Rest Length, Coordinate Length, and an argument for True Length

[PAllen]

I am going to argue a little on both sides of the 'true length' debate. Using Bobc2's 3d analogies, anyone who thinks alternate cross sections aren't 'real' will look like a fool getting furniture into their house or apartment without exploiting the ability to tilt it. On the other hand, we do consider a particular way of measuring objects as most generally useful.

Consistent with this point of view, I have argued in other posts that length contraction is more than an optical illusion. On the other hand, I do see value in defining a preferred set of dimensions for a rigid body. This has nothing to do with the aeither frame, instead with the rest frame of the rigid body. Further, I propose we can define 'invariant spatial dimensions' for a Born rigid object (but not for more realistic objects; but realistic objects approximate Born rigid objects for many purposes).

A Born rigid object has the feature that, no matter what its state of motion, all world lines of its consitituents are parallel (relative to the 4-metric). Thus, there is a unique spacelike, flat, hypersurface that is orthogonal to the world tube of the rigid object. Proper length dimensions computed in this hypersurface are taken to be invariant dimensions of the rigid object. Succinctly, invariant dimensions of a Born rigid object are the proper length dimensions computed in the flat spacelike hypersurface orthogonal to the object's world tube. (These dimensions are the same no matter where the world tube is sliced, by definition of Born rigidity).

 

[GrayGhost]

It's the simplest for a quick "shut up and calculate" exercise; the same simplicity creates boggled minds ("paradoxes") for non-inertial motion and such questions as "which rod is truly shorter".

Well, for a shut up and calculate exercise, LET and SR should be identical since they use the very same LTs.

Indeed though, SR is more complex when considering the POV of he who undergoes proper acceleration.

Far as paradoxes go, they arise only because of misunderstanding, which generally is the result of (unknowingly) injecting absolute simultaneity midstream during a relativistic scenario analysis.

GrayGhost

 

[harrylin]

Well, for a shut up and calculate exercise, LET and SR should be identical since they use the very same LTs.

Indeed though, SR is more complex when considering the POV of he who undergoes proper acceleration.

Far as paradoxes go, they arise only because of misunderstanding, which generally is the result of (unknowingly) injecting absolute simultaneity midstream during a relativistic scenario analysis.

GrayGhost

In this discussion there appears to be a misunderstanding that is due to Minkowski. Based on his presentation of SR, some people here think that length contraction (and as a consequence, time dilation) "just relates to differences in cross-section views" so that "the rod itself is completely unaffected".

A mere difference of view (like length under an angle, or frequency with classical Doppler) cannot result in a difference between identical objects under the same view; Einstein's moved clock would then, when brought back, appear to indicate the same time as the one in rest. That is a wrong prediction, contrary to SR.

 

[GrayGhost]

In this discussion there appears to be a misunderstanding that is due to Minkowski. Based on his presentation of SR, some people here think that length contraction (and as a consequence, time dilation) "just relates to differences in cross-section views" so that "the rod itself is completely unaffected".

The cross-sectional views presented by BobC2 was a simplistic method of conveying the general notion. His analogy is fundamentally correct. However wrt SR, the POV differentials do not arise from one guy being here and the other over there, but rather is due to the relative velocity between observer and rod. A relative velocity in 4-space is analogous to different viewing locations in 3-space. Minkowski's modeling of time (duration) as another spatial axis (length) allows for BobC2's analogy.

A mere difference of view (like length under an angle, or frequency with classical Doppler) cannot result in a difference between identical objects under the same view; Einstein's moved clock would then, when brought back, appear to indicate the same time as the one in rest. That is a wrong prediction, contrary to SR.

What you say would be true if the contractions were due to relative location alone. however they are not. They are due to relative velocity. The clocks are still moving when they pass each other again, and so the views remain rotated wrt one another, so they each continue to witness the other ticking differently. If the accelerating clock arrives back into the other clock's frame on reunion, then no angular POV differentials exist (since v=0) and so no relativistic effects exist. However, they did exist while in transit until reunion. Over that (spacetime) interval, the accelerating clock ticks slower than the inertial clock at the center. Therefore, its reading must lag the inertial clock on reunion, any way you slice it.

GrayGhost

 

[GrayGhost]

BobC2,

Indeed, the moving contracted length is a projection from the proper frame (thru 4-space) into the observer's 3-space in his own instant. This the result of angular orientation differentials within the 4-space that arise with relative motion. That said, it's not really about what is special about the proper frame, but rather whether what is measurable is less-than-real or not.

