# Acceleration and the twin paradox

#### Demystifier

Science Advisor
2018 Award
Thank You and I am sorry for lack of clarity. Under the effect of memory is meant: the coming behavior of a body depends on the previous history of this body. In common worldview it is enough to know position and velocity of a body to know its future evolution. Here it is not the case, am I right? I understand, that it is not easy question for the rapid answering. Author of this "memory" remark is Dmitri Martila.
The time integral appearing in the paper is not really a memory effect. It is just a summation of infinitesimal increments, not much different from a non-relativistic formula for a traveled distance as function of time $$x(t)=\int dt\,v(t),$$ where $v(t)$ is a time-dependent velocity.

#### PhoebeLasa

More specifically, I think the key to it is understanding what coordinate transformations do to a "now" (simultaneity plane), and from that how you have two completely different definitions of "now" right before and right after switching reference frames (acceleration).
I think you're right. But I'm not sure that the terminology "switching reference frames" is the best way to phrase it (even though that's commonly-used terminology). You can regard the accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip. It's not an inertial reference frame, but it is a reference frame. And that reference frame is such that, when he is accelerating toward the home twin, he will say that the home twin is rapidly getting older. That is the key to understanding the traveler's perspective in the twin paradox.

#### stevendaryl

Staff Emeritus
Science Advisor
I think you're right. But I'm not sure that the terminology "switching reference frames" is the best way to phrase it (even though that's commonly-used terminology). You can regard the accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip. It's not an inertial reference frame, but it is a reference frame. And that reference frame is such that, when he is accelerating toward the home twin, he will say that the home twin is rapidly getting older. That is the key to understanding the traveler's perspective in the twin paradox.
In the special case of constant acceleration, there is a nice coordinate system, Rindler coordinates, but if the acceleration is nonconstant, then it's pretty hopeless to come up with a coordinate system in which the rocket is always at rest. For instance, if the rocket accelerates for a while, drifts for a while, decelerates, then drifts back to where it started, there is no good way to describe that using a noninertial coordinate system. The way you have to describe situations like that is either to use inertial coordinates, where the rocket is not at rest, or else use charts, which are coordinate systems that only apply to small regions of spacetime. (Then you have to worry about relating coordinates of one chart with coordinates of the other).

#### Dale

Mentor
I'm not sure that the terminology "switching reference frames" is the best way to phrase it (even though that's commonly-used terminology)
I truly hate that terminology. It conveys the idea that a reference frame is something which has a physical existence and some limited spatial extent, both of which I think lead to conceptual errors in novice students.

#### PhoebeLasa

In the special case of constant acceleration, there is a nice coordinate system, Rindler coordinates, but if the acceleration is nonconstant, then it's pretty hopeless to come up with a coordinate system in which the rocket is always at rest. For instance, if the rocket accelerates for a while, drifts for a while, decelerates, then drifts back to where it started, there is no good way to describe that using a noninertial coordinate system.
[...]
Brian Greene, in his book and NOVA series called "The Fabric of the Cosmos", gave an example where the acceleration wasn't constant (and wasn't even one-dimensional, it was small, slow circular motion at an extremely great distance). And the result was that the person who was riding (a bicycle) around in a small circle would say that the local time at a place an extremely great distance away was varying back and forth over several centuries, once for each completed circle.

Somewhere in some old posts on this forum, I've seen a description of how those kinds of calculations are done, but I haven't been able to find them lately.

#### PAllen

Science Advisor
Brian Greene, in his book and NOVA series called "The Fabric of the Cosmos", gave an example where the acceleration wasn't constant (and wasn't even one-dimensional, it was small, slow circular motion at an extremely great distance). And the result was that the person who was riding (a bicycle) around in a small circle would say that the local time at a place an extremely great distance away was varying back and forth over several centuries, once for each completed circle.

Somewhere in some old posts on this forum, I've seen a description of how those kinds of calculations are done, but I haven't been able to find them lately.
Many physicists have criticized this statement. In fact, it describes an invalid coordinate system (the same distant event has multiple coordinates, while a well formed coordinate chart is one-one with events). This issue is discussed in the Nikolic paper referenced by Demystifier, which argues that non-inertial frames are only locally physically meaningful. This description by Greene comes from trying to give an inconsistent global interpretation to non-inertial frames.

