Can time run backwards in an accelerating frame?

[Gumby The Green]

TL;DR Summary

In the twin paradox, if the traveling twin keeps track of the proper time of a stationary observer who's farther away than the earthbound twin, how can he avoid concluding that the observer's time ran backwards at some points during the journey?

I'm trying to make sure I understand how the traveling twin tracks the time of his stationary earthbound sibling and the time of another stationary observer who's farther away. From what I've understood until now, it's pretty straightforward with the earthbound twin: In the traveler's frame, the Earth's time moves more slowly during each leg of the journey even though he ends up being younger than his sibling when he returns, but what makes up the difference is the fact that—again, in his frame—the Earth's time leaps forward during the turnaround. My first question is: Is this accounting of the Earth's time physically real in the traveler's frame? In other words, is it the one and only sensible description of the way that Earth's time runs in his frame? If not, what's an example of another description that would be sensible and consistent with our best theories?

Now about that other stationary observer who's farther away... In the traveler's accounting of her time, everything is the same except that her time leaps forward even more than the Earth's does in the traveler frame since he's farther away from her than he is from the earth. This would imply that she ages even more than the earthbound twin does during the journey, which can't be true since they're both stationary, so the only way it all works is if her time runs backwards in the traveler's frame at some point during the journey to undo the excess leap forward. You may be asking how time could move backwards like that but it turns out that there are reasons to think that it could and would during the two times when the traveler accelerates away from the stationary observers. So my second question is the same as the first one but with the distant observer's time instead of the Earth's time.

For more details on the scenario and my reasoning, please see the question I've posted on the Physics StackExchange, especially the spacetime diagram and the three paragraphs below it (and consider giving it an upvote if you think it's worthwhile so more people will see it). Dale (who I assume you all know) generously gave a detailed answer and suggested that I come here with my follow up questions, so that's what I'm doing. The main question I have about his answer right now is this: He says "A correct resolution that is purely physical is that the invariant proper time on [the traveler]’s clock is given by [insert formula]... this holds in all reference frames, and nothing further is either needed or 'physical'." But if there's a single correct way to track the time on the traveler's clock in the stationary frame, shouldn't there also be a single correct way to track the time on the stationary clocks in the traveler's frame (even though that "frame" is technically multiple frames pieced together)?

 

[Orodruin]

There is no unique way to ”track” the time of any object that is not colocated due to the relativity of simultaneity. You can make conventions such as the typical Einstein synchronization in an inertial frame but these have no real physical significance. This is the reason time dilation works both ways (both observers finding the other’s clock running slow - they have different conventions regarding what is ”the same time”).

The correct way to track the time on a clock is frame independent.

I suggest you read this as it makes analogies to Euclidean geometry, which is hopefully more familiar: https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

 

[anuttarasammyak]

In xy plane you are at the Origin. There is a bar in some distance. You observe its x length X and y length Y.
You spin in some angle together with your xy coordinates. Again you observe its x length X' and y length Y'.
I am sure you do not claim the difference between X and X', and Y and Y' because you are accustomed that spinning in some angle change the observation. This is xy story and SR tells xt story in a similar way. We can accustomed to that also.

 

[Gumby The Green]

You can make conventions such as the typical Einstein synchronization... they have different conventions regarding what is ”the same time”).

Thanks, I'll read that, but first, are you saying that the relativity of simultaneity is the same as the conventionality of simultaneity? If so, that really conflicts with what I've studied. From what I understand, those are two separate things (as John Norton explains at the beginning of this page on the conventionality of simultaneity)—the relativity of simultaneity still applies after a convention has been chosen and is what's relevant here. So can we just say that the same convention applies in every frame and move past that?

It's true that simultaneity is different in each frame (due to the relativity of simultaneity), but each frame still has a single truth about which events are simultaneous, doesn't it? The only thing I care about here is what's true in the frame of the traveling twin, i.e., what clock times for the stationary observers does he conclude are simultaneous to his at each point during his journey?

 

[PeroK]

Summary:: In the twin paradox, if the traveling twin keeps track of the proper time of a stationary observer who's farther away than the earthbound twin, how can he avoid concluding that the observer's time ran backwards at some points during the journey?

