Question about Lorentz contraction

[drkitten]

What about a 12" deep jar flying through space, mouth towards a 12" ruler flying directly at it? Would the ruler fit all the way inside, fit exactly to the rim or not fit? Common sense tells me it would fit to the rim (all relative effects cancelling out).

The jar would break.

Not to be too sarcastic,.... but if you have two objects moving together at relativistic speeds, there's going to be a hell of a crash.

The question you really wanted to ask was whether or not the crash (when the near tip of the ruler hit the bottom of the jar) happens before or after the far tip of the ruler clears the mouth of the jar. And the answer to that depends on where you are and how you are moving.

An observer at the mouth of the jar, and stationary relative to the jar, will see the tip of the ruler pass him before the crash happens.

An observer at the tip of the ruler will see the crash happen before he enters the jar.

An observer exactly balanced between the two, such that the relative speeds are identical (although in opposite directions) will see the ruler just fit -- he will see the crash and the entrance has happening simultaneously.

Which one will be correct? All three, of course.

 

[Ziggurat]

I'd argue that the deformation indicates that the bond is pulling tighter along the direction of motion; pulling tighter requires more energy; and so forth. That was my general line of argument.

Yes, but that argument can't work, because the angle won't matter if you calculate the energy using simple lorentz transforms. For the angle to matter, the simple relativistic expressions using just rest mass would have to be wrong.

<Shrug> I still say that the bonds will represent more energy if they are compressed than not, and that this extra energy will show up as a mass increase. But whatever.

The energy of a moving H2 molecule does not depend upon its orientation. If the H2 molecule is moving perpendicular to the bond direction, the bond length is unchanges. What you are positing implies that there be an angle dependence to the energy of a moving H2 molecule - I see no way around that.

The bond will also be a mass decrease, not an increase, because the bound state is lower energy than the free state.

 

[Dr. Trintignant]

Sometimes it helps to get the math definitions nailed down solid before trying to get an inuitive grasp of what they mean.

Thanks for the explanation. I'm fairly good with this since I actually do 3D graphics for a living. That extra minus sign indeed makes it a weird kind of rotation--maybe one day I'll write a little program to simulate it in 1+1 or 2+1 dimensions.

In special relativity, its:
s² = x² - (ct)²Note the minus sign. That's what makes all the difference.

Could one claim that time is actually an imaginary dimension, at least in the mathematical sense? In other words, we're really still using Euclidean distance, but time itself has a -1^1/2 folded into it, giving the minus sign outside of the squaring?

I suppose it's just two ways of looking at the same thing, but it's a bit of a strange thought.

- Dr. Trintignant

 

[Ziggurat]

Could one claim that time is actually an imaginary dimension, at least in the mathematical sense?

Some people do, though I don't like that treatment. The problem with that is that using s² = (ct)² - x² works just as well - both conventions get used, and the choice seems to be mostly one of taste. So do we make time or distance "imaginary"? The answer would depend upon what's essentially an arbitrary choice of which metric you use. So even if you want to go with imaginary time axis, make sure you don't invest any deep meaning in that designation.

 

[Dr. Trintignant]

So do we make time or distance "imaginary"?

There are 3 spatial dimensions and only 1 time dimension, so space wins the right to be "real" .

- Dr. Trintignant

 

[Yllanes]

The famous Gravitation by Misner, Thorne, and Wheeler (1973) has this to say about imaginary time

