Another Twins Paradox question

[Physical_Anarchist]

I don't know why a position diagram is relevant. A speed graph would be more appropriate and I think it would look like this:
.../\
../...\
/...\...__________
...\.../
....\../
.....\/

versus
..._____
.../...\
../....\
/....\
......\.../
.....\.../
......\_____/

 

[JesseM]

I don't know why a position diagram is relevant. A speed graph would be more appropriate and I think it would look like this:
.../\
../...\
/...\...__________
...\.../
....\../
.....\/

versus
..._____
.../...\
../....\
/....\
......\.../
.....\.../
......\_____/

Either a speed graph or a position graph can be used to determine the total spacetime "length" of each path (ie the proper time along each path). Similarly, if you have two paths drawn on an ordinary piece of graph paper with x and y coordinate axes drawn on, then if you can describe the paths in terms of the y-position as a function of x-position, like y(x), or if you can describe the slope of each path at every point along the x-axis with the function S(x), then you can use either one to find the total length of the path between the two points.

You can see in your diagram above that the speed graphs for the two paths look quite different, and that your addition of that third acceleration at the very tail end of the last graph didn't make their overall shapes the same (also, if that acceleration was very brief compared to the overall time spend in space, the last upward section should be much shorter along the t-axis, but maybe they weren't meant to be to scale). In terms of figuring out the proper time, if you know the functions for speed as a function of time v(t) in a particular inertial reference frame, and you know the times of the two endpoints t_0 and t_1 in that frame, then to find the total proper time you'd integrate \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt. The value of this integral for the second path will be only barely changed if you add a brief acceleration immediately before the time t_0 when they reunite.

So again, what's important is the precise nature of the position vs. time or speed vs. time functions for each one. The "whichever twin accelerates will have aged less" is not meant to be a general answer that covers all cases, it's only meant to cover the case where one twin moves inertially between the two endpoints while the other does not (as long as this is true, then no matter what specific position vs. time or speed vs. time function you pick for the second twin, you will find that his proper time is less). But in the case where both twins accelerate, you obviously can't apply this rule, you have to consider the two paths in a more specific way.

All this is directly analogous to the example of two paths drawn on an ordinary 2D piece of paper; if one is straight while the other has a bend in it, you can say "whichever path has the bend will be longer", but this is not a general rule that would cover all cases, in an example where neither path is straight you have to consider the specific shape of each path.

edit: I just noticed something about your graphs--is there a reason that the changes in speed in the first graph are sharp, while the changes in speed in the second graph have those flat intervals? If the flat intervals are meant to be extended periods of time at rest relative to the earth, so that both ships spend the exact same amount of total time at rest relative to the Earth from beginning to end, and also the same amount of time moving away from Earth at speed v and the same amount of time moving towards it at speed v, then in that case they will be the same age when they reunite. If this is what you meant all along and I misunderstood, then sorry for the confusion.

edit 2: OK, another thing I missed, but shouldn't constant speed always be a flat section of the graph? On a graph of speed vs. time, a non-flat slope would be a period of acceleration (speed changing at a constant rate), is that what you meant the sloped parts of the graph to represent? I thought in your example the idea was that each ship spent most of the time moving at constant velocity, with only brief periods of acceleration.

 

[Al68]

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c. The Earth's apparent superluminal velocity comes about because the ship is not moving inertially. The v <= c restriction applies to velocities of objects observed in a single inertial reference frame. (There's probably a more precise way to state this, but I can't think of it off the top of my head.)

Why would we not call this v > c genuine? If we are calling the v < c restriction for the ship's velocity (on the way to star system) relative to Earth's position genuine. If we do not call Earth's position of 5 light years away from the star system "genuine", then our velocity of v = 0.866c would not be "genuine". Are you just saying this v > c is not genuine because this is not a restriction for accelerating observers?

Also, I have read all of the referenced explanations on the internet (except the one on jstor, I don't have access), and none of them address the questions I have. That's probably because I've read them all before, and no longer have the questions that they do address.

Also, it's my understanding that Einstein initially thought he should be able to consider the ship at rest with the Earth and star system moving back and forth relative to the ship, and still resolve the Twins Paradox in SR. But he gave up on this and tried to resolve it with GR. And physicists generally consider his GR resolution erroneous. Is this correct? I think this is what wikipedia says, also.

