that the case is movement under expansion of space and not due to ordinary kinematical motion. If you are interested in discussing it even though you don't know the right answer, reply your thoughts.)

 

[shihab-kol]

Well, I don't know much about this either but I will still try.
Time dilation occurs due to the curvature of space time and has a mathematical formula .
But if the space is expanding then both the bodies are moving away from each other.
According to my understanding of the matter, time dilation will be experienced in both as they are moving away relative to each other .

I am a novice so, my understanding may be wrong.
It is up to the more experienced members to say whether I am right or wrong.

 

[jbriggs444]

Consider two bodies A &B are moving apart with a velocity V due to the expansion of space. According to an observer in A the body B is moving away and an observer in B feels the body A is moving apart. Can some one answer in which body the time dilates and why?. ( I am specifying once again that the case is movement under expansion of space and not due to ordinary kinematical motion. If you are interested in discussing it even though you don't know the right answer, reply your thoughts.)

The subject line mentions special relativity. Let us start there. In special relativity, if you ask which of two moving bodies in relative motion are time-dilated, the correct answer is "it depends". It depends on what frame of reference you choose. If you choose the frame of reference in which body A is at rest, body B's time is dilated. If you choose the frame of reference in which body B is at rest, body A is time dilated. The thing that allows both to be true is the relativity of simultaneity. In order to assess time dilation, we need to have a notion of simultaneity. In special relativity, the natural simultaneity convention is Einstein clock synchronization. Einstein clock synchonization requires one free choice: A frame of reference.

But you make it clear that you are not interested in ordinary kinematic time dilation in special relativity.

If we consider two objects that have a recession velocity due to the expansion of the universe we still need to make a choice on clock synchronization in order to determine time dilation. But the choice of synchronization convention is not quite so simple. In our universe, one can choose to use "co-moving" coordinates. This is a coordinate system in which an observer at rest sees the universe as homogenous and isotropic. In particular, such an observer will see the same cosmic microwave background radiation in all directions. In co-moving coordinates, two objects that are both at rest are not time dilated relative to one another.

In other coordinates, time dilation can, of course, be different. There are no global inertial frames of reference in general relativity. The choice of a coordinate system/frame of reference is not as simple as picking one object to treat as being at rest.

 

[Ibix]

According to my understanding of the matter, time dilation will be experienced in both as they are moving away relative to each other .

No - neither experiences time dilation. Both say that their clocks are ticking normally. Both say that the other fellow's clock is ticking slowly.

As @jbriggs444 notes, though, this only applies in special relativity and making a conventional choice of simultaneity criterion. In general relativity it's more complex. Making the "obvious" choice of simultaneity criterion in the cosmological case he appears to be interested in, the two clocks both regard the other as ticking normally and all of the redshift is due to the expansion of space.

 

[EdgarLOwen]

Consider two bodies A &B are moving apart with a velocity V due to the expansion of space. According to an observer in A the body B is moving away and an observer in B feels the body A is moving apart. Can some one answer in which body the time dilates and why?. ( I am specifying once again that the case is movement under expansion of space and not due to ordinary kinematical motion. If you are interested in discussing it even though you don't know the right answer, reply your thoughts.)

Rafeek,

1. During the relative motion each observer sees the other's coordinate time running slower relative to the proper time of his own comoving clock. The two observers disagree.

2. However if the relative motion then stops then both observers will agree their clocks are now running at the same rate but that one observer's clock shows less actual elapsed time than the other (in the general case). The question is why is that? The answer is that the observer with less elapsed time on his clock has moved further in space thus he has moved less far in time.

Edgar L. Owen

 

[Rafeek AR]

No - neither experiences time dilation. Both say that their clocks are ticking normally. Both say that the other fellow's clock is ticking slowly.

As @jbriggs444 notes, though, this only applies in special relativity and making a conventional choice of simultaneity criterion. In general relativity it's more complex. Making the "obvious" choice of simultaneity criterion in the cosmological case he appears to be interested in, the two clocks both regard the other as ticking normally and all of the redshift is due to the expansion of space.