#### PAllen

Science Advisor
In the special case of constant acceleration, there is a nice coordinate system, Rindler coordinates, but if the acceleration is nonconstant, then it's pretty hopeless to come up with a coordinate system in which the rocket is always at rest. For instance, if the rocket accelerates for a while, drifts for a while, decelerates, then drifts back to where it started, there is no good way to describe that using a noninertial coordinate system. The way you have to describe situations like that is either to use inertial coordinates, where the rocket is not at rest, or else use charts, which are coordinate systems that only apply to small regions of spacetime. (Then you have to worry about relating coordinates of one chart with coordinates of the other).
You can generally use one physically motivated chart over any arbitrary non-inertial world line by restricting its scope to a world tube around the world line. The more extreme the mix of accelerations, the narrower the world tube must become to remain a valid chart. Basically, I'm speaking of Fermi-Normal coordinates. If you try to extend them far from the origin world line, you are forced to use multiple, overlapping charts. However, if you restrict them to a narrow world tube, the a single chart is possible. This is equivalent to the approach in the Nikolic paper, I believe.

#### stevendaryl

Staff Emeritus
Science Advisor
You can generally use one physically motivated chart over any arbitrary non-inertial world line by restricting its scope to a world tube around the world line. The more extreme the mix of accelerations, the narrower the world tube must become to remain a valid chart. Basically, I'm speaking of Fermi-Normal coordinates. If you try to extend them far from the origin world line, you are forced to use multiple, overlapping charts. However, if you restrict them to a narrow world tube, the a single chart is possible. This is equivalent to the approach in the Nikolic paper, I believe.
Okay, but I think what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.

#### Nugatory

Mentor
Okay, but I think what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.
It does not, if the other twin is outside the tube. Nor will any other coordinate system, as there's an assumption about simultaneity embedded in the word "when" in any question that starts "When the travelling twin is X years old...."

One of the keys to getting people through the twin paradox is getting them to understand that the question is ill-formed except when both twins are at the same place at the same time.

#### ghwellsjr

Science Advisor
Gold Member
...You can regard the accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip. It's not an inertial reference frame, but it is a reference frame. And that reference frame is such that, when he is accelerating toward the home twin, he will say that the home twin is rapidly getting older. That is the key to understanding the traveler's perspective in the twin paradox.
There are other ways for an "accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip" than the one you are assuming and they can have the home twin getting older at different rates. Don't assume that the non-inertial reference frame that Brian Greene promotes is the only way to do it. In other words, there is more than one key to understanding the traveler's perspective in the twin paradox.

#### ghwellsjr

Science Advisor
Gold Member
In the special case of constant acceleration, there is a nice coordinate system, Rindler coordinates, but if the acceleration is nonconstant, then it's pretty hopeless to come up with a coordinate system in which the rocket is always at rest. For instance, if the rocket accelerates for a while, drifts for a while, decelerates, then drifts back to where it started, there is no good way to describe that using a noninertial coordinate system. The way you have to describe situations like that is either to use inertial coordinates, where the rocket is not at rest, or else use charts, which are coordinate systems that only apply to small regions of spacetime. (Then you have to worry about relating coordinates of one chart with coordinates of the other).
Radar coordinates work just fine with non-constant accelerations. They also work equally fine with constant accelerations. They also work equally fine with no acceleration (inertial observers). They work equally fine with multiple observers/objects accelerating in any arbitrary manner. They work fine in all circumstances.

#### PAllen

Science Advisor
Okay, but I think what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.
No, it won't. You would need some other coordinate chart. One based on radar simultaneity will provide an answer for any situation where the the origin world line is inertial before some event, and inertial again after some event, no matter what happens in between. However, without this restrictions, radar simultaneity also fails for arbitrary non-inertial world lines.

#### PAllen

Science Advisor
Radar coordinates work just fine with non-constant accelerations. They also work equally fine with constant accelerations. They also work equally fine with no acceleration (inertial observers). They work equally fine with multiple observers/objects accelerating in any arbitrary manner. They work fine in all circumstances.
Unfortunately, this is not true. For an eternally accelerating observer, radar coordinates have exactly the same limited coverage as Rindler coordinates.