The traveling twin could set up a coordinate system where the earthbound twin's coordinates (worldline) are parameterised by his/her proper time. In this case, we would have a normal parameterisation where the proper time ran from $0$ to the end time continuously. There would be no jumps or loops in the proper time parameter.

For example, the earthbound twin could send a radio signal every second (this would effectively be a broadcast of his/her proper time). The traveling twin could use that in his/her calculations. The traveling twin would in all cases receive the signals in the order they were send: i.e. with increasing proper time.

 

[PeterDonis]

From what I understand, those are two separate things (as John Norton explains at the beginning of this page on the conventionality of simultaneity)

I have no idea where Norton's "conventionality of simultaneity" comes from. Part of the definition of an inertial frame is a particular simultaneity convention, based on Einstein clock synchronization. A so-called "inertial frame" in which some different simultaneity convention is adopted is not an "inertial frame" by any definition I've ever seen.

I note that Norton is not a physicist but a philospher, and his web page is certainly not a physics textbook or a peer-reviewed paper in physics. I suspect that if he ever did try to get the content of the page you referenced published in a physics paper, it would come back rejected from peer review.

 

[PeterDonis]

It's true that simultaneity is different in each frame (due to the relativity of simultaneity), but each frame still has a single truth about which events are simultaneous, doesn't it?

As I said in my previous post, part of the definition of an "inertial frame" is a particular simultaneity convention. So once you've picked out a particular inertial frame, what events are simultaneous in that frame is uniquely defined.

I would not say this counts as a "single truth" about which events are simultaneous because simultaneity is not something that is directly observable. It's a way of labeling spatially separated events as "happening at the same time" for convenience in calculation, not because there is any "true" fact of the matter about which spatially separated events happen "at the same time". There isn't.

 

[Orodruin]

Thanks, I'll read that, but first, are you saying that the relativity of simultaneity is the same as the conventionality of simultaneity? If so, that really conflicts with what I've studied. From what I understand, those are two separate things (as John Norton explains at the beginning of this page on the conventionality of simultaneity)—the relativity of simultaneity still applies after a convention has been chosen and is what's relevant here. So can we just say that the same convention applies in every frame and move past that?

You are mixing two things here. A convention such as the Einstein convention does not fix a single simultaneity because it does not single out a particular rest frame. However, there may be other conventions that do. The relativity of simultaneity is a wider concept relating to any foliation of spacetime into spacelike surfaces.

So can we just say that the same convention applies in every frame and move past that?

No, because you are dealing with accelerated frames and Einstein synchronization deals only with inertial frames. Apart from that, it is counter productive to understanding relativity to place too much value on ”keeping track” of the time for distant observers precisely because ”now” is completely conventional.

but each frame still has a single truth about which events are simultaneous, doesn't it?

Yes, but it is completely conventional and carries no physical significance whatsoever. Calling it ”truth” is probably to give it too much importance.

The only thing I care about here is what's true in the frame of the traveling twin.

Which, again, is an accelerated frame so you cannot use Einstein synchronization. The answer will depend on whatever coordinates you happen to choose.

 

[Orodruin]

A so-called "inertial frame" in which some different simultaneity convention is adopted is not an "inertial frame" by any definition I've ever seen.

In the sense of inertial frames as defined by Newton’s first law, essentially any affine coordinate system with one timelike and three spacelike basis vectors would probably qualify. That’s just word wrangling though and the most convenient choice is of course the typical Minkowski coordinates.

 

[Gumby The Green]

For example, the earthbound twin could send a radio signal every second... The traveling twin could use that in his/her calculations.

I'm going to replace "earthbound twin" with "distant observer" because that's whose time I think can run backwards in the traveler's frame. Now I think your proposed method would basically cause the traveler's accounting of the distant observer's time to be dictated by the relativistic Doppler shift, which wouldn't be consistent with the principles of relativity. Doesn't the time dilation formula tell us that, in the traveler's frame, stationary clocks run slower by the gamma factor? How can the traveler just decide to ignore that principle and track time in some other arbitrary way?