Farewell to 'ict'
One sometime participant in special relativity will have to be put to the sword: "x4 = ict." This imaginary coordinate was invented to make the geometry of spacetime look formally as little different as possible from the geometry of Euclidean space; to make a Lorentz transformation look on paper like a rotation; and to spare one the distinction that one otherwise is forced to make between quantities with upper indices (such as the components p* of the energy-momentum vector) and quantities with lower indices (such as the components of the energy-momentum 1-form). However, it is no kindness to be spared this latter distinction. Without it, one cannot know whether a vector (§2.3) is meant or the very different geometric object that is a 1-form (§2.5). Moreover, there is a significant difference between an angle on which everything depends periodically (a rotation) and a parameter the increase of which gives rise to ever-growing momentum differences (the "velocity parameter" of a Lorentz transformation; Box 2.4). If the imaginary time-coordinate hides from view the character of the geometric object being dealt with and the nature of the parameter in a transformation, it also does something even more serious: it hides the completely different metric structure (§2.4) of + + + geometry and - + + + geometry. In Euclidean geometry, when the distance between two points is zero, the two points must be the same point. In Lorentz-Minkowski geometry, when the interval between two events is zero, one event may be on Earth and the other on a supernova in the galaxy M31, but their separation must be a null ray (piece of a light cone). The backward-pointing light cone at a given event contains all the events by which that event can be influenced. The forward-pointing light cone contains all events that it can influence. The multitude of double light cones taking off from all the events of spacetime forms an interlocking causal structure. This structure makes the machinery of the physical world function as it does (further comments on this structure in Wheeler and Feynman 1945 and 1949 and in Zeeman 1964). If in a region where spacetime is flat, one can hide this structure from view by writing

ds² = (dx¹)² + (dx²)²+ (dx¹)³ + (dx⁴)²
with x4 = ict, no one has discovered a way to make an imaginary coordinate work in the general curved spacetime manifold. If "x4 = ict" cannot be used there, it will not be used here. In this chapter and hereafter, as throughout the literature of general relativity, a real time coordinate is used, x° = t = ct_conv (superscript 0 rather than 4 to avoid any possibility of confusion with the imaginary time coordinate).

On the choice of metric (t^2 - x^2 or x^2 - t^2): Workers in general relativity all use the metric with -+++ signature, one of the reasons is that it reduces to the Euclidean metric in the non relativistic limit. In QFT it has been 'traditional' to use the metric with +--- signature and there's certainly no other reason than tradition to make that choice. The only time a metric with +--- signature is better is when working with two-spinors.

 

[Schneibster]

Two good arguments.

Yes, but that argument can't work, because the angle won't matter if you calculate the energy using simple lorentz transforms. For the angle to matter, the simple relativistic expressions using just rest mass would have to be wrong.

I'm not sure I follow this argument. Angle of what?

The energy of a moving H2 molecule does not depend upon its orientation.

From whose point of view? From the point of view of the H2 molecule? Correct. From the point of view of an outside observer? Incorrect. IMO, so far. If you want to get into it, clarify what you meant above.

If the H2 molecule is moving perpendicular to the bond direction, the bond length is unchanges. What you are positing implies that there be an angle dependence to the energy of a moving H2 molecule - I see no way around that.

Ahhh, I think I begin to see what you mean. You're arguing the symmetry of physical law over rotation. Well, you see, this is an asymmetric situation: from the point of view of an observer, who defines themself as motionless, the moving object has ceased to be symmetric in this fashion; it possesses a special direction, that is, a gauge has been fixed, and that direction is the direction it is moving in. The presence of this asymmetry causes an asymmetry in the results of physical law over rotation, in the eye of the observer. I'll guarantee you this, though: whatever the observer sees, it will all be accounted for by the transforms, allowing it to be related back to the POV of the H2 molecule, which sees itself as symmetric and motionless.

The bond will also be a mass decrease, not an increase, because the bound state is lower energy than the free state.

What about the strength of the field necessary to account for the closer bond? Fields are energy, and the EM field explicitly is- its exchange particle is the photon, the quantum of energy.

 

[Schneibster]

Could one claim that time is actually an imaginary dimension, at least in the mathematical sense? In other words, we're really still using Euclidean distance, but time itself has a -1^1/2 folded into it, giving the minus sign outside of the squaring?

I suppose it's just two ways of looking at the same thing, but it's a bit of a strange thought.

- Dr. Trintignant

Yllanes points out that it leads to a difficulty with Minkowski's interval formula, and I'll point out a different problem.

