# No definite viewpoint for the accelerating traveler?

1. Feb 14, 2013

### Alain2.7183

If an accelerating traveler, at some given instant in his travels, is told that his question "How old is my home twin right now?" has no unique, definite answer, then wouldn't he have to regard that statement as implying that she must not really EXIST at that instant at all? If she DOES exist right then, wouldn't he have to believe that she must be DOING something definite right then?

2. Feb 14, 2013

### Staff: Mentor

She is doing something definite at every instant, no questions or doubt there. We can even label each of those instants with the time that herr wristwatch reads at that instant, say things like "When her wristwatch read 3:00 she sneezed; when it read 3:01 she wiped her eyes, ...." and so forth.

But when the accelerated traveler speaks of what she's doing "right now", we have to ask him what "right now" means. Which time on her wristwatch is the accelerated traveler talking about when he says "right now"? That question has no unique definite answer; so "right now" has no unique definite meaning, and therefore the question "What is she doing right now?" also has no unique definite answer.

Last edited: Feb 14, 2013
3. Feb 14, 2013

### ghwellsjr

In addition to everything Nugatory said about the traveler's inability to assign a "right now" to the home twin, the home twin also cannot assign a "right now" to the accelerating traveler. And it's not because the traveler is accelerating. Even if he stops accelerating, the problem still exists. And even if he stops at a point distant from the home twin such that he is stationary with respect to the home twin the problem still exists.

But Special Relativity provides a way to deal with the problem by defining an Inertial Reference Frame (IRF). You can pick any IRF you want and then you can talk about "right now" in a meaningful way. But you can pick another IRF and have a different meaning to "right now" that is just as valid as the first one.

4. Feb 14, 2013

### Staff: Mentor

As Nugatory mentioned, the phrases "right now", "at that instant", "right then", etc. have no unique meaning. So no question with any such phrase has a unique answer. Whether the rest of the question is about their age, what they are doing, or their existence doesn't remove the ambiguity in the question.

Now, once you specify a coordinate system then the questions become uniquely defined. Frame variant quantities are perfectly legitimate things to ask questions about, you just have to be clear what frame they refer to.

5. Feb 18, 2013

### 1977ub

Once we're in GR, and we're comparing a "Far out in space" twin with a "surface of the earth" twin, do we still have these same ambiguities?

6. Feb 18, 2013

### PAllen

Even more so. You can't compare vectors at a distance in GR; you can in SR.

7. Feb 18, 2013

### 1977ub

People speak with confidence about the rate of clocks in orbit vs on Earth.

But they can do so without having a definite opinion about which tick of a satellite is simultaneous with a particular time EST, I take it.

8. Feb 18, 2013

### PAllen

Correct. You are using a family of clocks to establish a particular coordinate system to great precision. To make this coordinate system work, you find that precise adjustments are needed to the rate of orbiting clocks. The direct observables in this situation are sequences of signals sent from one 'clock' to another. This direct observable is invariant. It would be 'explained' differently in different coordinate systems, but the result of the observation would always be predicted to be the same.

9. Feb 18, 2013

### Staff: Mentor

Yes. One way of thinking about it: When EST changes over to EDT, we all reset our clocks but that doesn't change the rate at which the clocks tick.

If I want to compare the rate at which someone else's clock is ticking relative to my own, I don't need to worry about what time the other guy thinks it is, nor how much time he thinks has passed since some event in his past. All I need to do is count how many times my clock ticks in a given interval, count how many times his clock ticks in the same interval, and compare. The trick, and the place where the simultaneity convention comes into play, is in deciding what "the same interval" means (and I hope that alarm bells went off in your mind when you saw those words).

10. Feb 18, 2013

### 1977ub

Yes the point observer rec'vs tick-pulses but these don't translate into an origination time without confidence on how far the pulses traveled.

Does the GR earth-surface observer experience the same ambiguity deciding the distance to an orbiting satellite that the AO SR observer does an RF source?

11. Feb 18, 2013

### PAllen

Distance in SR or GR is defined by integrating invariant interval along spaceilike path, i.e. along a curve of a simultaneity convention. Thus, within these theories, distance has no possible meaning without a simultaneity convention.

