The scope of inertial time dilation?

[mysearch]

I am trying to resolve a number of questions concerning the scope of time dilation within the confines of special relativity, i.e. flat spacetime. For example, if I simply state that 2 inertial frames of reference pass each other with a relative and constant velocity [v], it is my understanding that special relativity cannot give precedence to either one of these inertial frames. As such, either could claim that time in the other frame ticks slower and even if we stopped these 2 systems, to compare time, we have not established any frame of reference by which to resolve the relative ticking of their respective clocks.

So can we consider similar example in which net time can be compared?

The example I was considering is essentially an extension of the twin paradox, which I know this forum has addressed countless times, so let me make it clear that I am not questioning the accepted position on this issue. However, the extension involves both twins going on an identical journey relative to planet Earth, which we might simply reference as [A]. Twin-1 travels a path ABCA, while twin-2 travels a reverse path ACBA. As such, both twins experience identical acceleration and velocity such that ABCA represents the mirror image of ACBA. However, as twin-1 goes from B to C and twin-2 goes from C to D, they pass each other, appearing as 2 inertial frames of reference with a combined relative velocity [v], such that the situation might appear as initially described above.

So how does time dilation affect the twins in this system?

Based on the resolution of the normal twin paradox, where one twin remains on Earth, the Earth is resolved as the inertial frame of reference from which the second twin accelerates away and returns. So, by the same argument, I am presuming that both twins in this extended example will age more slowly than a person remaining on Earth. However, unlike the initial description, when this set of twins both return to Earth, they have a common frame of reference against which they can compare the net rate of time experienced and their age.

So what is their relative age to each other?

As far as I can see, time dilation based on the Lorentz transforms would suggest that they must be the same age, as the reversing of acceleration and velocity does not seem to affect the outcome of the transforms. However, would appreciate any further in-sights before raising anymore issues. Thanks

 

[Fredrik]

So can we consider similar example in which net time can be compared?

A clock measures the proper time of the curve in spacetime that represents its motion, so if you know the curves you can calculate the time they're showing.

So how does time dilation affect the twins in this system?

I'm picturing them going in a circle, with positions labeled as on a clock. They start at the 6 o'clock position, go in opposite directions, and meet the first time at 12 o'clock. They will be the same age at all pairs of events where they have traveled the same distance.

If we choose to define their "points of view" as their comoving inertial frames, both will describe the other as aging slower than themselves at first. When they get to the 3 and 9 positions, they will describe each other as the same age and aging at the same rate. When they meet at 6, both will describe the other as the same age and aging slower than themselves. When they get to 3 and 9 again, same age, same aging rate.

There's nothing contradictory about this, because the comoving inertial frame is a different coordinate system at each point on the circle. Their simultaneity planes are tilted in different directions at different points. As they approach the 3 and 9 positions the first time, the effect of the changing "tilt" of the simultaneity planes is much larger than the effect of time dilation. That's why their ages can be the same even though time dilation has been "trying" to increase the age difference the whole time.

The most important lesson we can learn from examples like this is that it's much easier to just calculate the proper time of the curve directly, than indirectly by adding up contributions from time dilation while compensating for the changing tilting angles of the simultaneity planes.

So what is their relative age to each other?

They are the same age, and you're right that they're younger than people on Earth. Their world lines have the same proper time, and that's a larger proper time than that of Earth's world line between the same two events.

 

[mysearch]

Many thanks for the insights, it is much appreciated. While your comments have raised a number of questions in my mind, I want to consider the implications by drawing a few spacetime diagrams and working through a few practical examples. Thanks again.

 

[mysearch]

Many thanks again for the help, it is much appreciated, but could I try to clarify some of the points raised and some of my open issues.