IOWs, the question is not whether the body changes in and of itself per others who accelerate, but rather whether said contractions exist even though the body has not changed in and of itself. The key point ... contractions of moving bodies are measurable, and must exist per the math of the theory.

I'd say that it seems that everyone is on the same page here, and that the discussion is about semantics alone. However, Greg has mentioned that it cannot be proven that contractions are real. This suggests that he believes that what is measured is not necessarily real. So, what then is one's definition of real?

Is the moving length truly contracted? Does the proper length not change, even though moving others record the length contracted? I submit that the last 2 are concurrently true. That said, Greg's preferred usage of the word "real" is a personal choice of his own. I could equally say that what is measured dictates what is real, and therefore "both" the proper and contracted lengths are real. That is, one is not a contradiction of the other. Yet I agree in that the proper POV is "special", although it is not a preferred frame.

IMO, it's best to say "proper" vs "real". Folks know what a stationary proper length is, and folks know what a moving contracted length is. The important thing is to realize these lengths are verifiable by measurment. Everyone agrees that viewing a rod does not change its proper length.

GrayGhost

 

[harrylin]

[..]
What you say would be true if the contractions were due to relative location alone. however they are not. They are due to relative velocity. The clocks are still moving when they pass each other again, and so the views remain rotated wrt one another, so they each continue to witness the other ticking differently. If the accelerating clock arrives back into the other clock's frame on reunion, then no angular POV differentials exist (since v=0) and so no relativistic effects exist. However, they did exist while in transit until reunion. Over that (spacetime) interval, the accelerating clock ticks slower than the inertial clock at the center. Therefore, its reading must lag the inertial clock on reunion, any way you slice it.

GrayGhost

I pointed out that an absolute physical effect can not be due to a mere difference of POV and your assertion that "relativistic effects exist" and that "the accelerating clock ticks slower" affirms what I said. Why do you think that we disagree?

I also gave the example from http://adsabs.harvard.edu/abs/2004AmJPh..72.1316G (about a tilted mirror, therefore I called it a "prism").
If the proper dimensions of such a "moving" mirror or prism were "true" in the way the OP suggested, then the speed of light would be truly isotropic relative to the mirror and therefore "truly not" isotropic relative to the "rest" frame - or the laws of optics are messed up.

Note: I disagree with the way that article formulates it, it seems to make the inverse error as the OP by suggesting that a moving prism is "really" contracted ("our assumption was correct and the moving mirror really has a different inclination angle than the angle for the stationary mirror").

 

[Dale]

the same simplicity creates boggled minds ("paradoxes")

So what? The solution to that is better scientific education, not worse scientific theories.

 

[harrylin]

So what? The solution to that is better scientific education, not worse scientific theories.

This isn't a matter of scientific theories but about their interpretation. The solution to such boggled minds is physical explanations that make sense and that are consistent with the theories.

 

[Dale]

This isn't a matter of scientific theories but about their interpretation. The solution to such boggled minds is physical explanations that make sense and that are consistent with the theories.

IMO SR does make sense.

 

[Dale]

I think Greg might have done well to argue about the "true" length of a simple 3-D beam sitting on the floor with everyone standing around with different slanted views, etc. (no relativity involved at all). If he could get everyone to buy into a "true" length for that situation, then we could go on to the implications of the relativistic situation. However, I don't think he has you on board even at that level.

Correct, on both counts.

 

[Dale]

IMO, it's best to say "proper" vs "real". Folks know what a stationary proper length is, and folks know what a moving contracted length is. The important thing is to realize these lengths are verifiable by measurment. Everyone agrees that viewing a rod does not change its proper length.

I agree with this completely. It makes sense to invent new words for new concepts, but when a concept is not new then you should learn and use the existing word for it. In this case "proper length".

 

[bobc2]

In this discussion there appears to be a misunderstanding that is due to Minkowski. Based on his presentation of SR, some people here think that length contraction (and as a consequence, time dilation) "just relates to differences in cross-section views" so that "the rod itself is completely unaffected".

It is good to see that you seem to acknowledge our point of view.

A mere difference of view (like length under an angle, or frequency with classical Doppler) cannot result in a difference between identical objects under the same view; Einstein's moved clock would then, when brought back, appear to indicate the same time as the one in rest. That is a wrong prediction, contrary to SR.