12. Feb 19, 2013

### Alain2.7183

In the case of the traditional "Twin Paradox", to make the GR explanation analogous to the SR explanation (via the equivalence principle), the inertial home twin needs to be considered to be floating in space, with no real gravitational fields anywhere.

In the several descriptions I've seen that use a fictitious gravitational field to resolve the twin paradox from the traveler's perspective, there was no ambiguity anywhere ... the procedure always gave a specific (unique) answer to the question of how much the home twin ages during the traveler's turnaround (according to the traveler). And that answer was always the same answer that is given by the SR analysis that uses the momentarily co-moving inertial reference frames. See, for example, the Wikipedia page on the twin paradox, and in particular, their section on the traveler's perspective.

13. Feb 19, 2013

### PAllen

As has been pointed out to you in another thread, any specific method will give a specific answer. In no way does that mean there is a unique answer to the amount the distant twin ages during turnaround. The two methods you mention agree because pseudo-gravity is dependent on (metric expressed in) coordinates. The specific method you refer to uses coordinates based on the simultaneity of instantly comoving observers (even if this is not made explicit by the writer). Since they are both based on the same simultaneity convention, it is no surprise they agree. However, if a different simultaneity convention were used, you would get a different metric, and a different answer for distant twin age as function of traveling twin's clock; what must agree is the total differential aging. You should also be aware that for a somewhat more complex twin trajectory, you can't use either of these methods - because the lines of simultaneity intersect. No problem, just use a different convention.

The idea that 'one method is presented' implies there is a unique preferred answer, is your (invalid) inference. It is not stated in explanations using this simplest approach to pseudo-gravity.

14. Feb 20, 2013

### Alain2.7183

I don't think that the intersections of the traveler's lines of simultaneity affect the usefulness or legitimacy of that coordinate system, FOR HIM. I think that the ONLY thing that is important to the traveler, is that at each instant in his life, his coordinate system tells him the current position and the current age of every object in the entire (assumed flat) universe. In particular, his coordinate system has no need to be a GR "chart" (as defined by Wald): GR charts DO need to be invertible, because they must be capable of knitting together the multiple charts that are necessary to cover the entire (curved) universe. There is no such requirement for the traveler in SR, because his single coordinate system covers the entire flat universe ... no "knitting" is required. And the fact that a spacetime point may not determine a unique age of the traveler is of no importance to the traveler at all: he would just say "That's strange, but it's just the way nature works, like it or not".

15. Feb 20, 2013

### PAllen

Any coordinates system has the requirement that it doesn't give two labels to the same point. That's got nothing to do with GR. As to physics, what meaning do you think there is to the statement:

Both at 3pm on my watch and at 4pm on my watch, an earth home clock read 7pm ? (in between, it advanced to 8pm).

That is, can you describe any way at all to correlate this statement with any observation you could make? (You cannot; not only that, all direct observation contradicts such a description - the earth clock is seen to move monotonically forward, throughout any traveler journey). Given that no observation correlates to this, why should a sane person believe it? It NOT a required implication of SR; in fact I recently researched that Einstein, for example, never used such lines of simultaneity in any of his SR or GR work. So you would posit that Einstein never understood SR?

Last edited: Feb 20, 2013
16. Feb 20, 2013

### Staff: Mentor

Which is something that a "coordinate system" in which the lines of simultaneity intersect fails to do, right?

17. Feb 21, 2013

### jbriggs444

I may have lost context here. As I understand it we are trying to imagine a coordinate system attached to the traveller which assigns a time coordinate to remote events by waiting until the traveller's hyper-plane of simultanity crosses the remote event and then using the traveller's "time now" as the time coordinate for the remote event.

So it seems to me that if lines of simultaneity intersect, that is not a "hole" in space time that is unmapped by the traveller's coordinate system. Rather it is an area of space time that is multiply mapped; assigned more than one time coordinate.