A clock measures the proper time of the curve in spacetime that represents its motion, so if you know the curves you can calculate the time they're showing. ]I'm picturing them going in a circle, with positions labeled as on a clock. They start at the 6 o'clock position, go in opposite directions, and meet the first time at 12 o'clock. They will be the same age at all pairs of events where they have traveled the same distance. If we choose to define their "points of view" as their comoving inertial frames, both will describe the other as aging slower than themselves at first. When they get to the 3 and 9 positions, they will describe each other as the same age and aging at the same rate. When they meet at 6, both will describe the other as the same age and aging slower than themselves. When they get to 3 and 9 again, same age, same aging rate. There's nothing contradictory about this, because the comoving inertial frame is a different coordinate system at each point on the circle. Their simultaneity planes are tilted in different directions at different points. As they approach the 3 and 9 positions the first time, the effect of the changing "tilt" of the simultaneity planes is much larger than the effect of time dilation. That's why their ages can be the same even though time dilation has been "trying" to increase the age difference the whole time. The most important lesson we can learn from examples like this is that it's much easier to just calculate the proper time of the curve directly, than indirectly by adding up contributions from time dilation while compensating for the changing tilting angles of the simultaneity planes. I used the ABCA versus ACBA description because it allowed me to draw a parallel with the case of 2 arbitrary inertial frames passing each other. While these paths imply acceleration and deceleration, I assumed that the effects applied equally to both twins, such that the journey could be described essentially in terms of constant velocity. They are the same age, and you're right that they're younger than people on Earth. Their world lines have the same proper time, and that's a larger proper time than that of Earth's world line between the same two events.

Your example made me realize that I should try to draw the respective journeys as spacetime diagrams. I have attached an attempt, more by way of general reference, based on a simplified journey, where one twin goes ABA, while the other goes ACA. Again, the distances, acceleration and velocity are all identical mirror images.

One of the purposes of the identical journeys was the assumption that any effects of a brief (and hypothetical) period of ultra-high acceleration in the paths ABA and ACA would cancel, such that we could essentially describe the journey in terms of the constant velocity. As such, the attached diagram is only an approximation based on a constant velocity [v=0.6c] with respect to A, i.e. Earth.

In Earth frame-A, the twins both travel 6 LYs in 10 years at an ‘average’ velocity [v=0.6c]. In contrast, the proper time recorded by the clock accompanying each twin is 8 years. Due to space contraction of the paths ABA and ACA reduce from 6 LYs to 4.8 LY’s, such that each twin maintains the same perception of velocity [v=0.6c].

I have drawn one set of diagonal lines (45 degree] that align to [c], which as far as I can see is the only way the twins could exchange their local (proper) time. However, this exchanged time seem to reflect the effects of the propagation delay rather than anything meaningful in terms of time dilation.

So while I agree that the “comoving inertial frame is a different coordinate system” it also appears to be localised and isolated. While the view depicted by the spacetime diagram cannot be observed, it does seem to suggest that time would tick at the same rate for both twins throughout the journey, based on the assumptions of the diagram. Therefore, I am not sure I necessarily understand the intended implication of the statement “They will be the same age at all pairs of events where they have traveled the same distance”.

As a side note, there appears to be the suggestion that should 2 frames of reference be passing pass each other, with constant velocity, it does not automatic imply that the tick of the clock is really different. For example, when the 2 twins reach their initial destination B and C respectively, they are separated by 6 LY’s, although they can only account for 4.8 LY’s. However, the initial issue being raised for clarification is that they would be the same age and would appear to have been the same age throughout their journey. Again, would appreciate any further clarifications. Thanks

 

[Fredrik]

So while I agree that the “comoving inertial frame is a different coordinate system” it also appears to be localised and isolated.

Pick any point P on either curve. The comoving inertial frame at that point has the origin at P, a t axis that's tangent to the world line, which has slope 1/v in the diagram, and an x-axis that would have slope v if you drew it in the diagram. (That means that it makes the same angle with a horizontal line as the world line makes with a vertical line). The simultaneity lines are parallel to the x axis. Note what happens to the simultaneity lines at B and C. An event on the world line, just before the the turnaround, is simultaneous with a much later event on the other guy's world line, than an event just after the turnaround.