Sorry, Harrylin, I just don't understand your straw man. For two identical rods, A & B, moving in opposite directions we have shown spacetime diagrams that easily account for the observer moving with rod A seeing a different length for rod B -- and manifestly accounted for by the different cross-section views of the rods. We have shown the spacetime diagram for the twin paradox (in fact we sketched this two different ways--one using the rest system of one twin, and again choosing a rest system that is symmetric for the first leg of the trip). The clocks for the twins do not agree at the end of the trip.

 

[JesseM]

The "world without time" is causing more trouble than it's worth, so I'll retract it. Instead, I'll simply define that the rod's length is independent of and therefore unaffected by time. With that definition in mind, I don't see any circularity in the argument of the OP.

It's circular for the reasons I've already pointed out and you never really addressed, like the fact that you define "view" in such a way as to guarantee that the rest frame has "one view" while other frames have a "composite of many views". I see you repeat essentially the same circular argument in this post as well:

In the rest frame, all locations on the rod are at the same time. When observed from the moving frame, each location on the rod is at a different time. Imprecisely put, the moving frame doesn't get a good reading of the rod.

Of course, this statement assumed we can compare the time on atoms at different locations on the rod to decide whether they are "at the same time" or "at a different time", but the atoms themselves don't behave in any way that indicates they have an opinion about simultaneity, simultaneity is a human concept and it's up to us to define which event on the worldline of atom #2 is at the "same time" as an event on the worldline of atom #1. There is no non-circular reason that we can't use the definition of simultaneity in the frame F where the rod is moving at 0.99c, and therefore say "in frame F, all locations on the rod are at the same time. When observed from the rest frame, each location on the rod is at a different time".

I'm afraid I can't do any better than that right now. I'm late for work. This will have to be my last post for a while. I don't get enough sleep if I spend time here. I get sleepy at work, can't think properly, and have to clock out and take a nap. Not good.

OK, don't wear yourself out! Still if you continue to think you can defend your position as something other than an aesthetic opinion or personal intuition, perhaps you can find a little time to respond to posts on weekends.

 

[ghwellsjr]

I'm afraid I can't do any better than that right now. I'm late for work. This will have to be my last post for a while. I don't get enough sleep if I spend time here. I get sleepy at work, can't think properly, and have to clock out and take a nap. Not good.

OK, don't wear yourself out! Still if you continue to think you can defend your position as something other than an aesthetic opinion or personal intuition, perhaps you can find a little time to respond to posts on weekends.

Then again, maybe he's working on the weekends, too:

I worked 58 hours last week.

 

[GrayGhost]

I pointed out that an absolute physical effect can not be due to a mere difference of POV and your assertion that "relativistic effects exist" and that "the accelerating clock ticks slower" affirms what I said. Why do you think that we disagree?

OK, maybe I misundertood you then. I think I see what you were saying. You interpreted BobC2's illustration and comments to suggest that relativistic effects are "equivalent to optical effect" and thus illusionary, yes? If so, then I'd agree with you, because there should then be no aging differentials between the twins.

I'm supposing however, that BobC2 considers his 3D measured contractions as optical illusionary effects, whereas (in analogy) the SR 4D measured contractions are instead considered as geometrical effects. The geometry being Minkowskian. In that I would have no argument, as no one here disagrees that the relativistic effects are the result of angular orientation differentials within fused spacetime between the POVs, the result of relative motion. Each body exists as a limited projection (across 4-space) within the other's POV, and hence the contractions. And, neither body ever changes in and of itself.

My sticking point is that the relativistic effects are directly measurable. Substituting the word "real" for "proper" suggests to many that contractions are illusionary effect. They are not.

GrayGhost

 

[PAllen]

OK, maybe I misundertood you then. I think I see what you were saying. You interpreted BobC2's illustration and comments to suggest that relativistic effects are "equivalent to optical effect" and thus illusionary, yes? If so, then I'd agree with you, because there should then be no aging differentials between the twins.

I'm supposing however, that BobC2 considers his 3D measured contractions as optical illusionary effects, whereas (in analogy) the SR 4D measured contractions are instead considered as geometrical effects. The geometry being Minkowskian. In that I would have no argument, as no one here disagrees that the relativistic effects are the result of angular orientation differentials within fused spacetime between the POVs, the result of relative motion. Each body exists as a limited projection (across 4-space) within the other's POV, and hence the contractions. And, neither body ever changes in and of itself.