The subject line mentions special relativity. Let us start there. In special relativity, if you ask which of two moving bodies in relative motion are time-dilated, the correct answer is "it depends". It depends on what frame of reference you choose. If you choose the frame of reference in which body A is at rest, body B's time is dilated. If you choose the frame of reference in which body B is at rest, body A is time dilated. The thing that allows both to be true is the relativity of simultaneity. In order to assess time dilation, we need to have a notion of simultaneity. In special relativity, the natural simultaneity convention is Einstein clock synchronization. Einstein clock synchonization requires one free choice: A frame of reference.

But you make it clear that you are not interested in ordinary kinematic time dilation in special relativity.

If we consider two objects that have a recession velocity due to the expansion of the universe we still need to make a choice on clock synchronization in order to determine time dilation. But the choice of synchronization convention is not quite so simple. In our universe, one can choose to use "co-moving" coordinates. This is a coordinate system in which an observer at rest sees the universe as homogenous and isotropic. In particular, such an observer will see the same cosmic microwave background radiation in all directions. In co-moving coordinates, two objects that are both at rest are not time dilated relative to one another.

In other coordinates, time dilation can, of course, be different. There are no global inertial frames of reference in general relativity. The choice of a coordinate system/frame of reference is not as simple as picking one object to treat as being at rest.

Lets consider your first answer." It depends". If " it depends" on either frame of reference then that fact can be used to create a new "TWIN PARADOX". Consider again that both the twins are not coming back together. A return in the journey could only be possible with the help of " local kinematics" and that seems resolve this paradox but it didn't.
let's consider the other argument that both the frames are experiencing time dilation but the very saying does unknowingly imply a "preferred reference frame " according to which that statement is correct. But then again if the preferred reference frame is placed in the universe itself, the question of " whether time dilates in the preferred reference?" does exist. So a prfered reference frame is impossible in the case of expansion of universe.
The third option is " time doesn't dilates in any frame of reference". I don't know what considerations have he took to said that. The expansion of "space" is purely spatial and so the time has to dilate.

 

[Bandersnatch]

I am specifying once again that the case is movement under expansion of space

The point is, you can't use special relativity to describe it. SR only works in static space-times, and expanding universe is an example of a non-static space-time.

 

[Rafeek AR]

Rafeek,

1. During the relative motion each observer sees the other's coordinate time running slower relative to the proper time of his own comoving clock. The two observers disagree.

2. However if the relative motion then stops then both observers will agree their clocks are now running at the same rate but that one observer's clock shows less actual elapsed time than the other (in the general case). The question is why is that? The answer is that the observer with less elapsed time on his clock has moved further in space thus he has moved less far in time.

Edgar L. Owen

In the case of expansion the relative motion doesn't stops. It seems absolutely relative.

 

[Rafeek AR]

The point is, you can't use special relativity to describe it. SR only works in static space-times, and expanding universe is an example of a non-static space-time.

SR does works in expanding universe. What official reference do you have to prove your statement.

 

[Nugatory]

SR does works in expanding universe.

Special relativity assumes the existence of global inertial frames, which exist only in flat spacetime. In fact, that's why it's called "special" relativity; it applies to the special case of flat spacetime. Our expanding universe is not a flat spacetime, so special relativity does not apply across the entire spacetime.

 

[Ibix]

What official reference do you have to prove your statement.

Any relativity textbook, I should think. Certainly any general relativity textbook.

 

[weirdoguy]

What official reference do you have to prove your statement.

Umm, most of the books that deal with general relativity?

 

[sweet springs]

Hi. Co-moving coordinate time of Universe is what you expect to know. All the things that stick to expanding universe share it. Best.

 

[Rafeek AR]

Special relativity assumes the existence of global inertial frames, which exist only in flat spacetime. In fact, that's why it's called "special" relativity; it applies to the special case of flat spacetime. Our expanding universe is not a flat spacetime, so special relativity does not apply across the entire spacetime.