While the view depicted by the spacetime diagram cannot be observed, it does seem to suggest that time would tick at the same rate for both twins throughout the journey, based on the assumptions of the diagram.

Except for an enormous jump at the turnaround events. Since we're defining their "points of view" using the comoving inertial frames (this isn't the only option, but it is the simplest), from one twin's point of view, the other is much younger than him at the event just before the turnaround, and much older than him at the event just after the turnaround.

Therefore, I am not sure I necessarily understand the intended implication of the statement “They will be the same age at all pairs of events where they have traveled the same distance”.

In this diagram, pairs of events where they have traveled the same distance always lie on the same horizontal line. At two such events, they are the same age. However, from one twin's point of view at such an event, the other is either younger or older at that time, because the phrase "at that time" refers to coordinate time, and demands that he compare two events on the same simultaneity line of the comoving frame (slope v remember), not the same horizontal line.

As a side note, there appears to be the suggestion that should 2 frames of reference be passing pass each other, with constant velocity, it does not automatic imply that the tick of the clock is really different. For example, when the 2 twins reach their initial destination B and C respectively, they are separated by 6 LY’s, although they can only account for 4.8 LY’s.

I think there's some inappropriate language here. The first sentence is suggesting that there's such a thing as "real" time, and the second that there's a "real" distance.

 

[mysearch]

Fredrik, I appreciate the knowledgeable feedback, which is proving very useful as I am in still in the process of reading up on this subject and some of the implications of spacetime diagrams you are raising are new to me. Do you know of any good on-line tutorials that go into the details that you are raising? Despite the possibility of using some inappropriate language, in a technical sense I hope, I have added a few comments and attached another diagram in the hopes of clarifying my understanding of the points being highlighted.

Pick any point P on either curve. The comoving inertial frame at that point has the origin at P, a t axis that's tangent to the world line, which has slope 1/v in the diagram, and an x-axis that would have slope v if you drew it in the diagram. (That means that it makes the same angle with a horizontal line as the world line makes with a vertical line). The simultaneity lines are parallel to the x axis. Note what happens to the simultaneity lines at B and C. An event on the world line, just before the the turnaround, is simultaneous with a much later event on the other guy's world line, than an event just after the turnaround. Except for an enormous jump at the turnaround events. Since we're defining their "points of view" using the comoving inertial frames (this isn't the only option, but it is the simplest), from one twin's point of view, the other is much younger than him at the event just before the turnaround, and much older than him at the event just after the turnaround.

See attached diagram. I have added the lines of simultaneity to which I think you are referring; however, this is meant as question not a statement of fact. To be honest, if I have drawn the lines correctly just before and after B and C, I am not sure how they are to be interpreted in any overall context. However, I will wait for confirmation that I haven’t got the wrong idea.

In this diagram, pairs of events where they have traveled the same distance always lie on the same horizontal line. At two such events, they are the same age. However, from one twin's point of view at such an event, the other is either younger or older at that time, because the phrase "at that time" refers to coordinate time, and demands that he compare two events on the same simultaneity line of the comoving frame (slope v remember), not the same horizontal line.

If I have drawn the lines of simultaneity correctly for B, then I believe your statement above is referring to –B and +B as shown on the attached diagram. At these 2 points, i.e. just before and after B, the distances reflected in –B and +B and associated with B occur at equal time, i.e. t=t’. Again, this is a question rather than a statement.

I think there's some inappropriate language here. The first sentence is suggesting that there's such a thing as "real" time, and the second that there's a "real" distance.