There is an exactly equivalent form of the Lorentz transform that does not use the gamma term:

[latex]$$ \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} $$[/latex]

Everyone knows that part of the Lorentz transform; some references call the denominator "tau" (τ, a Greek letter). But you get exactly the same results if you use hyperbolic trig sinh and cosh functions as you would to describe normal rotation, applied to the direction of motion and the time dimension, rather than the two axes that form the plane of ordinary rotation. I've explained elsewhere that velocity is a rotation, and you've probably heard it said elsewhere before; it was not and is not a metaphor, it is a realistic description of physical fact.

We live in a 3-D world, and as a result of that happenstance, we tend to think that axes of rotation are somehow associated with dimensions. They are not. In fact, it is best not to think of axes of rotation, but of planes of rotation. In other words, not the axis that threads the object, but the plane that cuts it in half, normal to that axis; we would then speak not about the Earth's axis, but about its plane of rotation, which would cut it at its equator. There is a reason for this: axes of rotation are imaginary, but planes of rotation are real. Let me prove it to you.

Imagine a 2-D world. Now imagine a rotating 2-D object in that world. In what direction does the axis of rotation point? The direction it points doesn't exist in that world. But note that the plane of rotation does.

So how many planes of rotation are there in different dimensionalities? Turns out it depends on the number of different pairs of dimensions there are. So 1-D rotation is impossible because you can't define the plane of rotation. 2-D rotation is possible, but there is only one possible plane of rotation. 3-D permits rotation in three different planes: x-y, x-z, y-z. So what does adding a fourth dimension do? Most people assume that it only adds one extra axis of rotation; but this is a mistake. It actually adds three: x-t, y-t, and z-t, to the existing three. So in 4-D spacetime, there are six planes of rotation. And that's the other reason that it's better to talk about planes of rotation than axes: what direction does the axis of something rotating in the x-t plane point, and how is that direction different from the direction that the axis of something rotating in y-t points? But stick to the planes, and you'll have no trouble.

So what is rotation in, say, x-t? Why, it's nothing but velocity in x. The actual technical name for this type of rotation is rapidity. It actually talks about the angle; but the meaning of the word "angle" is somewhat different in this case, because the geometry of spacetime is not circular as the geometry of space is; it is hyperbolic, which is why we have to use hyperbolic trig to figure the rotation angle. The unusual features of hyperbolic trig that are highly relevant in this case are, hyperbolic trig is defined so that there exist directions one cannot reach by any finite amount of rotation. There are two such regions: the one that no hyperbolic function can describe, which approximately conceptually corresponds to the right angle in circular geometry, and the other half of the hyperbola, the axis of which points in the opposite direction from the one the angle one is currently measuring is referenced to, which can be described but not reached by any finite rotation.

This precisely corresponds to the situation we find ourselves in with regard to spacetime; first, the direction of time is immutable, it moves always forward, never backward. Even matter that is moving at relativistic speed agrees with an observer on the "direction" time increases in; it may not be moving in that direction as fast as we are, but it's not moving backward in time. "Backward in time" corresponds to the opposite region of the hyperbola; there's no way to rotate something so that it is moving backward in time. And the right angle, the one that has the angular measure, "infinity," that is the speed of light, the rapidity no material object can be rotated to.

The physical implications of this are simple: space is curved with respect to time, into a hyper-hyperboloid. Thinking of any one spatial dimension with respect to time, drawn on a piece of paper, we see the time axis as a straight line and the spatial dimension as a hyperbola.

That is the best way to visualize the geometry of spacetime, and the physical meaning of velocity, that I have yet found. I may find a better; and you may find it useless. It works for me; YMMV. I have nothing else to offer for your birthday, so I hope you like it.

 

[Ziggurat]

Ahhh, I think I begin to see what you mean. You're arguing the symmetry of physical law over rotation. Well, you see, this is an asymmetric situation: from the point of view of an observer, who defines themself as motionless, the moving object has ceased to be symmetric in this fashion; it possesses a special direction, that is, a gauge has been fixed, and that direction is the direction it is moving in. The presence of this asymmetry causes an asymmetry in the results of physical law over rotation, in the eye of the observer.