If in comoving distance there exist no global reference frame as you said, then you have to resolve in which frame of reference does the time dilates. Or can you specifically solve this issue with enough mathematics? ( in GR ofcourse).

 

[Rafeek AR]

Hi. Co-moving coordinate time of Universe is what you expect to know. All the things that stick to expanding universe share it. Best.

In FLRW matrices the spatial dimensions are multiplied by a factor a(t). If this factor have to multiply uniformly along all the three spatial dimensions then the problem of time dilation do exist isn't it?. Do my understanding have something faulty in it. I would like if someone could solve it with enough mathematics.

 

[Dale]

If in comoving distance there exist no global reference frame as you said, then you have to resolve in which frame of reference does the time dilates. Or can you specifically solve this issue with enough mathematics? ( in GR ofcourse).

In GR, the usual solution describing a homogenous and isotopic universe, the one consistent with the data, is called the FLRW metric. It is not a static metric, so there is no global gravitational time dilation. However, there is still local kinematic time dilation. Any non-comoving observer is time dilated relative to a local comoving observer, but two non-local comoving observers cannot be unambiguously compared.

 

[Rafeek AR]

The point is, you can't use special relativity to describe it. SR only works in static space-times, and expanding universe is an example of a non-static space-time.

If we can't use SR to solve it, does that implies the " twin paradox" we can create with this time dilation does need to be solved using GR.If so it violates a statement of Einstein himself " twin paradoxes are actually solvable within special relativity itself".

 

[jbriggs444]

If we can't use SR to solve it, does that implies the " twin paradox" we can create with this time dilation does need to be solved using GR.If so it violates a statement of Einstein himself " twin paradoxes are actually solvable within special relativity itself".

As has been pointed out, special relativity applies in flat space-time. The twin paradox of special relativity is resolvable within special relativity. A "twin paradox" within a non-flat space time would not be resolvable within special relativity.

Note that you have not posed a "twin paradox" yet. So there is nothing yet to resolve.

 

[Rafeek AR]

. However, there is still local kinematic time dilation. Any non-comoving observer is time dilated relative to a local comoving observer, but two non-local comoving observers cannot be unambiguously compared.

does your statement implies that any two non-local comoving observers "cannot" be compared with the help of even FLRW matrice or if we compared to know " time dilation" the FLRW matrice won't give an answer. Does that implies the " paradox" does exist or does not?.

 

[jbriggs444]

does your statement implies that any two non-local comoving observers "cannot" be compared with the help of even FLRW matrice or if we compared to know " time dilation" the FLRW matrice won't give an answer. Does that implies the " paradox" does exist or does not?.

You should first understand that the FLRW "metric" is a global feature of our expanding space-time. It will indeed resolve any "twin paradox" within our universe. It will do this by allowing one to unambiguously compute the elapsed time along any trajectory that either twin could follow (i.e. any timelike trajectory).

If you have two twins that start together, separate and follow different paths and then re-unite, this will allow you to unambiguously predict how much each twin has aged during their trips.

Note that a "metric" is not the same thing as a "frame of reference". A metric is a global feature. For any two events in space-time, the metric tells you how far apart those two events are. This terminology is made more clear in the mathematical field known as "topology". The metric measure of distance is a scalar function. It takes as input two events in space-time. It produces as output a scalar value. That value is either a positive real number (the two points have a timelike separation -- one is unambiguously after the other), an imaginary number (the two points have a spacelike separation -- you cannot get from one to the other at less than the speed of light) or zero (the two points are separated by a null interval -- a light signal could get from one to the other).

None of this requires a coordinate system to be set up. The inputs to this metric function are events.

A frame of reference or "coordinate chart" can be applied to label each event in space time with 4 dimensional coordinates (x, y, z, t). [More generally, we allow space-time to be separated into multiple patches with a different coordinate chart for each -- that's a manifold]. Given a coordinate chart, you could express the metric as a function that takes two coordinate tuples as input and produces a scalar as output. It's the same metric. It just takes coordinates as inputs instead of events.