I think I understand your concern, because I assume that you believe I am making some reference to ‘absolute’ time and distance? To be honest, I can’t help thinking that some aspects of the spacetime in the local frame appear illusionary, e.g. the effects at the turnaround points. At one level we know that the twins start and end the journey at the same age and, as such, this seems to be a real effect. Should this journey take place at some future time, then the effects of time dilations on the twins with respect to Earth time could also be proved to be ‘real’. How you prove the 'reality' of local time with respect to each of the moving twins is not clear to me. Equally, the issue of space contraction along ABA and ACA seems to be problematic, at least in my mind at this stage, because the distance before and after the journey is 6 LYs, but is required to be 4.8 LY’s on-route in order to maintain sub-light velocity in the moving frames. Again, thanks for the help.

 

[Fredrik]

Do you know of any good on-line tutorials that go into the details that you are raising?

Online, no, but there are a few books. I haven't read Taylor & Wheeler myself, but it's the one that gets the most recommendations. I like Schutz myself (the SR sections in his GR book), which according to the author was influenced by Taylor and Wheeler.

See attached diagram. I have added the lines of simultaneity to which I think you are referring;

Looks good.

If I have drawn the lines of simultaneity correctly for B, then I believe your statement above is referring to –B and +B as shown on the attached diagram. At these 2 points, i.e. just before and after B, the simultaneous distances reflected in –B and +B and associated with B occur at equal time, i.e. t=t’. Again, this is a question rather than a statement.

In the comoving inertial frame at event -B (the one that's called B- in the diagram), -B is simultaneous with -B', so when the twin on the right is at -B, his "point of view" is that his brother is much younger (and aging at a slower rate). In the comoving inertial frame at event +B, +B is simultaneous with +B', so when the twin on the right is at B+, his "point of view" is that his brother is much older (and aging at a slower rate).

Of course he can't know any of this at those events, since his brother might have had a rocket malfunction or something like that. A person's "point of view" isn't known to him at the time, at any time. It has to be calculated after the fact.

To be honest, I can’t help thinking that some aspects of the spacetime in the local frame appear illusionary, e.g. the effects at the turnaround points.

I would say that they're neither "real" nor "illusory". There's just more than one way to describe a set of events, and none of those ways is more "right" than all the others. An object's entire existence in spacetime is a coordinate-independent fact, but different coordinate systems are "slicing" it up in different ways, and describing the "slices" separately. This gives us different descriptions of the same thing.

At one level we know that the twins start and end the journey at the same age and, as such, this seems to be a real effect.

Measurement results can be described as real. If your measuring device displays a specific result at some event E, that's a coordinate-independent fact. The time and place at which E occurs are however just numbers assigned by a function (a coordinate system).

How you prove the 'reality' of local time with respect to each of the moving twins is not clear to me.

The way I define SR includes taking the axiom for time measurements to be "A clock measures the proper time of the curve in spacetime that represents its motion". So I consider it an axiom, not a derived result, and not something you'd prove.

Equally, the issue of space contraction along ABA and ACA seems to be problematic, at least in my mind at this stage, because the distance before and after the journey is 6 LYs, but is required to be 4.8 LY’s on-route in order to maintain sub-light velocity in the moving frames.

In the frame comoving with the twin on the right at A (when he has reached full speed), an object that in the diagram would be stationary at the location of B, would be 2.4 light-years away, and moving towards him at speed 0.6c. That distance, at that speed, should take 2.4/0.6=4 years.

 

[mysearch]

Thanks for all the help, but having hopefully got some of the basics into better focus, could I raise a time dilation example that is a little more ‘cosmic’ in scope. Again, I have attached a diagram simply for general reference of what is possibly a frivolous and technically flawed scenario. The diagram is trying to represent 2 galaxies diverging from a point of close proximity along the timeline of an expanding universe. As such, this example is analogous to the earlier example of the twins going from AB and AC in an identical fashion.