Thanks,
Alan

 

[pervect]

Why would we not call this v > c genuine? If we are calling the v < c restriction for the ship's velocity (on the way to star system) relative to Earth's position genuine. If we do not call Earth's position of 5 light years away from the star system "genuine", then our velocity of v = 0.866c would not be "genuine". Are you just saying this v > c is not genuine because this is not a restriction for accelerating observers?

Here's a quote from Baez's et al paper on GR ("The Meaning of Einstein's equations" which helps explain this point.

http://math.ucr.edu/home/baez/einstein/node2.html

(also available at http://arxiv.org/abs/gr-qc/0103044 in pdf if you don't like the chopped-up version)

Before stating Einstein's equation, we need a little preparation. We assume the reader is somewhat familiar with special relativity -- otherwise general relativity will be too hard. But there are some big differences between special and general relativity, which can cause immense confusion if neglected.

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

This is the issue that you are running into with apparently FTL velocities. You are using a non-inertial coordinate system, and expecting it to act like an inertial coordinate system.

Note that because the underlying problem is in flat space-time, one actually CAN talk about the relative velocities of two particles. But in order to do so and get the right answer, one must restrict oneself to inertial coordinate systems.

Note that even in flat space-time, if the velocity between two objects is changing (because one of them is accelerating), the velocity of a distant object "at the same time" is ambiguous, because "at the same time" is an ambiguous concept in SR.

Thus the main problem is in your expectations. You are applying concepts which work in inertial coordinates and expecting them to apply in generalized coordinates.

You might also notice (or maybe you haven't) that the velocity of light is not constant in your accelerated coordinate system. Thus when you say that the distant object is moving "faster than light", it is actually not moving faster than light moves at that particular location. In your non-inertial coordinate system, light does not have a constant coordinate velocity, and in the region where the Earth appears to be moving faster than 'c', light appears to be moving even faster than the moving Earth.

Also, I have read all of the referenced explanations on the internet (except the one on jstor, I don't have access), and none of them address the questions I have. That's probably because I've read them all before, and no longer have the questions that they do address.

I also do not have access to the Jstor article.

As far as I can tell, you are trying to run before you can walk. It is possible to understand and work with non-inertial coordinates in relativity, but it requires some sophisticated mathematical techniques, such as the process of "parallel transport" that Baez alludes to.

It is both easier and more productive (IMO) to start to learn about relativity in a coordinate independent manner. This means learning about 4-vectors, and space-time diagrams. You need to have a firm grasp on SR, especially on the relativity of simultaneity (which you apparently still are struggling with from what I can infer from your remarks) before you can go on to handle GR and arbitrary coordinate systems.

You might also give some thought to the philosophical idea that coordinate systems are not the fundamental basis of reality.

Rather than treat coordinates as the basis of reality, think of the arrival of signals, and the readings of clocks, as being the fundamental basis - after all, that is actually what you can observe. You do not directly perceive the coordinates of some distant object, you percieve signals from that object.

The "coordinate" of a distant object are just something that you compute. Coordinates are supposed to be a convenience to make your life easier (and not a millstone around your neck that drags you into confusion). What you actually physically observe are signals emitted from and sent to said object (such as radar signals, or observations you make with a telescope).

The Doppler explanation of relativity, for instance, tells you all about how to compute the arrival time of such signals.

If you get into a muddle, think not about coordinates, but think instead about physical signals - when they were sent (and by whose clock that time deterimnation was made!), and when they arrived. Think about things that you actually could directly observe (i.e. NOT coordinates, which are things that you compute, not observe).

Also, it's my understanding that Einstein initially thought he should be able to consider the ship at rest with the Earth and star system moving back and forth relative to the ship, and still resolve the Twins Paradox in SR. But he gave up on this and tried to resolve it with GR. And physicists generally consider his GR resolution erroneous. Is this correct? I think this is what wikipedia says, also.

Thanks,
Alan

I can't make heads or tails of this remark. If you could provide a specific quote from the Wikipedia I might be able to say more.

 

[Al68]

pervect,

I was only objecting to the use of the phrase "not genuine". And I note that using observation as a basis for reality is objected to by some on this forum. I have my own reasons for asking these questions. I don't have a problem with apparent FTL velocities. I was just remarking that observations should not be referred to as "not genuine". And I would think that, after your comments here, that you would agree.

And here is a link to that wikipedia article: http://en.wikipedia.org/wiki/Twin_paradox#Resolution_of_the_Paradox_in_General_Relativity

It is part of their Twins Paradox article.