My sticking point is that the relativistic effects are directly measurable. Substituting the word "real" for "proper" suggests to many that contractions are illusionary effect. They are not.

GrayGhost

Actually, I've argued in an earlier post that the 3d analogs of different cross section are no more illusory than the SR 4d case. In one orientation, you cannot fit a beam through an opening. Tilt it, and you can. I find this quite analogous to length contraction. The moving rod is able to be 'momenarily' contained within barriers that shoot up and recede quickly, that are closer than its rest length. However, in the rod's rest frame, what is perceived is that the front barrier went up and down first, then the back barrior moved a ways, and then went up and down. The perceptions are different and equally valid, and as real as anything matters in physics.

[Edit: and continuing the 3-d analogy, when you tilt a beam to get it through and opening, one end goes through ahead of the other, rather than both at the same time.].

 

[GrayGhost]

Actually, I've argued in an earlier post that the 3d analogs of different cross section are no more illusory than the SR 4d case. In one orientation, you cannot fit a beam through an opening. Tilt it, and you can. I find this quite analogous to length contraction. The moving rod is able to be 'momenarily' contained within barriers that shoot up and recede quickly, that are closer than its rest length. However, in the rod's rest frame, what is perceived is that the front barrier went up and down first, then the back barrior moved a ways, and then went up and down. The perceptions are different and equally valid, and as real as anything matters in physics.

[Edit: and continuing the 3-d analogy, when you tilt a beam to get it through and opening, one end goes through ahead of the other, rather than both at the same time.].

Well, I do understand what you are saying.

Also, I do recognise that the 3D guy rotating his ruler angularly to take the beam's length measurement colinearly, is analogous to the moving observer dropping back into the rod's rest frame to take the beam's length measurement colinearly.

Basically, the contracted length is apparent while not illusionary effect, and is measurable. I mean, in the 3D world, the contracted length is an illusionary effect only if one assumes the proper length has truly contracted. If one does not assume such, then there is no illusionary effect.

Sounds reasonable to me. Let me think on that awhile, see how it sits.

GrayGhost

 

[harrylin]

IMO SR does make sense.

SR makes sense to me - that's not even a discussion point.

 

[Dale]

SR makes sense to me - that's not even a discussion point.

Then I don't get your point about "boggled minds".

 

[harrylin]

[..]
Sorry, Harrylin, I just don't understand your straw man.

I didn't see any straw man... but such remarks kill serious discussions.

For two identical rods, A & B, moving in opposite directions we have shown spacetime diagrams that easily account for the observer moving with rod A seeing a different length for rod B -- and manifestly accounted for by the different cross-section views of the rods. We have shown the spacetime diagram for the twin paradox (in fact we sketched this two different ways--one using the rest system of one twin, and again choosing a rest system that is symmetric for the first leg of the trip). The clocks for the twins do not agree at the end of the trip.

Obviously! Now, this thread is about fitting interpretations; in particular the claim that such physically different clocks are compatible with the interpretation that nothing happened to both clocks.
And the same with the mirror example: the claim that a moving mirror is truly undeformed must be compatible with SR's predictions in order to be a correct interpretation of SR.

 

[harrylin]

Then I don't get your point about "boggled minds".

Paradoxes and debates about what a theory means often indicate a lack of correct understanding.
The OP makes a claim about SR, based on a long argument; and I cited a contrary claim in a physics journal, also based on a long argument. At best one of them can be right.

Here's an extreme example of a boggled mind and paradoxes: Dingle taught SR and wrote books about it, only to discover that he had never really understood it.
- http://en.wikipedia.org/wiki/Herbert_Dingle

 

[harrylin]

Actually, I've argued in an earlier post that the 3d analogs of different cross section are no more illusory than the SR 4d case. In one orientation, you cannot fit a beam through an opening. Tilt it, and you can. I find this quite analogous to length contraction. The moving rod is able to be 'momenarily' contained within barriers that shoot up and recede quickly, that are closer than its rest length. However, in the rod's rest frame, what is perceived is that the front barrier went up and down first, then the back barrior moved a ways, and then went up and down. The perceptions are different and equally valid, and as real as anything matters in physics.

[Edit: and continuing the 3-d analogy, when you tilt a beam to get it through and opening, one end goes through ahead of the other, rather than both at the same time.].