 

[PeroK]

I'm going to replace "earthbound twin" with "distant observer" because that's whose time I think can run backwards in the traveler's frame. Now I think your proposed method would basically cause the traveler's accounting of the distant observer's time to be dictated by the relativistic Doppler shift, which wouldn't be consistent with the principles of relativity. Doesn't the time dilation formula tell us that, in the traveler's frame, stationary clocks run slower by the gamma factor? How can the traveler just decide to ignore that principle and track time in some other arbitrary way?

You are confusing coordinate time with proper time. Every autumn, we set our clocks back an hour, but that doesn't mean that time runs backwards. That's simply a discontinuous change in coordinate time.

Note that in SR the principle of relativity applies to inertial reference frames (IRF). There are inherent difficulties, in general, in patching together instantaneous IRF's into a consistent accelerating (non-inertial) reference frame. These difficulties are solved using more advanced mathematical machinery than is needed to study SR using IRF's only.

All I've proposed is that the radio signals from any source (be it an earthbound twin or distant observer) can be tracked by anyone, regardless of their state of motion. And that those signals are always received in the same order as they were sent. In that sense, proper time can never be said to "run backwards".

 

[Orodruin]

whose time I think can run backwards in the traveler's frame

If it does, it is not really a physical effect but a result of using a bad coordinate system.

 

[Ibix]

I'm going to replace "earthbound twin" with "distant observer" because that's whose time I think can run backwards in the traveler's frame.

Time doesn't run backwards in the traveler's frame. What you are doing here is defining two inertial frames that each cover part of spacetime, but overlap. Then you are chaining them together, looking at the overlapping bit and saying "time is running backwards". Where that goes wrong is that chaining together overlapping frames isn't a legitimate operation.

If you let parts of your frame cover the same region of spacetime twice you get silly results, just like you do if you let a topographic map cover the same region twice. Go to Google maps and print a map of your town that includes your house. Then scroll a bit and print a different map that also includes your house. Now you have two maps covering overlapping regions. What you are trying to do by chaining frames is exactly the same as laying those two maps next to each other and looking surprised that you have two houses on your map. You don't have two houses - you have a bad map that doesn't honestly represent reality.

If you want to construct a coordinate system in which an observer who accelerates is treated as at rest at all times then you need to use curved coordinates. Simply chaining together two orthonormal systems won't work because they will overlap in places and miss out regions in other places.

 

[Sagittarius A-Star]

Summary:: In the twin paradox, if the traveling twin keeps track of the proper time of a stationary observer who's farther away than the earthbound twin, how can he avoid concluding that the observer's time ran backwards at some points during the journey?

That the time of the other stationary observer A, who's further away, formally runs backward is an artifact of the fact, that he is behind the Rindler horizon of C's frame, while the traveler C accelerates "effectively infinite" away from the stationary earthbound sibling B when starting the journey. Therefore, this scenario cannot be described correctly in the rest frame of C. It makes sense, that you use the OP tag "Rindler horizon".

At the turnaround, such an effect cannot happen, because then C is accelerating towards A:

For the traveling twin at turnaround, this gravitational field fills the universe. In a weak field approximation, clocks tick at a rate of t' = t (1 + Φ / c²) where Φ is the difference in gravitational potential. In this case, Φ = gh where g is the acceleration of the traveling observer during turnaround and h is the distance to the stay-at-home twin.

Source:
https://en.wikipedia.org/wiki/Twin_paradox#Viewpoint_of_the_traveling_twin

Behind the Rindler horizon, $\Phi / c^2 < -1$.

 

[jbriggs444]

For the standard twin paradox the traveling twin's hyperplane of simultaneity sweeps forward across the stay-at-home twin's world line as the traveler accelerates toward his twin.

If you want "time travel into the past", you want the Andromeda paradox where a small velocity change away from the remote object results in the hyperplane of simultaneity sweeping backward across the remote world line.

As others have pointed out, both results are artifacts of a particular choice of coordinates.

 

[Dale]

Is this accounting of the Earth's time physically real in the traveler's frame? In other words, is it the one and only sensible description of the way that Earth's time runs in his frame? If not, what's an example of another description that would be sensible and consistent with our best theories?

Not only is it not the one and only sensible description of the way that Earth’s time runs in his frame, it is not even a valid description.