I understand the assymetry of the situation, but I do not think that can affect the result, because if it did, it would mean some pretty basic relativity stuff is wrong.

Say I have an object of rest mass m. Doesn't matter how it's constructed, doesn't matter what its internal structure is, if its rest mass is m, then its rest energy is E = mc². Now let's say I change reference frames. What's the object's energy now? Well, it's
E = mc²/(1-v²/c²)
That answer is sufficient and complete, according to special relativity. It does not depend upon the shape of the object, its internal structure, or anything else: all it depends upon is the rest mass and the velocity. So unless that equation is wrong, the orientation of the object cannot matter. If the orientation of the object does matter, then that equation is wrong.

What about the strength of the field necessary to account for the closer bond? Fields are energy, and the EM field explicitly is- its exchange particle is the photon, the quantum of energy.

It's not merely the strength of the fields which are relevant, it's also the shape. And the shape for the fields of both the protons and the electrons is squished in the direction of motion. So I don't think the field need be any stronger at the point in question because of the angle. If one were to actually solve the energy in the fields explicitly, one should find that it is indeed independent of the orientation of the H2 molecule (with respect to its motion). If that were NOT the case, then that would mean the equation I posted above would be wrong (since energy in the field contributes to the total energy of the object).

 

[Schneibster]

I understand the assymetry of the situation, but I do not think that can affect the result, because if it did, it would mean some pretty basic relativity stuff is wrong.

I disagree strongly. I think it would mean that relativity stuff is right, but applies differently to things that are small (i.e., quanta) than things that are big (i.e., macroscopic objects).

Say I have an object of rest mass m. Doesn't matter how it's constructed, doesn't matter what its internal structure is, if its rest mass is m, then its rest energy is E = mc². Now let's say I change reference frames. What's the object's energy now? Well, it's
E = mc²/(1-v²/c²)
That answer is sufficient and complete, according to special relativity. It does not depend upon the shape of the object, its internal structure, or anything else: all it depends upon is the rest mass and the velocity. So unless that equation is wrong, the orientation of the object cannot matter. If the orientation of the object does matter, then that equation is wrong.

If the object is made of atoms and molecules and they are randomly oriented as such pretty much always are, then the lobes of the waveforms that make up the standing waves of the electrons will point in random directions. Some will be compressed, and some will not. If the object is rotated, then different ones will be compressed and uncompressed, but if the object is made of enough atoms to be macroscopic, that is, subject to the ordinary Second Law of Thermodynamics, rather than the statistics of quantum mechanics, then its overall energy will be unchanged. But that does not mean that the energy of the atoms within it will not be changed.

Your mistake is attempting to reason in the quantum realm using logic that only works for macroscopic objects. Quanta are not macroscopic objects, and they do not obey the rules that macroscopic objects do; they have their own rules, and they obey them.

It's not merely the strength of the fields which are relevant, it's also the shape. And the shape for the fields of both the protons and the electrons is squished in the direction of motion. So I don't think the field need be any stronger at the point in question because of the angle.

I point out the implicit self-contradiction in this pair of statements.

If one were to actually solve the energy in the fields explicitly, one should find that it is indeed independent of the orientation of the H2 molecule (with respect to its motion). If that were NOT the case, then that would mean the equation I posted above would be wrong (since energy in the field contributes to the total energy of the object).

I'll say it again: you can't use macroscopic reasoning to talk about the behavior of quanta. Try again. You'll get it.

 

[Cuddles]

I disagree strongly. I think it would mean that relativity stuff is right, but applies differently to things that are small (i.e., quanta) than things that are big (i.e., macroscopic objects).

Evidence? Exactly the same relativistic equations are used when dealing with electrons as with spaceships. If the laws were different for different scales I'm pretty sure we'd have noticed by now. Whether a system consists of one particle, two particles or lots of particles makes no difference. If you accelerate the system to a certain velocity its energy increases by the same amount, no matter how many particles it is made of. Your argument that things are pointing in random directions only works for large systems, but the relativistic equations are valid for all systems, therefore that cannot possibly be the correct reasoning.