From one perspective, each galaxy might be seen to have a velocity of [v=0.6c], but from the ‘stationary’ perspective of an occupant within either galaxy, the net velocity is 0.88c, as calculated from the Lorentz transform. Within each galaxy is an Earth-like planet on which life has been evolving for millions of years. Of course, these galaxies have been diverging for billions of years and the actual rate of expansion will not have been linear, but for the sake of simplicity it is assumed that the divergent velocity is constant for the period of evolution on the Earth-like planets, such that gamma approximates to [2.1]

Based on previous posts, the line of simultaneity would suggest that time in the ‘other’ galaxy was ticking slower and therefore life would be less evolved than on the local planet. However, on one of these planets, a very clever scientist discovers a ‘worm-hole’ that allows near instantaneous transport to the other planet. The question is:

What will be the state of evolution on this other planet?

A: the same; B: lagging behind; C: advanced:?

 

[Fredrik]

This is a much more difficult problem. Cosmic expansion is usually studied in FLRW coordinates. This is a coordinate system in which every galaxy stays at fixed spatial coordinates, but the proper distance between them in a hypersurface of constant time coordinate (i.e. "space") is growing with increasing time coordinate. But the galaxies themselves, are described as not moving in this coordinate system.

The derivative with respect to t, of the proper distance between two such galaxies, can be >c without violating local Lorentz invariance. Many of the galaxies we see through telescopes are "moving away from us" faster than c in this sense (while not moving at all, in terms of spatial FLRW coordinates). If the rate of expansion is slower than linear expansion, it's possible in principle to aim your rocket at the other galaxy, accelerate for a while, turn off the engine, and get there in a finite time. (It helps to picture an ant walking on an inflating balloon).

In GR, the concept of "comoving inertial frame" is replaced by the concept of "normal chart". Unfortunately, there's more than one kind (Riemannian normal charts, Fermi normal charts,...). However, in a small region of spacetime around on a point on the world line of (either) one of the galaxies, I think they all agree (approximately), both with each other and with the FLRW coordinates. So there's nothing really weird about defining both observers' points of view using the same FLRW coordinates, and if we do, they agree about simultaneity, and therefore, there's no time dilation.

This is however just one of several ways to define their points of view. Another one is to use normal charts. If we do, I think the situation will be pretty similar to a SR scenario involving two rockets going in opposite directions, but I don't know how similar, because I've never done the calculations. I also don't know if the results depend significantly on what kind of normal chart we're using. But I'm pretty sure that they all involve a disagreement about simultaneity and an associated time dilation.

Regarding that wormhole, the phrase "near instantaneous transport" implies that the time coordinate of arrival is only a little higher than the time coordinate of departure, but you didn't say in which coordinate system.

 

[mysearch]

This is a much more difficult problem. Cosmic expansion is usually studied in FLRW coordinates. This is a coordinate system in which every galaxy stays at fixed spatial coordinates, but the proper distance between them in a hypersurface of constant time coordinate (i.e. "space") is growing with increasing time coordinate. But the galaxies themselves, are described as not moving in this coordinate system.

My knowledge of modern cosmology is only cursory, but I have taken a quick look at the FLRW and the Schwarzschild metrics, which also raise some equally interesting problems about spacetime. In the context of flat spacetime [k=0] and the general assumption of homogeneous space, i.e. no wonderful gravitational anomalies like black holes, I am assuming the FRW coordinates can be related to the following reduced form of the FRW metric: [tex]ds^2 = dt^2 + a^2(t)dr^2[/tex]. If so, it seems to be describing the expansion of space [dr] as function of acceleration, which is itself a function of time (t).

The derivative with respect to t, of the proper distance between two such galaxies, can be >c without violating local Lorentz invariance. Many of the galaxies we see through telescopes are "moving away from us" faster than c in this sense (while not moving at all, in terms of spatial FLRW coordinates). If the rate of expansion is slower than linear expansion, it's possible in principle to aim your rocket at the other galaxy, accelerate for a while, turn off the engine, and get there in a finite time. (It helps to picture an ant walking on an inflating balloon).