First, the reason it is not valid: one of the requirements of a coordinate system is that it must be invertible, or one-to-one. See https://arxiv.org/abs/gr-qc/9712019 pages 33-37. While time running “backwards” isn’t a problem, time switching between “backwards” and “forwards” is a problem. It means that the same event will have multiple time coordinates. So the coordinate chart is no longer invertible, we cannot go from an event to a unique coordinate, which is not valid.

Now, you might try to “fix” this issue by patching several different coordinate systems together. That certainly is possible but it is actually quite difficult to do correctly because of the care that you have to take on the boundaries of the patches. More importantly, it is unclear how such a patchwork would represent “the traveler’s frame”, particularly in regions where more than one of the patches could conceivably cover the same events. These regions are exactly the ones where you get the “backwards” time you are describing.

The second issue is uniqueness. If Ingrid is an inertial observer and if Nancy is non-inertial then when we say “Ingrid’s frame” there is a unique convention for what that means. However, when we say “Nancy’s frame” there is no such established meaning. If we want to use “Nancy’s frame” then we have to define in exceptional clarity what we mean by that. And other authors are free to use a different meaning. All that is required of us is to follow the basic rules of coordinates described by Carroll above (invertible and smooth).

One such alternative meaning is Dolby and Gull’s radar coordinates. See https://arxiv.org/abs/gr-qc/0104077 especially figure 9. Note that in these coordinates the inertial observer’s time never “goes backwards” in the non-inertial observer’s frame. Furthermore, these radar coordinates have the advantage that they respect the second postulate even in the non-inertial frame, and they reduce to standard inertial coordinates when applied to inertial observers. In my opinion, they are a good convention for a non-inertial observer’s frame. They do not have time going backwards. So that is certainly not a necessary feature.

 

[Gumby The Green]

You are confusing coordinate time with proper time.

I don't think I am. I'm not making any claims about the proper time of the stationary observers, just about the times on their clocks that are simultaneous with the traveler at each point during his trip.

Note that in SR the principle of relativity applies to inertial reference frames (IRF).

I was only referring to inertial frames when I mentioned the time dilation formula, etc. I was giving a simple example of a part of the journey—the inertial legs—where using the relativistic Doppler shift clearly wouldn't give the traveler an accurate accounting of the time on the stationary clocks.

those signals are always received in the same order as they were sent. In that sense, proper time can never be said to "run backwards".

I doubt that that's a valid analysis. If the arrival times of the signals don't tell the recipient anything about the speed of the sender's time, why would they tell him anything about the direction of her time?

 

[PeroK]

I don't think I am.

Yes, you are. Take my example of clocks going back in the autumn. That doesn't mean time goes backwards in any physical sense. Only that we've chosen to reset the current time.

The situation with relativity of simultaneity regarding distant events is more subtle, but ultimately you are simply choosing to reset the time on a distant clock to reflect a changing simultaneity convention.

The concept that simultaneity is relative is one of the hardest things to digest in SR. For students and philosophers!

I'm not making any claims about the proper time of the stationary observers, just about the times on their clocks that are simultaneous with the traveler at each point during his trip.

"Simultaneous" is not an absolute concept. In fact, it's possible to do away completely with the concept of simultaneity. It's not actually a physical thing at all.

I doubt that that's a valid analysis. If the arrival times of the signals don't tell the recipient anything about the speed of the sender's time, why would they tell him anything about the direction of her time?

The "speed of the sender's time" is not a very helpful concept. Those signals tell you that physically time is not running backwards and that you are elevating an arbitrary coordinate effect into something of physical significance.

Moreover, physics is ultimately about what can be measured; and, in terms of someone on Mars, say, all you can measure are signals received from Mars (if you think about it). Everything else is interpretation and inference based on the signals you receive. Unless you go to Mars (or get someone there to do local measurements and send you the results), then all you have to work with are signals.

 

[Gumby The Green]

physics is ultimately about what can be measured... Everything else is interpretation and inference based on the signals you receive.

By that measure, reciprocal time dilation is on equally poor footing as the time inversion I'm talking about, isn't it?

 

[PeterDonis]

using the relativistic Doppler shift clearly wouldn't give the traveler an accurate accounting of the time on the stationary clocks.

