Frequent flyer miles for the travelling twin

[neopolitan]

This is a "twins paradox" related question, but I am not interested in the ageing of either twin, per se.

One of our closer neighbours is a star called Delta Pavonis. It is interesting for two reasons, firstly it is close to a nice round 20 light years away and secondly because it is on the verge of becoming a red giant.

Say one of a pair of twins (Mary) travels off to visit Delta Pavonis at 0.8c, performs a series of observations (say for a year), then travels back to Earth at 0.8c.

How far does Mary travel? (0.8c * 50year = 40 ly) or (0.8c * 30year = 24 ly)?

The reason I ask is that the ship must expend some fuel to get up to 0.8c, so Mary will know she is cruising at that speed (relative to the Earth at blast off). 15 years of shipboard time later she arrives at Delta Pavonis so a quick calculation will show her that she has traveled 12 ly. (The same applies on the way back, hence the total apparent distance traveled is 24 ly.)

How is this resolved in a real universe example? It seems unlikely that the universe shrinks when Mary is in motion relative to the Earth. Has she somehow used a variant of "warp speed"? (If Mary mixes frames it will seem to be so, she knows that Delta Pavonis is 20 ly away before blasting off, then she blasts off and travels for 15 years of shipboard time before arriving at Delta Pavonis. 20ly / 15 years = 1.3333c. "Warp factor 1.3333. Engage!")

cheers,

neopolitan

 

[JesseM]

How is this resolved in a real universe example? It seems unlikely that the universe shrinks when Mary is in motion relative to the Earth.

I'd say the answer is just that there is no objective frame-independent physical truth about "length" in relativity, just like there is no objective frame-independent truth about velocity. In the inertial frame where Mary is at rest during her journey, the distance was always 12 light years, even before Mary left Earth, so no "shrinkage" occurred (and in this frame she was traveling at 0.8c while on Earth, then she decelerated and came to rest while Delta Pavonis continued to move towards her at 0.8c). In the Earth's inertial frame of course the distance remains 20 light years and we explain the 15 years of shipboard time in terms of time dilation of Mary's clock. If you wished you could come up with a non-inertial coordinate system where Mary is at rest both before and after accelerating, and in this coordinate system perhaps the distance from Earth to Delta Pavonis would shrink, although there isn't really any single "correct" way to define the rest frame of a non-inertial observer, you can choose different non-inertial coordinate systems which say different things about simultaneity and distances. And keep in mind that when you analyze things from the perspective of a non-inertial frame, you can't assume that the usual rules of inertial frames, like the speed of light being c or moving objects being shrunk by the gamma-factor, still apply.

 

[neopolitan]

So Mary would have no leg to stand on if she claims 40ly worth of Frequent Flyer miles, because there is no objective truth about "length"? Would Mary's provider of these miles be able to justify paying 24ly worth or would they be able to say "there is no objective truth about length, so you get no Frequent Flyer miles at all"?

The interstellar space between Earth and Delta Pavonis is not completely empty is it? The term used for the gas, dust etc in interstellar space is thehttp://en.wikipedia.org/wiki/Interstellar_medium" .

How is the density of the interstellar medium affected? In other words, when Mary is traveling through interstellar space, is that density higher (and would always have been higher)?

cheers,

neopolitan

 

[JesseM]

So Mary would have no leg to stand on if she claims 40ly worth of Frequent Flyer miles, because there is no objective truth about "length"? Would Mary's provider of these miles be able to justify paying 24ly worth or would they be able to say "there is no objective truth about length, so you get no Frequent Flyer miles at all"?

In a relativistic universe, an organization awarding Frequent Flyer miles would have to specify in advance what frame they were using to compute "distance traveled" (probably just Earth's frame, or the solar system's frame). This is just like how, in a relativistic universe, if the Guinness Book of World Records wanted to have a record for the fastest any human had traveled, they would have to specify what frame or object they were measuring speed relative to (again, probably the Earth's frame).

The interstellar space between Earth and Delta Pavonis is not completely empty is it? The term used for the gas, dust etc in interstellar space is thehttp://en.wikipedia.org/wiki/Interstellar_medium" .