Thanks,
Alan

 

[pervect]

I might quibble with the exact way Jesse worded his remark, but I agree with the spirit. There may be some way to explain it more concisely than the rather long quote I gave, but I'm not sure how to do it :-(.

As far as the Wikipedia goes, I agree that there are some issues with the "gravitational time dilation" explanation of the twin paradox, but those problems don't actually come up in the twin paradox itself in my opinion.

My personal opinion is that the explanation IS good enough to explain the twin paradox, but has troubles further down the road.

I suspect you are traveling down that road right now, so I'll give you some idea of where I see the roadblock occurring. The roadblock is that "the" coordinate system of an acclerated observer does not cover all of space and time. It's only good locally.

This is covered in various textbooks, unfortunately I've never seen a textbook that covers this well that does not use tensors.

The problem can be illustrated with a simple diagram without the math, though. Basically, if you draw the lines of simultaneity for "the" coordiante system of an accelerated observer, they eventually cross.

Example: accelerated observers follow a "hyperbolic" motion, whose equations are just:

x^2 - t^2 = constant

If you draw the "lines of simultaneity" for such an accelrated motion, all of them cross at the origin of the coordinate system.

(I've got a picture of this somewhere, can't find the post though).

The fact that all of the coordinate lines of simultaneity cross leads to a coordinate system that is good only in the region before the lines cross. After the crossing occurs, one point has multiple coordinates - the origin, for instance, has an infinite number of "time" coordinates. This is very bad behavior, it does not meet the standards that every point must have one and only one set of coordinates.

The fact is closely related to the existence of the "Rindler horizon" for an accelerated observer. Another reason the acclerated observer does not have a coordinate system that covers all of space-time is that he cannot see all of space-time. An observer who accelerates away from Earth at 1 light year/year^2 will never actually see any event on Earth that occurs at a time later than 1 year, unless he stops accelerating, for example. All events more than 1 light year behind the accelerating observer are "behind" his Rindler horizon.

Detailed treatments of this do exist (my favorite is in MTW's "Gravitation")- unfortunately, as I 've said, most of them use tensor notation.

 

[JesseM]

I was only objecting to the use of the phrase "not genuine". And I note that using observation as a basis for reality is objected to by some on this forum.

If you're referring to me, my objection was not to "using observation as a basis for reality", but rather to the notion that the v>c thing constitutes an "observation" in the first place. In an inertial frame, what you "observe" is not what you actually see at the time, but what you reconstruct in retrospect based on the assumption that light always moves at c in your frame. For example, if I look through my telescope in 2006 and see the image of an explosion 10 light-years away as measured in my frame, and then in 2016 I look through my telescope again and see the image of an explosion 20 light-years away in my frame (which is the same one as before since I'm moving inertially), I can do a calculation of speed/distance for each and say that I "observed" these explosions to happen simultaneously in my frame, even though I certainly didn't see them happen simultaneously.

Now, have you thought about what type of calculation an accelerating observer would have to make to say in retrospect that he "observed" the Earth moving faster than c? For one thing, it would involve using a different set of rulers to measure distance at each point during his acceleration. And if you want things to work out so that he always "observes" an event's distance at a given moment to be identical with the distance in his instantaneous inertial rest frame, then he can't just take the time he actually sees the event and divide the distance in his inertial rest frame at the moment he saw it by c like in the inertial case, because his inertial rest frame at the moment he sees the event will be different from the inertial rest frame he was in at earlier moments when the light was on its way. I'd suggest that you take a shot at figuring out just how he could calculate in retrospect when he "observed" different events based on the moments he actually sees the light from them, and then perhaps you will change your mind about whether this highly abstract (and not very well-motivated physically, unlike the inertial case) calculation really deserves the commonsense word "observation".

 

[jtbell]

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c.

Why would we not call this v > c genuine?

I shouldn't have said "not genuine." It has loaded connotations. A better statement is simply that coordinate velocities (and other physical quantities based on coordinate measurements) behave differently in non-inertial coordinate systems than in inertial ones. Restrictions that apply in inertial coordinate systems don't necessarily apply in non-inertial ones.

Consider a different, but somewhat similar example. Suppose you are studying the behavior of a distant star in a coordinate system that has your head as its origin, and its x-axis sticking out straight ahead from your nose. Turn your head. The coordinate system is now rotating. In that coordinate system, the star now has a huge velocity, many times the speed of light, perpendicular to the x-axis.