You have it backwards. The Doppler shift is what the traveler actually sees. And the traveler actually sees the distant observer's clock always running forwards. He never sees it running backwards. That is an "accurate accounting" of the direction in which the distant observer's clock "flows". (The "rate" at which the traveler actually sees the distant observer's clock "flow" will of course vary as he travels, but it will always be greater than zero.)

This whole idea of "the time on the stationary clocks" is the part that's not accurate; more precisely, it's not telling you anything about the actual physics or what is actually happening to the distant observer. All it is telling you is how you chose your coordinates. The "correction" you apply to the Doppler shift the traveler actually sees, to get what you are (incorrectly) calling "the time on the stationary clocks", is coordinate-dependent and has no physical basis.

I doubt that that's a valid analysis.

Your doubt is seriously misplaced.

If the arrival times of the signals don't tell the recipient anything about the speed of the sender's time

You're wrong. They do. See above.

why would they tell him anything about the direction of her time?

Because they do. See above.

 

[PeroK]

By that measure, reciprocal time dilation is on equally poor footing as the time inversion I'm talking about, isn't it?

Yes, time dilation is purely a coordinate effect. It has no physical significance in itself.

Compare with differential ageing, which is physical.

As an aside, once you get past the basics of SR, time dilation, length contraction and simultaneity tend to disappear from physics. The importrant concepts are energy-momentum, four-vectors and invariant quantities.

 

[PeterDonis]

what you are (incorrectly) calling "the time on the stationary clocks"

To further elucidate this point:

I'm not making any claims about the proper time of the stationary observers

Yes, you are. See below.

just about the times on their clocks that are simultaneous with the traveler at each point during his trip.

You are very confused. By "the times on their clocks", you mean the times the stationary observers actually read on their actual clocks at some event that you are saying is "simultaneous with the traveler" at some point during the traveler's trip. Those times are proper times. That's what the times the stationary observers actually read on their stationary clocks are: proper times.

And since these times are proper times, you cannot obtain them by applying "corrections" to the Doppler-shifted observations the traveler actually sees. Those "corrections" give you coordinate times in whatever coordinate system you have chosen, not proper times.

 

[Gumby The Green]

Compare with differential ageing, which is physical.

When the twins are moving inertially away from each other, which of them is aging faster? The answer depends on which of their frames you're in, doesn't it?

 

[PeroK]

When the twins are moving inertially away from each other, which of them is aging faster? The answer depends on which of their frames you're in, doesn't it?

"Aging" itself is not a well-defined physical concept without further specification. Again, you are trying to take a coordinate effect (time dilation) and force a physical interpretation (aging). I would say that time dilation does not equate to aging. If aging is measured locally, then (by the laws of physics) all objects age at the same rate!

You can talk about differential aging if two objects meet twice, having taken different paths through spacetime. Then you can say that one object is absolutely older than the other. Or, that more proper time has elapsed for one than the other between meetings. That's a physical, invariant quantity that all observers agree on: object A aged $t_A$ seconds and object B aged ##t_B# seconds between the two meetings, which are well-defined events.

 

[Ibix]

When the twins are moving inertially away from each other, which of them is aging faster? The answer depends on which of their frames you're in, doesn't it?

It depends on your coordinate choice, yes. You aren't required to use the time of the frame in which you are at rest.

As a side note "which frame you are in" is seriously misleading. You're in all frames. The only question is which one you choose to use.

 

[Gumby The Green]

The Doppler shift is what the traveler actually sees. And the traveler actually sees the distant observer's clock always running forwards.

I understand that.

That is an "accurate accounting" of the direction in which the distant observer's clock "flows". (The "rate" at which the traveler actually sees the distant observer's clock "flow" will of course vary as he travels, but it will always be greater than zero.)

What is that based on though? If the signals don't accurately depict the exact speed of the distant clock's "flow", how are you sure that they accurately depict its direction in the traveler's frame at all times (even while its behind the traveler's Rindler horizon)?

it's not telling you anything about the actual physics or what is actually happening to the distant observer

I'm not making any claims about what's actually happening to the distant observer in their own frame. All I care about is what's true in the Traveler's frame.