How is the density of the interstellar medium affected? In other words, when Mary is traveling through interstellar space, is that density higher (and would always have been higher)?

Yes, the particle density would be higher in a frame with a relativistic velocity relative to the average rest frame of the particles in the medium.

 

[neopolitan]

The interstellar space between Earth and Delta Pavonis is not completely empty is it? The term used for the gas, dust etc in interstellar space is thehttp://en.wikipedia.org/wiki/Interstellar_medium" .

How is the density of the interstellar medium affected? In other words, when Mary is traveling through interstellar space, is that density higher (and would always have been higher)?

cheers,

neopolitan

Yes, the particle density would be higher in a frame with a relativistic velocity relative to the average rest frame of the particles in the medium.

Not quite what I meant. But then I don't know the basis on which the density is calculated. Do you? Is the density calculated "relative to the average rest frame of the particles in the medium" or relative to the Earth, since they are based on observations made from the Earth?

Your answer is tailored very well to avoid the potential for an absolute rest frame. You are saying, I assume, that the density in "the average rest frame of the particles in the medium" is the minimum density one could ever measure. You couldn't measure a lower density from any other frame (which one might try to do if one had hopes of finding an absolute rest frame in relation to which the average rest frame of the interstellar medium was in motion).

(Quick pre-emptive comment, I am not saying there is a such an absolute rest frame because even if there were an absolute rest frame, like LET seems to be based on, a defining characteristic of it would be that it is undetectable. A detectable absolute rest frame doesn't seem to work with any theory and I certainly don't champion one. Even if there was an absolute rest frame, the density of the ISM would still be higher in that frame than in "the average rest frame of the particles in the medium".)

cheers,

neopolitan

 

[JesseM]

Not quite what I meant. But then I don't know the basis on which the density is calculated. Do you? Is the density calculated "relative to the average rest frame of the particles in the medium" or relative to the Earth, since they are based on observations made from the Earth?

No, I'm not sure how the density is calculated experimentally, although some quick googling turned up this paper which says in section IIB:

The Milky Way system is not only made of stars; it also contains significant amounts of tenuous matter, inhomogeneously spread out throughout interstellar space. The interstellar matter, which exists in the form of gas (atoms, molecules, ions, and electrons) and dust (tiny solid particles), manifests itself primarily through obscuration, reddening, and polarization of starlight, through the formation of absorption lines in stellar spectra, and through various emission mechanisms (both over a continuum and at specific wavelengths).

The rest of that section talks more about the type of evidence that convinced astronomers of the existence of the interstellar medium and its composition, but it isn't too specific about how the density is estimated. Section IIIB on p. 5 seems to talk more about how this is estimated, but I couldn't really follow the details; it seems to have something to do with spectrography though. It might be possible to determine the Earth's velocity relative to the medium using spectrography too, since the Doppler effect causes a shift in the position of absorption lines for whatever element you're looking at if the atoms are moving relative to you. I'm not sure if the velocity is actually large enough for this to be a measurable effect, though; I would guess that the solar system's velocity relative to the interstellar medium is a pretty miniscule fraction of light speed, since the sun's velocity relative to the center of the galaxy is only around 220 km/s (0.0007c), and I would imagine the interstellar medium in the sun's neighborhood is rotating around the center of the galaxy in the same general direction so the relative velocity would probably be significantly lower than that.

Your answer is tailored very well to avoid the potential for an absolute rest frame. You are saying, I assume, that the density in "the average rest frame of the particles in the medium" is the minimum density one could ever measure. You couldn't measure a lower density from any other frame (which one might try to do if one had hopes of finding an absolute rest frame in relation to which the average rest frame of the interstellar medium was in motion).

Yes, that's right.

 

[Saw]

Just to say that I like the example very much. I think it's analogous to the issue dealt with in the thread I initiated called "The show of the duel". As you know, Jesse, I think that, when faced with a human practical problem (in the other case, "is the duel fair?"; here, "is the traveller entitled to being credited with more or less frequent flyer miles"), physics is obliged to provide a solution, a single solution. It is not a question of saying: "the guy who invented the game should choose a frame from which the judgment should be made and that is it, it is an arbitrary rule". No, the rule cannot be arbitrary: if it is to be a good rule, which cannot be challenged and which duly addresses the problem, it must be a unique solution, valid for all frames.