Take two hydrogen atoms and accelerate them. Now work out their total relativistic mass. At what point in those workings does either the distance between them or the direction of motion become involved? It doesn't. Therefore the bond energy cannot contribute to the relativistic mass.

 

[Ziggurat]

I disagree strongly. I think it would mean that relativity stuff is right, but applies differently to things that are small (i.e., quanta) than things that are big (i.e., macroscopic objects).

I don't see why it should apply differently, nor have you really explained HOW it should apply differently (only claimed that it should). And if you could demonstrate that it does not apply the same way in such a case, I think you'd actually have shown something quite remarkable. Furthermore, the problem doesn't actually depend upon the quantum mechanical nature of the hydrogen atom: the issue is the same even if you use classical charged spheres. You can't just appeal to quantum mechanics without further detail and expect that to be taken as anything more than speculation.

If the object is made of atoms and molecules and they are randomly oriented as such pretty much always are,

Often the case, but quite certainly not always. For example, single crystals with layered structures (which aren't hard to come by). Bond energies in different directions can be markedly different, and apply over macroscopic sizes. So you can't claim that this is irrelevant beyond the quantum scale, because if it's relevant at the quantum scale, it's relevant at the macroscopic scale. There's no escape clause here.

Try again. You'll get it.

OK, I'll try a different argument.

What is the energy cost of changing the orientation of a hydrogen molecule in the molecule's rest frame? Zero. That's true whether you want to treat it classically or quantum mechanically. What is the energy cost in every other reference frame? Zero. If it's zero in one frame, it MUST be zero in every other frame. If it isn't, you've violated conservation of energy. If the energy cost to change the orientation is zero in every reference frame, then every orientation must have the same energy in every reference frame.

 

[~enigma~]

Yes, the front end will bend into the hole under the force of gravity (and I think the bending also results in a deceleration of the front), but let's assume an "ideal ruler". Much like an ideal string is massless, this ideal ruler is absolutely rigid; it doesn't bend.

Causality does not change.

 

[Dr. Trintignant]

I have nothing else to offer for your birthday, so I hope you like it.

Thank you for the detailed explanation--it's much appreciated.

"Planes of rotation" is a fascinating way of looking at rotations. Although it's "intuitive" to me that a rotation vector always has one and only one dual, which is the plane containing all vectors which are orthogonal to the original vector, it's clear that this doesn't work for hyperbolic geometry.

From when I first started learning about relativity (even before I understood the equations), I had this sense that movement through space somehow "subtracted" from movement in time; that there was some total quantity that had to be maintained. Now I know: it's rapidity, and spatial movement "subtracts" from movement in time through the rotation.

So thanks again. I'm fairly visual-minded, and so to fully understand what raw equations do, I really like to have some kind of visualization of what's going on, even if it's simplified.

- Dr. Trintignant

 

[Dr. Trintignant]

What is the energy cost of changing the orientation of a hydrogen molecule in the molecule's rest frame? Zero.

What if there were something else to account for it, though? If the bond energy changed, but something else precisely subtracted that amount, the net energy would remain zero.

Is there any chance that time could have anything to do with it? In one rotated state, it might be that a clock on each atom would appear synchronized (to an outside observer). In another rotated state, they might not. I'm not sure how this would make a difference to the total energy, but there's really not much to account for. The bond energy must be completely dominated by that of compressing the nucleus, or even the nucleons themselves.

- Dr. Trintignant

 

[Ziggurat]

What if there were something else to account for it, though? If the bond energy changed, but something else precisely subtracted that amount, the net energy would remain zero.

That's concievable (though I don't see any candidates for that cancelation - your suggestion of unsynchronized "clocks" doesn't make sense to me since it's a basically static situation), but then it just becomes a bookkeeping issue in terms of changing how you allocate the total energy between two different categories.