 

[Al68]

Jesse,

I was not referring to you. And I wasn't referring to a direct observation of a measurement of Earth's velocity relative to the ship, while the ship decelerated. Just the observation by the ship's observer that the Earth's apparent position relative to the ship changed by a few light years in a matter of days, according to the ship's clock. I was only suggesting that the observation of Earth at 5 light years from the star system prior to deceleration, then the observation of Earth at 10 light years from the star system after deceleration should be considered genuine. After all, if we cannot call this length expansion genuine, how could we say the ship ended up 10 light years from earth, while its clock only showed 5.77 years since it left earth?

Thanks,
Alan

 

[JesseM]

Jesse,

I was not referring to you. And I wasn't referring to a direct observation of a measurement of Earth's velocity relative to the ship, while the ship decelerated. Just the observation by the ship's observer that the Earth's apparent position relative to the ship changed by a few light years in a matter of days, according to the ship's clock.

But again, this won't be a straightforward "observation". If he looks through his telescope, he won't see the position of the Earth change by a few light years in a matter of days (or if he does, it'll only be in the sense that he has switched which set of rulers he is using to define distance, the apparent size of the Earth as seen in his telescope won't change significantly). Like I said above, even in SR, "observation" involves a process of calculating dates of events in retrospect, long after they were observed. And the calculation for an accelerated observer needed to insure that what he "observes" matches his instantaneous inertial reference frame at each moment would be very complicated and not too physically well-motivated. Again, I invite you to figure out what he will actually see through his telescope during the acceleration, and then to figure out exactly what sorts of calculations he'd have to do on the dates he saw things to get the dates he "observed" them to work out the way you want them to.

I was only suggesting that the observation of Earth at 5 light years from the star system prior to deceleration, then the observation of Earth at 10 light years from the star system after deceleration should be considered genuine.

Why? Again, please try to figure out what measurements and calculations he'd have to do to get this "observation", and explain why this series of complicated calculations should be considered more "genuine" than some different set of calculations corresponding to a different non-inertial coordinate system.

After all, if we cannot call this length expansion genuine, how could we say the ship ended up 10 light years from earth, while its clock only showed 5.77 years since it left earth?

This comment is only true in the Earth's inertial reference frame, it's not an objective coordinate-independent statement about reality, any more than the statement "the velocity of the Earth is zero". Even in other inertial frames, it is not true that the ship "ended up" 10 light years from earth. And there's certainly no reason to think this would have to be true in a non-inertial coordinate system.

 

[Al68]

Jesse,

I was only referring to the fact that the ship's observer could look at his clock when he arrived at the star system, it would read 5.77 years. He could then use radar to measure his distance to earth, it would be 10 light years. He could calculate how far apart the Earth and star system were in his frame before he decelerated as 0.866c * 5.77 years = 5 light years. This would be approximate, neglecting the distance he would have needed to decelerate.

Please don't read more into this than I intended. I'm not under the impression that the ship could observe Earth's position in real time. And as far as calculating the magnitude of the ship's decceleration relative to Earth's apparent position, and Earth's apparent position relative to the ship as a funtion of ship's time, as observed by the ship, I don't think I'll try it right now. Especially since this was my point. These are the kinds of questions I have. I'm only asking them because they are not covered in most textbooks. And textbooks are not interactive (neither are most professors for that matter).

And I think it is normally considered reality that, theoretically, a ship's crew could end up 10 light years from earth, at rest with earth, after an elapsed proper time of 5.77 years by the ship's clock. Is this not considered objective reality by physicists?

Thanks,
Alan

 

[JesseM]

Jesse,

I was only referring to the fact that the ship's observer could look at his clock when he arrived at the star system, it would read 5.77 years. He could then use radar to measure his distance to earth, it would be 10 light years. He could calculate how far apart the Earth and star system were in his frame before he decelerated as 0.866c * 5.77 years = 5 light years. This would be approximate, neglecting the distance he would have needed to decelerate.

OK, but now you're comparing distances in two different inertial reference frames. So he did not "observe" that the Earth moved from 5 light years to 10 light years, he just figured out that the distance to the Earth is different in two different inertial frames. The fact that each frame happened to be his instantaneous rest frame at that moment is irrelevant, it doesn't imply something like "the Earth moved from 5 light years to 10 light years from his point of view" unless you define his "point of view" as a non-inertial coordinate system which defines distances at each moment to be identical to the distance in his instantaneous inertial rest frame at that moment.

And I think it is normally considered reality that, theoretically, a ship's crew could end up 10 light years from earth, at rest with earth, after an elapsed proper time of 5.77 years by the ship's clock. Is this not considered objective reality by physicists?