#### ghwellsjr

Science Advisor
Gold Member
Okay, but I think what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.
Most people may want an answer to the question of how old the home twin is for any age of the traveling twin but they should be told it's a multiple-choice problem where the last choice is "all of the above".

#### pervect

Staff Emeritus
Science Advisor
Okay, but I think what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.
I'm not sure there is an answer to this question.

Let's talk about a specific case where the issue arises. Suppose we have a classic twin paradox set up, with travelling twin, accelerating at a constant 1g. Said twin asks "what time is it on Earth now" when when he's one light hear away from Earth, at the point where the Earth is at or past the accelerating twin's Rindler horizion, such that light signals emitted from the Earth will no longer be able to catch up with the accelerating twin, assuming he keeps accelerating at the same rate.

The most careful answer is to point out the constraints on the size of an accelerated frame, and discuss how simultaneity is relativeI think, though I suspect in many cases people want and expect an answer to the issue of simultaneity as stevendaryl points out. However, it i s reasonably likely that the people expecting/demanding an answer don't fully accept the fact that simultaneity is relative, part of their expectation of an answer is a carry over from the concepts of absolute time where simultaneity was universal rather than relative.

I don't have a better answer at this point, I think a lot of popularized answers in relativity have come about because it's easier than trying to explain to people that there is no more absolute simultaneity in SR. After a while of trying to explain that simultaneity is relative, I can see why people don't want to go through the hassle, especially when the people asking their questions are asking something that seems unrelated without realizing that the relativity of simultaneity even enters into a complete answer to their seemingly unrelated question.

#### PeterDonis

Mentor
what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin?
And the correct answer to this question is "mu"; as Nugatory pointed out, the question is ill-formed. IMO it's better to just face that up front, rather than trying to salvage people's pre-relativistic intuitions in some form. Understanding why the question is ill-formed is a key part of understanding relativity.

#### PeterDonis

Mentor
I can see why people don't want to go through the hassle, especially when the people asking their questions are asking something that seems unrelated without realizing that the relativity of simultaneity even enters into a complete answer to their seemingly unrelated question.
But part of understanding relativity is understanding how these things that don't seem related, to one's pre-relativistic intutions, actually are related. Again, IMO it's better to get that out on the table up front, to just bluntly say "your pre-relativistic intuitions are wrong and you need to unlearn them to really understand relativity", instead of going in circles trying to explain without really explaining (because the real explanation requires giving up the intuitions that you're going in circles trying to preserve in some form).

#### pervect

Staff Emeritus
Science Advisor
But part of understanding relativity is understanding how these things that don't seem related, to one's pre-relativistic intutions, actually are related. Again, IMO it's better to get that out on the table up front, to just bluntly say "your pre-relativistic intuitions are wrong and you need to unlearn them to really understand relativity", instead of going in circles trying to explain without really explaining (because the real explanation requires giving up the intuitions that you're going in circles trying to preserve in some form).
On the whole, I agree. Certainly I'm not going to oppose anyone who tries to explain things more fully, though I may not always feel motivated enough personally, especially if the target audience seems unreceptive or appears to lack needed background.

My current thinking is that it is good to point out that textbooks (specifically, MTW) do say that there are constraints on the size of an accelerated frame, and the reason for this restriction is to ensure that every event in space-time has one and only one set of coordinates in the chart - this is basically your position as well, if I'm reading your posts correctly.

Also I think it's worth pointing out that accelerating observers may not be able to exchange light signals (or any other sort of signals) with non-accelerating observers in certain cases, and to give some examples of such cases, such as the rocket accelerating at 1g who is 1 light year away (in the Earth frame). It's also needed for relevance to point out that this lack of ability to exchange any sort of signals does makes "clock synchronization" rather problematic. On the whole I don't think I need to say any more than that, really.