18. Feb 21, 2013

### PAllen

Multiple mapping is prohibited for coordinates by definition. Do you really think it makes sense to say that NYC exploded at 3pm on my watch and also at 4PM on my watch, even though I only see it explode once, and no observation I can make is consistent with it being simultaneous to two points on my world line? Instead, don't you think it is better to say that these completely unobservable lines of simultaneity that Einstein never used have limits on their applicability. They are useful (as are other simultaneity conventions) when they don't lead to absurdities.

(It would be different if there was some observation consistent with multiple simultaneity. But there is none.)

19. Feb 21, 2013

### jbriggs444

Yes, I do think it makes sense to say that NYC exploded at two distinct coordinates. And you are incorrect about this giving rise to inconsistencies. It's just a coordinate. It's not an observable.

Last edited: Feb 21, 2013
20. Feb 21, 2013

### PAllen

There are accepted definitions of coordinates. They label a point on a manifold (physically, an event) once. Thus, such a system is not a coordinate system.

Tell me, why would I want to use such a non-coordinate system reflecting a non-observables in a way the does lead to nonsense that has no observable or logical basis? Instead, I can use any valid coordinate system to compute any observable, and conceptually model reality in a consistent way.

[edit: Why absurd? I know that NYC blows up once; every possible observation I can make indicates it blows up once; I have a plethora of valid coordinates consistent with SR that model it blowing up once. Why should I choose a method that constructs mathematically invalid coordinates and models it as blowing up at two different times of my history? Really??]

Last edited: Feb 21, 2013
21. Feb 22, 2013

### jbriggs444

Not all definitions of coordinates require that the things being labelled be manifolds. Not all "coordinate systems" on a manifold need be bijective, topology-preserving or even cartesian.

That said, I can certainly understand how you would want those properties to hold for useful coordinate systems. And I can understand that you might want to adopt terminology requiring this to be the case.

I can use polar coordinates to refer to the north pole without worrying overmuch about nonsense ensuing. Mind you I agree that using such coordinates to label the north pole is one thing. Using them to model physics at the north pole would be more difficult.

Topology is not my strong suit, but what I think you are saying is that you want a "mathematically valid" coordinate system to be one that embodies a homeomorphism between a manifold and cartesian n-space.

Any coordinate system which assigns multiple coordinates to the same point cannot (of course) be a homeomorphism because it fails to be a bijection.

The question you posed did not ask whether you should use a coordinate system that happens to have multiple coordinates for a single event. You asked whether I thought that it would make sense. I think that it does make sense. It's not an example that lends itself easily to a coordinate system that labels the same event twice, but one can contrive a labelling that does so.

Suppose that I am driving east when the NYC blows up. Let's say that it blows up at 2:30 pm EST. I glance at my clock and see that it reads 1:30 pm CST. But I am not paying careful attention and don't know whether I've crossed the time zone line yet.

I can label the NYC blow up at both 1:30 or 2:30 using "my personal time zone" coordinates. This does not entail that NYC blew up twice.

Last edited: Feb 22, 2013
22. Feb 22, 2013

### Staff: Mentor

Do you reset your watch as you're driving? If you do, you're changing from one coordinate system to another. If you don't, there's only one label you can assign to the blowup event, and that's the time on your watch when the blowup happens.

23. Feb 22, 2013

### jbriggs444

I have one hour of coordinate values to play with. If I choose to use them to double-label events, that's my business, not yours.

Last edited: Feb 22, 2013
24. Feb 22, 2013

### PAllen

Why do you think it makes sense to use a model that has a feature that is counter-factual to all observations, especially what there are a plethora of lmodels consistent with both observations and SR to choose from?

As for definitions, the following is one common definition of coordinates:

"In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space"

To a mathematician, the pole in polar coordinates is not covered by the coordinate system. In fact, this feature, in the case of a sphere, is the quintessential example used to show that there are manifolds such that no single coordinate system (patch) can cover the whole object.

Last edited: Feb 22, 2013
25. Feb 22, 2013

### Staff: Mentor

As long as you don't call them "coordinates", sure. You can even call them "coordinates" if you want, but....

Q: If we call the tail a leg, how many legs does a horse have?
A: Four. Calling a tail a leg doesn't make it a leg.