 

[jsfisher]

Happy birthday, Dr. Trintignant.

 

[boooeee]

It's nice to see a debate without too much invective.

I go back and forth between Ziggurat's and Schneibster's points. From a purely classical sense, it would seem that Lorentz contraction would change the energy content of a particular charge configuration. For example, imagine two positive charges connected by a rod of length d (like a barbell). If that rod is travelling at a relativistic speed along the axis of the rod, to a stationary ovbserver, the two charges would appear closer together, and thus have a higher energy (I think).

Then I go back to Ziggurat's point about rotating the configuration. If the barbell was rotated so that the axis is perpendicular to the direction of motion, the barbell "uncontracts", and it would appear to a stationary observer that energy has been released. However, to an observer travelling with the barbell, there is no energy change. The barbell is merely pointing in a different direction.

Is it possible that the Abraham-Lorentz force could somehow account for the bookkeeping? In classical electrodynamics, a charge "resists" a change in acceleration due to the impact of it's own electrodynamic field on itself. So, rotating the barbell in the rest frame would still "cost" something.

 

[Ziggurat]

I go back and forth between Ziggurat's and Schneibster's points. From a purely classical sense, it would seem that Lorentz contraction would change the energy content of a particular charge configuration. For example, imagine two positive charges connected by a rod of length d (like a barbell). If that rod is travelling at a relativistic speed along the axis of the rod, to a stationary ovbserver, the two charges would appear closer together, and thus have a higher energy (I think).

It's not that simple. The electric field of a moving charge is no longer a simple 1/r² form: it has an angular dependence. It is weaker along the direction of motion (compared to the non-moving case), and stronger perpendicular to the direction of motion. In fact, the fields are stronger for the perpendicular case, not the parallel case. Let's do the math. For a moving charge, the field is (dropping the force constants):

[latex] E = \frac{q*(1 - v^2/c^2)}{R^2*[1 - (v^2/c^2)sin^2\theta]^{3/2}} [/latex]

(damn it - latex isn't being processed right) where theta is the angle from the direction of motion. theta = 0 for charges aligned along the direction of motion, and 90 for charges perpendicular. So we have:

[latex] E(\theta=0) = \frac{q}{R^2} (1 - v^2/c^2)[/latex]
[latex] E(\theta=90) = \frac{q}{R^2} \frac{1}{[1 - (v^2/c^2)]^{1/2}} [/latex]

Now the relevant R along the direction of motion is going to be contracted, and perpendicular to the direction it won't be. How much is it contracted? By a factor of (1-v²/c²)^1/2. That gets squared (with the R^2), so that the field strength along the direction of motion is actually unchanged at the point of interest compared to the stationary case. But the field in the perpendicular direction at the point of interest is actually stronger than in the stationary case. In other words, this appears to be backwards from the original supposition. Actually trying to do a full calculation of the energy in the fields in each case is going to be quite complicated, and you need to consider the magnetic fields as well (the total energy will be the same in both cases, but my hunch is the distribution between electric and magnetic energies may not be). Which is why it's easier to resort to general principles (such as conservation of energy) to guide us in this sort of problem.

Then I go back to Ziggurat's point about rotating the configuration. If the barbell was rotated so that the axis is perpendicular to the direction of motion, the barbell "uncontracts", and it would appear to a stationary observer that energy has been released. However, to an observer travelling with the barbell, there is no energy change. The barbell is merely pointing in a different direction.

Is it possible that the Abraham-Lorentz force could somehow account for the bookkeeping?

No, it isn't possible. We can make those forces arbitrarily small by doing the rotation slowly.

 

[boooeee]

You're correct. I forgot about the distortion of the fields.

I also suspect you're correct about the magnetic fields in the scenario that the rod is perpendicular to the direction of motion. Doing the calculation for point charges would be a pain. However, if you consider that parallel currents attract via their magnetic interaction, and that two point charges moving in the same direction are "like" two parallel currents, the attraction from the magnetic field most likely will exactly cancel the added repulsion due to the increased electrical field.