Obviously the metric function you get for one coordinate chart could be entirely different from the metric function you get using another coordinate chart. But it turns out that the path length (and elapsed time is a path length) determined using one coordinate chart and its associated metric function will be identical to the path length determined using another coordinate chart and its associated metric function. That is to say that path length is an invariant feature of space-time. It does not depend on your frame of reference.

[Note that pretty much everything I know about general relativity, I've picked up by osmosis, lurking in these forums over the years]

 

[sweet springs]

I learned it in section 110 of Landau Lifshitz 's Classical theory of field. They call velocity against expanding universe "proper motion of the body". You can read it from open resource. This velocity decreases as universe inflates. It is similar to conservation of angular momentum. Best.

 

[Dale]

does your statement implies that any two non-local comoving observers "cannot" be compared with the help of even FLRW matrice

I am referring to a general property of Riemannian geometry. There is simply no unambiguous way to compare non-local vectors because parallel transport is path dependent in a curved manifold.

Think of two vectors on a sphere, each pointing north, on opposite sides of the equator. If you compare them by transporting one along the equator then you will find that they are parallel. But if you transport it over the pole then you will find that they are anti-parallel.

Does that implies the " paradox" does exist or does not?

The paradox does not exist. There is no definable gravitational time dilation in the FLRW spacetime. So a "paradox" involving gravitational time dilation is undefined.

 

[Nugatory]

If we can't use SR to solve it, does that implies the " twin paradox" we can create with this time dilation does need to be solved using GR.If so it violates a statement of Einstein himself " twin paradoxes are actually solvable within special relativity itself".

You'll have to tell us what sort of "twin paradox" you're thinking about - exactly how are you arranging for the twins to separate and reunite? - before we can answer that question for sure.

However, it is quite likely that you are misunderstanding the quoted statement. All twin paradoxes that involve acceleration in flat spacetime are solvable within special relativity itself, and whoever made that statement (You attribute it to Einstein. Do you have a source so that we can see it in context?) was almost certainly speaking in that context.

As an aside: If you are not completely familiar with the Twin paradox FAQ, it's well worth reading.

 

[Mister T]

does your statement implies that any two non-local comoving observers "cannot" be compared with the help of even FLRW matrice or if we compared to know " time dilation" the FLRW matrice won't give an answer. Does that implies the " paradox" does exist or does not?.

I think the heart of your misconception is that the twin paradox illustrates something more than time dilation. Like the twin paradox, time dilation is illustrated by a comparison of clocks. But in the case of time dilation at most only one of those clocks will be measuring proper time for an observer. In the case of the twin paradox things are carefully arranged so that both clocks measure an elapse of proper time.

 

[Rafeek AR]

You'll have to tell us what sort of "twin paradox" you're thinking about - exactly how are you arranging for the twins to separate and reunite? - before we can answer that question for sure.

However, it is quite likely that you are misunderstanding the quoted statement. All twin paradoxes that involve acceleration in flat spacetime are solvable within special relativity itself, and whoever made that statement (You attribute it to Einstein. Do you have a source so that we can see it in context?) was almost certainly speaking in that context.

In the usual twin paradox, the twins reunite and re-uniting needs a change in velocity some where along the path. One of the two twins can thus feel an inertia along its path and can therefor concludes that time dilates for the twin to whom inertia is felt.

In the case of expansion of space we can't arbitrarily re-unites the twins, since the movement is only along expansion of space. So as you said a proper twin paradox formulation is impossible in the case of expansion. But that doesn't implies that - the bodies are not in relative motion. So let's formulate it in a different way. The bodies A and B are moving away from each other and both of them are sending light signals in every second. The light signals thus contains each of theirs age. The paradox is that according to A the frequency of the light signals he sends should be greater than the frequency of light signals he is receiving. And according to B the frequency of sending is again greater than the frequency he is receiving. This is the paradox as I mentioned or doubted. All of your answers does seems that when considering FLRW matrice , the comoving bodies doesn't or perhaps doesn't states in "WHICH BODY TIME DILATES? A OR B". Can we tell A or B? Or can we tell A &B? Or can we tell not A &notB. Those three are the possible answers. So a simple logic can tell that any answer for the above problem should finally state anyone of the three solution.