Yes indeed. There is a big difference between rotating an object so that it fits through the door opening, and bending your head so that you see the object rotated. In both cases you have the same perspective of it. However, in the first case, something happened to the object (with physical consequences) and in the second case, nothing happened to it.

 

[JesseM]

In this discussion there appears to be a misunderstanding that is due to Minkowski. Based on his presentation of SR, some people here think that length contraction (and as a consequence, time dilation) "just relates to differences in cross-section views" so that "the rod itself is completely unaffected".

A mere difference of view (like length under an angle, or frequency with classical Doppler) cannot result in a difference between identical objects under the same view; Einstein's moved clock would then, when brought back, appear to indicate the same time as the one in rest. That is a wrong prediction, contrary to SR.

I'm unclear why you think the "difference in cross-sectional views" explanation of length contraction is wrong, or what you mean when you say "as a consequence, time dilation". I would say that differences in the instantaneous rates of two clocks can understood be in terms of the amount of a worldline that can be "sandwiched" between two infinitesimally-separated surfaces of simultaneity, but obviously differences in total elapsed proper time for two worldlines in a twin paradox type scenario cannot be.

 

[GrayGhost]

I also gave the example from http://adsabs.harvard.edu/abs/2004AmJPh..72.1316G (about a tilted mirror, therefore I called it a "prism").
If the proper dimensions of such a "moving" mirror or prism were "true" in the way the OP suggested, then the speed of light would be truly isotropic relative to the mirror and therefore "truly not" isotropic relative to the "rest" frame - or the laws of optics are messed up.

Unfortunately, I am not a subscriber of the American Physics Journal, so I cannot read the article. So I'm not sure what your point was here? Too bad, because I really would like to know what that point was.

Note: I disagree with the way that article formulates it, it seems to make the inverse error as the OP by suggesting that a moving prism is "really" contracted ("our assumption was correct and the moving mirror really has a different inclination angle than the angle for the stationary mirror").

Well, there's that use of the word "real" again. That seems to be the biggest problem from what I see. Without having read the article, I'd bet that he equates "real with measured", which means the proper length is real per the proper frame and the contracted length is real per the frame in relative motion. GregAshmore equates "real with proper", period.

GrayGhost

 

[harrylin]

Unfortunately, I am not a subscriber of the American Physics Journal, so I cannot read the article. So I'm not sure what your point was here? Too bad, because I really would like to know what that point was.

The abstract is cited there but I'll copy it here:

"We derive a formula for the law of reflection of a plane-polarized light beam from an inclined flat mirror in uniform rectilinear motion by applying the Huygens-Fresnel principle. We then use this formula and the postulates of special relativity to show that the moving mirror is contracted along the direction of its motion by the usual Lorentz factor. The result emphasizes the reality of Lorentz contraction by showing that the contraction is a direct consequence of the first and second postulates of special relativity, and is not a consequence of the relativistic measurement of the length."

And with Google I also found the following link, it may work for you:
http://gluon.softcafe.net/gravity/reprints/AJP/Gjurchinovski-2004_SR-Reflection_AJP001316.pdf

My point was that different people make contrary claims relating to "real", all based on SR.

Well, there's that use of the word "real" again. That seems to be the biggest problem from what I see.

I agree, the very topic if this thread concerns the word "real" and what different people mean with it.

Without having read the article, I'd bet that he equates "real with measured", which means the proper length is real per the proper frame and the contracted length is real per the frame in relative motion. GregAshmore equates "real with proper", period.
GrayGhost

In fact, that AJP article claims (and I disagree) that "the [inclination] angle phi [which is affected by Lorentz contraction] is a real physical entity, which, by itself, has nothing to do with relativity. The value of phi is neither a result of an act of measurement, nor a result of an act of seeing."

In contrast, the OP appears to argue (and I also disagree with that) that a proper measurement is undistorted, so that the measurement with a rest system of the inclination angle of a moving mirror is distorted.

Cheers,
Harald

 

[Dale]

Paradoxes and debates about what a theory means often indicate a lack of correct understanding.

Agreed. But again, the solution to a lack of correct understanding is better education, not a new theory or interpretation. You can't go around revising your theories every time some uneducated person has a hard time in class.