In the duel example, we concluded that the key is looking at what SR finds as the invariant quantity, the proper time interval between each dueller firing and being fired. Fortunately, we found that all observers should agree that this proper interval is the same for both duellers and thus the duel is fair. Therefore, the fact that each observer had a different judgment about (i) whether the duellers received the light signals simultaneously or not and (ii) whether the duellers shot simultaneously or not, became… not thoroughly irrelevant (because it has an impact on how each observer makes his calculations to reach the invariant proper time), but at least non-decisive, in the sense that it is a sort of partial disagreement, concerning the path you choose to reach the target, not concerning the target itself (the proper time).

Here the question should be solved analogously. If what SR provides as invariant is the “proper time”, then we must look at it in order to guide the air Company on how to award the FF miles: you should get as many miles as the seconds of your proper time interval. That is fair. Thus the judgment of the different frames about the length traversed by the traveller is, just like simultaneity in the duel example, a sort of “interlocutory” decision: it is just a provisional, interim resolution, which is helpful for each observer to make the final judgment on the practical problem, but obviously cannot solve any problem by itself, if considered isolatedly.

In other words, if there is no “objective truth” about simultaneity, length and time, it is only because those concepts do not intend to make any final statement about the objective problem that we are facing (how many FF miles must be awarded): they contain, yes, a truth about the “interlocutory” result that each observer has measured during one of the steps that he must make to solve the problem, but not any objective truth about the latter.

 

[Saw]

I would like to retake this thread because:

First, the discussion looked very interesting and I'm sorry that after I intervened, it stopped.

Second, because now the solution I proposed to the practical issue of how many frequent flyer miles an Air Company should award to a given "traveller" (looking at her or his proper time) seems clearly wrong. We just have to think of this: a traveller flies away from the Earth at 0.5 c, whereas another one escapes at 0.99 c; they both return afterwards at the same speeds; the second will have experienced less proper time, but it’s clear that he/she should not be awarded fewer FF miles, since they traveled the same length (we can leave the issue of fuel consumed aside). What would be the solution to the problem: looking at proper length as measured from… an arbitrary reference frame?

 

[Dale]

That is why you have to read the fine print in your frequent flyer program. Most programs award based on the rest frame of the headquarters and do not provide any relativistic compensation.

 

[Saw]

That is why you have to read the fine print in your frequent flyer program. Most programs award based on the rest frame of the headquarters and do not provide any relativistic compensation.

Hmm... For a moment I liked your idea. I thought I could sue Air Companies based on that. But since coordinate length is always shorter, I am afraid I would not get a relativistic compensation but might end up having to pay a relativistic refund...

 

[yogi]

The actual distance flown should be measured in the proper frame - the distance to the target in the Earth frame (20 LY). This distance is independent of the speed or time measured by the traveling twin - so the total flying miles round trip is 40 ly.

 

[Saw]

The actual distance flown should be measured in the proper frame - the distance to the target in the Earth frame (20 LY). This distance is independent of the speed or time measured by the traveling twin - so the total flying miles round trip is 40 ly.

I agree that if all travellers are judged by that standard, they'll be treated equally and the system works. But you could take as reference any other inertial reference frame in the universe and the system would still work, wouldn't it? The number of FF miles awarded would be different, but the proportion among awards to travellers would not vary. Right?

 

[diazona]

It might vary if some of the destinations are moving with relativistic velocities with respect to each other.

Of course, in practice that's not the case since - unless you're talking about really long-range intergalactic travel - you're probably not going to have a large relative velocity between any two destinations. I don't think there are stars or planets flying around within our galaxy at half the speed of light. (That also suggests a standard reference frame to use to measure the distances traveled: just pick one that's nearly at rest with respect to all destinations in the system. The center-of-mass frame perhaps.)

 

[Saw]

It might vary if some of the destinations are moving with relativistic velocities with respect to each other.

Ah, thanks. So you mean that if in destination A two travelers get FF miles at the ratio 1:2, in destination B the proportion might be 1:3 or whatever... I'm trying to build an example with the relativistic velocity addition formula, but getting a bit lost. Any orientation?