 

[jbriggs444]

The paradox is that according to A the frequency of the light signals he sends should be greater than the frequency of light signals he is receiving. And according to B the frequency of sending is again greater than the frequency he is receiving. This is the paradox as I mentioned or doubted.

None of this is contradictory. Both A and B's predictions are correct. Both signals are red-shifted due to the expansion of the space between. This is what @Ibix pointed out in #4.

Edit: Note that it is "FLRW metric". Not matrice.

 

[russ_watters]

People make the very same error with regard to the normal twins paradox too; thinking by not reuniting the twins you create a paradox. You don't.

 

[sweet springs]

Hi. Emitted lights undertake red shift by expansion of univers. Hubble's law. Twin A see redder and younger than him figure of twin B and vice versa.

 

[Nugatory]

The paradox is that according to A the frequency of the light signals he sends should be greater than the frequency of light signals he is receiving. And according to B the frequency of sending is again greater than the frequency he is receiving.

There is no paradox here, any more than there is a paradox if the two twins were moving apart in the ordinary flat spacetime of special relativity. You're just seeing the Doppler effect at work; there's a good explanation in that FAQ that I linked to above.

If in comoving distance there exist no global reference frame as you said, then you have to resolve in which frame of reference does the time dilates. Or can you specifically solve this issue with enough mathematics? ( in GR of course).

This question is less well-specified than you may think.

First we have to be clear about exactly what it means to say that "time dilates". Let's say that I look at my wristwatch and see that it reads $T_1$, and at the same time that I look at my wristwatch and see $T_1$ some distant clock reads $T'_1$. I wait for a while, then look at my wristwatch again. Now I see that my wristwatch reads $T_2$; suppose that at the same time that I look at my wristwatch and see $T_2$ the distant clock reads $T'_2$. If $T'_2-T'_1<T_2-T_1$ then I say that the distant clock is running slow and is time-dilated.

But note that this is all critically dependent on what we mean by "at the same time". In flat spacetime there is a natural definition of "at the same time", which will lead you to the Lorentz transformations and the time dilation and length contraction formulas of special relativity - you just have to be careful to remember that observers moving relative to one another will end up with different notions of "at the same time", which explains most of the "paradoxes" of SR. However, in curved spacetime here is no such natural definition; all we have is "has the same time coordinate" which in turn depends on our completely arbitrary choice of how to assign time coordinates. If we choose to use comoving coordinates, we'll find that $T'_2-T'_1=T_2-T_1$; there is no time dilation and the Doppler shift between A and B is explained by expansion of space between them. However, if we use some other coordinate system we'll get some other result and come to a different conclusion about which if either clock is dilated.

[Edit: And to follow up on Russ's comment above: This would be a good time to review how relativity of simultaneity in ordinary special relativity leads to A's clock being slower than B's and B's clock also being slower than A's, when A and B are moving relative to one another]

 

[Dale]

The bodies A and B are moving away from each other and both of them are sending light signals in every second. The light signals thus contains each of theirs age.

This is perfectly fine.

The paradox is that according to A the frequency of the light signals he sends should be greater than the frequency of light signals he is receiving. And according to B the frequency of sending is again greater than the frequency he is receiving.

This is not a paradox. If you write it mathematically then you have $f_{AA}>f_{BA}$ and $f_{BB}>f_{AB}$ where the first letter indicates who is generating the signal and the second letter indicates who is measuring the signal.

Note that this is true in GR, SR, and Newtonian physics.

 

[Rafeek AR]

None of this is contradictory. Both A and B's predictions are correct. Both signals are red-shifted due to the expansion of the space between. This is what @Ibix pointed out in #4.