 

[GrayGhost]

Thanx for the link Harold,

"We derive a formula for the law of reflection of a plane-polarized light beam from an inclined flat mirror in uniform rectilinear motion by applying the Huygens-Fresnel principle. We then use this formula and the postulates of special relativity to show that the moving mirror is contracted along the direction of its motion by the usual Lorentz factor. The result emphasizes the reality of Lorentz contraction by showing that the contraction is a direct consequence of the first and second postulates of special relativity, and is not a consequence of the relativistic measurement of the length."

Well, I don't think that anyone here in this thread is suggesting that "taking the measurement" causes the Lorentz contraction. It's the 2 postulates in the presence of relative motion that causes this, and so its truly about the nature of spacetime. The act of holding the ruler up to take a measurement does nothing but measure (data collection).

My point was that different people make contrary claims relating to "real", all based on SR.

This is very true. I have found that in most cases though, they all understand the theory the same, and the assignment of "real" is as DaleSpam said ... analogous to a choice of convention, in which case it's about semantics. The only problem arises when someone says or mis-assumes that "the contraction isn't measurable".

In fact, that AJP article claims (and I disagree) that "the [inclination] angle phi [which is affected by Lorentz contraction] is a real physical entity, which, by itself, has nothing to do with relativity. The value of phi is neither a result of an act of measurement, nor a result of an act of seeing."

It would seem to me that the writer of the AJP article equates "measureable with real", which has always been my practice as well. Given such, the [inclination] angle phi is real, contracted or not. But then, that's just my opinion.

If the angle phi is rotated due to Lorentz contraction, then I am puzzled as to why you would think it has nothing to do with relativity. Relativity is not defined by the measurement, it's only verified by the measurement.

In contrast, the OP appears to argue (and I also disagree with that) that a proper measurement is undistorted, so that the measurement with a rest system of the inclination angle of a moving mirror is distorted.

I'd have to agree with the OP on this, personally. In my mind, to say something is distorted is to say relativistic effects are present, which requires v>0 technically, and v = luminal practically ... where undistorted is the view of everyday experience. Relativistic effects exist whether you take the measurement (to verify it) or not.

On the other hand, it seems that the use of the word "distortion" may be somewhat similar to the problem with using the word "real". Saying contrractions are not real suggests to many that it is illusionary effect, which is an error. We have relativistic effects. They are often referred to as distortions. Saying something is distorted suggests to many that it is illusionary effect, which is again an error. I prefer "proper as opposed to real", and "effects as opposed to distortions". What's important, is that all agree that the relativistic effects are verifiable by measurement, no matter how you label them.

GrayGhost

 

[harrylin]

Agreed. But again, the solution to a lack of correct understanding is better education, not a new theory or interpretation. You can't go around revising your theories every time some uneducated person has a hard time in class.

In the same post I showed how even educators can have a lack of understanding.

I also read somewhere the suggestion that it's better to teach yourself from the masters than from their followers. That goes perhaps too far, but for a correct understanding of a theory it certainly helps to study a few key texts from the originators.

 

[harrylin]

[..]
It would seem to me that the writer of the AJP article equates "measureable with real", which has always been my practice as well. Given such, the [inclination] angle phi is real, contracted or not. But then, that's just my opinion.

If one, like the OP, only perceives proper measurements as "real", then that "distorted" angle is not real...

If the angle phi is rotated due to Lorentz contraction, then I am puzzled as to why you would think it has nothing to do with relativity.

?? That's what that article argues; I wrote that I do not think so.

If I correctly understand them, then the OP and that author make contrary suggestions which both are, in different ways, incompatible with the PoR.

[..] What's important, is that all agree that the relativistic effects are verifiable by measurement, no matter how you label them.
GrayGhost

I don't think that that's a topic of discussion (or so I hope!) .

 

[GrayGhost]

harrylin,

Interesting, this topic of real vs apparent. It's the debate between ... (1) the proper POV presents what is real, versus (2) what is measurable is real, proper or not.

Let's say you and another fellow are standing on a train track, and a wonder train is whizzing toward you at v=0.866c. He who jumps first "is chicken", and loses the $1 bet. You ask yourself, what is the last moment I should jump off the track to win the bet and avoid being splattered? So you get your handy dandy calculator out, punch in some numbers right quick like, obtain the solution and the plan. You run your figures based on the contracted length, and the other fellow runs his based on the proper length. The only data you have is the train velocity, and the location of the train's center at any instant.

I'd bet $2 you win and he's chicken.

GrayGhost