I agree that expanding space can lead to >c without violating the principles of special relativity, but the question of movement in an expanding universe is an interesting issue. From a general definition of velocity [v=d/t], the issue of the relative velocity would appear unaffected by what was actually causing the increase in [d]. From an energy perspective, kinetic energy may be somewhat ambiguous, but I would have thought that a change in gravitational potential could be quantified, at least, on a local level between galaxies. However, this is probably a topic for another sub-forum, although it is an interesting one.

In GR, the concept of "comoving inertial frame" is replaced by the concept of "normal chart". Unfortunately, there's more than one kind (Riemannian normal charts, Fermi normal charts,...). However, in a small region of spacetime around on a point on the world line of (either) one of the galaxies, I think they all agree (approximately), both with each other and with the FLRW coordinates. So there's nothing really weird about defining both observers' points of view using the same FLRW coordinates, and if we do, they agree about simultaneity, and therefore, there's no time dilation. This is however just one of several ways to define their points of view. Another one is to use normal charts. If we do, I think the situation will be pretty similar to a SR scenario involving two rockets going in opposite directions, but I don't know how similar, because I've never done the calculations. I also don't know if the results depend significantly on what kind of normal chart we're using. But I'm pretty sure that they all involve a disagreement about simultaneity and an associated time dilation.

I would have thought the main difference to my example, shown attached to post #8, would be that acceleration factor in the FRW metric would cause the straight lines associated with constant velocity to become curves? However, I assumed that the accelerated expansion of the universe over ‘just’ the last 1-2 millions years of evolution would not be that significant, at least, on the scale of the universe as a whole.

Regarding that wormhole, the phrase "near instantaneous transport" implies that the time coordinate of arrival is only a little higher than the time coordinate of departure, but you didn't say in which coordinate system.

Yes, it another interesting point, albeit based on a completely hypothetical assumption of a wormhole. I guess at one level the wormhole would create a separate spacetime interval define by spatial distance [d] only; while in the normal, non-wormhole, route the spacetime interval would be defined in terms of the FRW metric, i.e. time + distance as a function of a(t). However, it is possibly only another illustration that the spacetime interval between 2 points depends on the path travelled. Anyway, I realize that I am wandering off topic and beyond the normal scope of this forum. Thanks

 

[mysearch]

I known PF doesn’t like speculative theories to be discussed within its main forums, so my intention is not necessarily to invite a discussion of the following papers, but rather to simply ask whether anybody knows the ‘status’ of the work of C. S. Unnikrishnan? I have quickly read through the first 7-page paper, which seems to raise some fairly reasonable questions about the twin paradox in respect to Einstein’s first explanation back in 1918 plus subsequent interpretations based on accelerated frames, lines of simultaneity and time encoded light beams etc.

http://www.ias.ac.in/currsci/dec252005/2009.pdf"

The second paper is 39-pages and I didn’t want to waste too much time at this stage, if more knowledgeable people had already reviewed the work. I suspect that mainstream may not have taken to the suggestion of some sort of absolute frame of reference linked to CMB.

http://arxiv.org/PS_cache/gr-qc/pdf/0406/0406023v1.pdf"

 

[Passionflower]

I agree that expanding space can lead to >c without violating the principles of special relativity,...

I do not agree with that.

In the theory of special relativity massive objects always move <c with respect to each other, no exceptions!

A FLRW spacetime is not flat and special relativity does simply not apply here.

 

[mysearch]

I do not agree with that. In the theory of special relativity massive objects always move <c with respect to each other, no exceptions! A FLRW spacetime is not flat and special relativity does simply not apply here.

I guess I was really looking for a response to the questions raised in https://www.physicsforums.com/showpost.php?p=2849795&postcount=11". Any help on this matter would be appreciated. Thanks.

 

[mysearch]

Inquiry about 'Cosmic Relativity'

I decided that it might be best to move my inquiry about the cosmic relativity paper written by C. S. Unnikrishnan to the ‘Beyond the Standard Model’ forum. This is the link to the thread if anybody is interested: https://www.physicsforums.com/showthread.php?p=2850780#post2850780"