Edit: Note that it is "FLRW metric". Not matrice.

The usual red shift is due to the fact that the light ray have to travel a distance. And if the distance increases that creates a red shift from the earlier light rays. But in relative motion time in a moving body does dilates with respect to an another stationary body. In the body where time is dilated the clock ticks slowly and that very slowing would result in a frequency variation. Yes you are right that expanding makes a red shift in light along with the variation in frequency due to the time dilation in the moving body. Consider a usual moving body ( not under expansion) , there too the variation in frequency can be obtained and both A and B will observe the shift due to the increasing space in between them. But in the ordinary case one of the two body is moving through space and the other is stationary in space there occurs a time dilation in the moving body and the frequency shift thus obtained is explicit to the frequency shift due the increase in distance. The frequency shift due to the increase in distance would be similar in two bodies when checking the incoming light, but in one body the time dilates and the frequency shift varies by that amount from the other body, and we all know that is how the usual twin paradox is solved.
In the case of expansion neither of the bodies are moving through space but according to each of them the other one is moving. Yes the red shift due to the increase in distance between them would be similar, but each of them will calculate an extra shift for the other body due to the time dilation in the other body that both the observers doesn't know if the movement of other body is due to the expansion of universe or ordinary motion. That extra shift necessarily have to contradict since both the observers include it.

 

[jbriggs444]

But in the ordinary case one of the two body is moving through space and the other is stationary in space there occurs a time dilation in the moving body and the frequency shift thus obtained is explicit to the frequency shift due the increase in distance.

There is no such thing as "moving through space". There is no such thing as being "stationary in space". There is no such thing as time dilation due to motion in space.

An observed frequency can be explained in different ways in different coordinate systems. None of these explanations are right and none of these explanations are wrong in any absolute sense.

In the case of expansion neither of the bodies are moving through space but according to each of them the other one is moving.

In co-moving coordinates, neither of the bodies claims that the other is moving. Only that the distance between them is increasing.

 

[Ibix]

But in the ordinary case one of the two body is moving through space and the other is stationary in space there occurs a time dilation in the moving body and the frequency shift thus obtained is explicit to the frequency shift due the increase in distance.

Here you have adopted a frame of reference in which one of the bodies isn't moving and the other is. Fine, but that's an arbitrary choice.And the whole point of special relativity's time dilation is that once you subtract out the effects of the movement of the source on the frequency you receive there is something left over; this is time dilation.

In the case of expansion neither of the bodies are moving through space but according to each of them the other one is moving.

If you start from the position that something is both moving and not moving you will inevitably get contradictions. Noone would claim that the other body is both moving and not moving at the same time. However, in general relativity you can have a phenomenon called metric expansion where an unmoving object is getting further away due to the time dependence of the metric. In other words, we chose a definition of distance that changes with our chosen definition of time. Then we attribute redshift to the changing distance, not to time dilation.

 

[Dale]

in "WHICH BODY TIME DILATES? A OR B". Can we tell A or B? Or can we tell A &B? Or can we tell not A &notB. Those three are the possible answers

There is always another possible answer: "mu", meaning that the question itself is flawed and needs to be un-asked. Here the question is flawed because it assumes that time dilation is defined in this scenario, which is not the case.

 

[Rafeek AR]

There is no such thing as "moving through space". There is no such thing as being "stationary in space". There is no such thing as time dilation due to motion in space..

All that I mean by moving through " space" is - in a usual time- distance graph ( one directional i,e through x axis) the time dilates in a body which have an actual projection on x- axis. The other body will not have a projection in x axis, that have I point by saying " stationary in space". Finding out which body does have more projection in the spatial dimension is the task involved in solving the " earlier twin paradox" which I understood that that body breaks the symmetry. symmetry breakage is the essential tool through which the twin paradox is solved. My question is " can we break the symmetry of motion due to expansion, such that time dilation can be attributed to any of the bodies or to both of the bodies or to none of the bodies".