#### ghwellsjr

Science Advisor
Gold Member
Radar coordinates work just fine with non-constant accelerations. They also work equally fine with constant accelerations. They also work equally fine with no acceleration (inertial observers). They work equally fine with multiple observers/objects accelerating in any arbitrary manner. They work fine in all circumstances.
Unfortunately, this is not true. For an eternally accelerating observer, radar coordinates have exactly the same limited coverage as Rindler coordinates.
You're right, I keep forgetting about those eternally accelerating observers. I should have limited my comments to the twin scenario that the OP specified, which is the one that stevendaryl was referring to.

#### dmitrrr

Nikolic has introduced no new postulates, he's further developing the implications of the two basic postulates upon which SR is based.
Thank You. Excuse me, I am old enough to remember, that the Physics is the same in all inertial reference systems. So I find it hard to believe, that the Physical laws are looking the same way in all systems: sir Newton has not used name "non-inertial system" in his three laws. Be well.

#### PAllen

Science Advisor
Thank You. Excuse me, I am old enough to remember, that the Physics is the same in all inertial reference systems. So I find it hard to believe, that the Physical laws are looking the same way in all systems: sir Newton has not used name "non-inertial system" in his three laws. Be well.
Who said they look the same in inertial versus non-inertial frames? Nobody in this discussion said this.

Newtonian physics certainly covers non-inertial frames (even if that word wasn't used). Centrifugal force, coriolis force, are new 'fictitious' forces that have to be added to the laws true in inertial frames to describe motion in non-inertial frames.

#### stevendaryl

Staff Emeritus
Science Advisor
And the correct answer to this question is "mu"; as Nugatory pointed out, the question is ill-formed. IMO it's better to just face that up front, rather than trying to salvage people's pre-relativistic intuitions in some form. Understanding why the question is ill-formed is a key part of understanding relativity.
Well, there are two different aspect to the claim that the question has no answer. The first aspect is that it's relative to the observer. For any inertial observers, there is a more-or-less unique, best answer to the question: "How old is twin A when twin B is age X?" But in the case of noninertial observers, "relative to the observer" doesn't even give a unique answer.

#### stevendaryl

Staff Emeritus
Science Advisor
Radar coordinates work just fine with non-constant accelerations. They also work equally fine with constant accelerations. They also work equally fine with no acceleration (inertial observers). They work equally fine with multiple observers/objects accelerating in any arbitrary manner. They work fine in all circumstances.
I suppose. I don't actually think of radar coordinates as being very meaningful though, because they make the answer to the question: "Relative to twin A, how old is twin B when twin A is age X?" dependent on the future behavior of the twins, right?

You say that event $e$ takes place at time $\tau$, according to twin A, if there is a time $\delta \tau$ such that a light signal sent from Twin A at proper time $\tau - \delta \tau$ will reach event $e$ and a return light signal from $e$ will reach Twin A at proper time $\tau + \delta \tau$. So in a sense, $e$ is halfway between times $\tau - \delta \tau$ and $\tau + \delta \tau$.

But the time at which the return signal from $e$ reaches Twin A depends on how A accelerates after time $\tau$. How old twin B is at time $\tau$ depends on what A does after time $\tau$.

#### m4r35n357

It does not, if the other twin is outside the tube. Nor will any other coordinate system, as there's an assumption about simultaneity embedded in the word "when" in any question that starts "When the travelling twin is X years old...."
Quite, and that's precisely my motivation for looking at "moving" clocks and visualization, IMO these should always be considered as part of the solution to the twin "paradox", at the very least via a spacetime diagram showing light paths and the doppler effect. At least then you can then show the novice what X and his clocks look like from Y's POV the whole time, and vice versa. You can talk sensibly and point out the rates of the clocks as well as the readings in a very clear visual manner.
Without tying all this together, beginners tend to (justifiably) come away with a vague unease that you have used some kind of mathematical trickery to make time disappear for one or both of the twins. This is particularly true of the "instantaneous turnaround" description. The acceleration argument attempts to mitigate this, but I've never seen a concise and simple description of the effect from this standpoint; I think acceleration is strictly for experts only!

#### A.T.

Science Advisor
Don't assume that the non-inertial reference frame that Brian Greene promotes is the only way to do it.
Isn't that true for both twins? The standard simultaneity convention for inertial frames is just a convention as well.

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