My impression is that the only thing a physicist would call a truly "objective reality" is something that is coordinate-independent, like the proper time along a worldline between two events on that worldline, or the answer to whether one event lies in the past light cone of another, or what the readings on two clocks will be at the moment they meet at a single point in spacetime. Any statement that's true in one coordinate system but not true in another can't be considered a simple physical truth, such statements can only be true relative to a particular choice of coordinate system (which means you have to specify which coordinate system you're talking about when making the statement).

 

[Al68]

OK, how about from the (non inertial) frame of the ship. Again, I know the ship cannot observe Earth in real time. When the ship accelerates away from earth, then cuts its engines, the ship's crew and Earth will disagree about what time the engines were cut. And they will disagree about the magnitude of the ship's acceleration. If the ship's crew calculates that their velocity relative to Earth is 0.866c after acceleration, will Earth also agree that the ship's velocity is 0.866c relative to earth? Is velocity invariant between inertial reference frames?

Thanks,
Alan

 

[pervect]

OK, how about from the (non inertial) frame of the ship. Again, I know the ship cannot observe Earth in real time. When the ship accelerates away from earth, then cuts its engines, the ship's crew and Earth will disagree about what time the engines were cut. And they will disagree about the magnitude of the ship's acceleration. If the ship's crew calculates that their velocity relative to Earth is 0.866c after acceleration, will Earth also agree that the ship's velocity is 0.866c relative to earth? Is velocity invariant between inertial reference frames?

Thanks,
Alan

The short answer is yes, everyone in an inertial coordinate system will agree that the velocity is .866 c after the acceleration stops, as long as inertial coordinates are used.

The procedure I would recommend to get a meaningful velocity from the coordinate system of an accelerting spaceship is rather involved, but it will give the same answer as the velocity measured by inertial observers, which is why I recommend it.

Let us say that we have an object, O, and that we want to measure it's velocity relative to the accelerating spaceship, using the coordinate system (viewpoint) of the spaceship.

Basically, one constructs a second space-ship, that maintains a "constant distance" away from the first.

This second space-ship is placed at the location of the object O.

Spaceship #2, at the location of object O, then measures the velocity of object O, using its own local set of clocks and rulers. (These will be different from the clocks and rulers in spaceship #1).

It can be shown that object O will measure the same velocity for spaceship #2 as spaceship #2 measures for object O, as long as measurements are made with standard clocks and rulers.

Technically one constructs a tetrad of "orthonormal basis vectors" at the second space-ship (even more exactly, an orthonormal tetrad of one-forms).

To make this work, one has to have a meaningful concept of what it means for spaceship #2 to be "stationary" (maintaining a constant distance) with respect to spaceship #1. If space-time is flat, or even if it's non-flat and static, this can be defined by the fact that light signals exchanged between spaceship #2 and spaceship #1 will always have the same travel time.

If the space-time is not static, there isn't any way that I'm aware of to define a meaningful concept of a "stationary" distant spaceship, and this procedure for defining relative velocity breaks down.

 

[JesseM]

OK, how about from the (non inertial) frame of the ship.

There is no single standard way to construct a coordinate system for a non-inertial object. If you specify how the coordinate system is constructed, then you can figure out how things will behave in that coordinate system.

Again, I know the ship cannot observe Earth in real time. When the ship accelerates away from earth, then cuts its engines, the ship's crew and Earth will disagree about what time the engines were cut. And they will disagree about the magnitude of the ship's acceleration.

It depends whether you're defining "acceleration" in terms of change in coordinate velocity, or in terms of the acceleration felt by the ship itself (which means acceleration at a given moment is the rate of change of velocity as seen in the ship's instantaneous co-moving frame at that moment). Often in relativity "acceleration" is used to refer to the latter, so "constant acceleration" would mean a constant G-force experienced by passengers, not a constant rate of change of velocity as seen in a single inertial frame. But I assume you were talking about the former type of acceleration, in which case different inertial frames will indeed disagree about the ship's acceleration.

If the ship's crew calculates that their velocity relative to Earth is 0.866c after acceleration, will Earth also agree that the ship's velocity is 0.866c relative to earth? Is velocity invariant between inertial reference frames?

Between the two rest frames of the objects who are in relative motion, yes. Other frames besides the rest frame of the ship and the rest frame of the Earth won't see the distance between the ship and the Earth increasing at 0.866 light-years per year, though.