The "correction" you apply to the Doppler shift the traveler actually sees, to get what you are (incorrectly) calling "the time on the stationary clocks"

I'm not applying any correction to the Doppler shift. I don't care about the Doppler shift or about anything the traveler sees. My argument is all about what can be logically deduced at the end of the journey from the most basic principles. I think you've misunderstood my argument, so please see the paragraph in my SE question that starts with "Here's my main argument".

 

[Gumby The Green]

Compare with differential aging, which is physical... You can talk about differential aging if two objects meet twice, having taken different paths through spacetime.

Are you implying that differential aging is 0% physical throughout the entire journey until the moment that the twins are standing in the exact same place (which is technically impossible), at which point it becomes 100% physical?

 

[Ibix]

Are you implying that differential aging is 0% physical throughout the entire journey until the moment that the twins are standing in the exact same place (which is technically impossible), at which point it becomes 100% physical?

There is no unambiguous way to compare clocks that are not at the same location, and you can get a huge range of values for "age now" unless the clocks are colocated. But for colocated clocks the recorded times must be agreed by all observers because this can have direct physical consequences. As for "technically impossible" you can have clock faces touching as they pass.

I would use "invariant" rather than "physical" because you can argue about what the second word means.

 

[PeroK]

My argument is all about what can be logically deduced at the end of the journey from the most basic principles.

The fundamental problem is that you've expended significant intellectual effort based on the assumption that there is a physical significance in a particular coordinate choice. In this case, that by choosing Rindler coordinates "time runs backwards" in some physically meaningful way.

Although your scenario is more sophisticated, ultimately it's no more worth analysing than the question of whether something is "really moving".

The difference between coordinate-dependent quantities and physically measurable invariant quantities is something that is worth learning. Your question is, sadly, despite the effort you've put into it, effectively meaningless in terms of physics.

Sorry, but there it is.

 

[PeroK]

Are you implying that differential aging is 0% physical throughout the entire journey until the moment that the twins are standing in the exact same place (which is technically impossible), at which point it becomes 100% physical?

No. I'm saying that you can only compare the ages of two objects unambiguously when they are colocated. With a strict definition, the question of relative age is meaningless until you have them back together.

Take for example two objects in the same orbit in opposite directions. There is continuous reciprocal time dilation between the objects, but yet no differential ageing when they meet. This exposes time dilation as a physically meaningless coordinate effect.

 

[Gumby The Green]

It depends on your coordinate choice, yes. You aren't required to use the time of the frame in which you are at rest.

"Coordinate choice" is synonymous with "reference frame choice", right? Are you saying that the twin who's moving at 0.6c does not have to conclude that the earthbound twin's rate of aging is only 80% of his own?

 

[PeroK]

Are you saying that the twin who's moving at 0.6c

This is meaningless. There is no concept of absolute velocity in physics.

does not have to conclude that the earthbound twin's rate of aging is only 80% of his own?

This is meaningless. As previously stated, aging is not synonymous with time dilation.

 

[Sagittarius A-Star]

If the signals don't accurately depict the exact speed of the distant clock's "flow", how are you sure that they accurately depict its direction in the traveler's frame at all times (even while its behind the traveler's Rindler horizon)?

It is physically impossible, that the traveler can receive light from behind the Rindler horizon.

 

[Ibix]

"Coordinate choice" is synonymous with "reference frame choice", right?

More or less. At this level, yes.

Are you saying that the twin who's moving at 0.6c does not have to conclude that the earthbound twin's rate of aging is only 80% of his own?

As PeroK says, your phrasing is sloppy. I think you mean "moving inertially at 0.6c with respect to the earthbound twin". Assuming that, sure. He could choose to work in the frame where both he and the stay at home are moving at equal speeds in opposite directions. They both have the same tick rate with that choice. His clock readings don't match with his coordinate time, but that's fine.

 

[Dale]

"Aging" itself is not a well-defined physical concept without further specification.

I am not sure I agree with that. Aging is the increase of proper time, particularly of an object whose internal state is a function of proper time.

On the other hand, aging “faster” or “slower” is not physical since it is a comparison between proper time and coordinate time.