Question about Lorentz contraction

[Dr. Trintignant]

Hi, all--just a lurker here, but I've been looking for a place to ask a question that's been bugging me for a while, and given that there seem to be a number of physics experts here, it seemed to be an appropriate forum.

On to the question, but first I want to mention that it's going to be a bit long because I want to go through some of the reasoning process behind it. It's entirely possible that there is some fault in my reasoning which makes the question irrelevant.

Ok, an extremely important principle in science is "the equivalence principle", which roughly states that the laws of physics are the same for all observers. The observers may disagree on what exactly is happening in a given experiment, but they will all agree that the laws of physics are the same and not being violated.

To get our bearings, let's start with Newtonian mechanics; specifically, the classic bouncing ball on a train. To the moving observer, the ball simply bounces up and down. Let's say once per second. The observer can confirm that all the basic laws of physics hold: conservation of energy, conservation of momentum, gravity, the equations governing potential and kinetic energy, etc.

A stationary observer disagrees about what is going on--from his perspective, the ball is bouncing in long parabolas, and has a velocity component in the X direction. If the train is traveling 25 m/s, the ball is impacting the floor of the train in 25 m intervals. The impact itself now follows a kind of reflection law--the impact angle is no longer 90 degrees, but something less. Nevertheless, the laws of physics hold perfectly--the input angle is the same as the exit angle, and of course all the other laws (conservation, etc.) all hold.

Now let's move to relativity. It's come to my attention fairly recently that magnetism is actually a relativistic phenomenon. One can call it any number of things--a relativistic pseudoforce, a projection of a 4-dimensional master force, or whatever--but the net result is the same. Magnetism is a direct result of relativity applied to moving charges.

Again, we have the situation where two observers fully agree on the laws of physics, but disagree on what is happening. An observer moving with a charged particle will claim that it is not creating any magnetic force, while a non-moving observer will not. However, the interactions between the particles will be identical, even if the forces those interactions results from are different.

Finally, let's move on to my question. Anyone familiar with relativity knows of the curious effect of Lorentz contraction. An object moving relative to an observer will appear shorter than it would if it were stationary.

It's especially curious if one asks what keeps such a moving object in that compressed state. Consider a simple H2 molecule, moving along the bond direction. The two atoms "want" to be a certain distance apart; a distance dictated by electromagnetic and other forces. And yet when moving relativistically, that bond distance appears shorter. It seems to me that even the normally spherical electron orbital of a lone H atom will appear squished into an ellipsoid.

It seems clear that some mysterious forces are at work--but what are they? It seems that if the equivalence principle is to work, some kind of force--if if it's "just" a pseudoforce--must come into play to keep objects in this compressed state. Otherwise all kinds of other laws would break down. The laws of chemistry dictate what kind of shape molecules come in, and Lorentz contraction changes that shape. There must be a new force that holds molecules into their distorted shapes.

Another thing comes to mind--there must be a tremendous amount of energy contained in that distortion. Electromagnetism is a very powerful force. Compressing any solid object along one axis must therefore require an enormous energy input. Is it possible that some--or even all--of the kinetic energy in an object is somehow "contained" in this distortion?

Well, that is all. As I said, it is quite likely that there is some fault in my reasoning process which makes the question irrelevant. If so, I'd like to hear where you think the fault lies. If there is no fault, am I right in thinking there must be a relativistic pseudoforce which results in Lorentz contraction? Thanks for the answers.

- Dr. Trintignant

 

[Ziggurat]

Ok, an extremely important principle in science is "the equivalence principle", which roughly states that the laws of physics are the same for all observers.

What you refer to is commonly called relativity (and there is in fact a classical version of this refered to as gallilean relativity). The term "equivalence principle" is generally used for something else (having to do with the relationship between gravity and inertial reference frames - irrelevant in the current context).

Now let's move to relativity. It's come to my attention fairly recently that magnetism is actually a relativistic phenomenon. One can call it any number of things--a relativistic pseudoforce, a projection of a 4-dimensional master force, or whatever--but the net result is the same. Magnetism is a direct result of relativity applied to moving charges.

That is correct.

It's especially curious if one asks what keeps such a moving object in that compressed state. Consider a simple H2 molecule, moving along the bond direction. The two atoms "want" to be a certain distance apart; a distance dictated by electromagnetic and other forces. And yet when moving relativistically, that bond distance appears shorter. It seems to me that even the normally spherical electron orbital of a lone H atom will appear squished into an ellipsoid.

It seems clear that some mysterious forces are at work--but what are they?

What you're missing is that magnetism is not the only relativistic effect in play. A spherically symmetric 1/r^2 field is valid for stationary charges, but it's actually NOT valid for moving charges. The electric field gets "compressed" along the direction of motion in addition to the emergence of magnetic fields (which also matter - the components of your H2 molecule are charged and moving). The potentials themselves are therefore no longer isotropic, and solving for a moving H2 molecule correctly will give you a different shape than for a stationary molecule. Actually trying to solve that would be quite difficult (it's much simpler to solve it in the rest frame and transform the solution), but if you did it correctly, you would get the contracted length as the equilibrium separation distance.

 

[Dr. Trintignant]

The term "equivalence principle" is generally used for something else (having to do with the relationship between gravity and inertial reference frames - irrelevant in the current context).

Thanks for the correction--it seems I misremembered what Wikipedia had said about it.

The electric field gets "compressed" along the direction of motion in addition to the emergence of magnetic fields (which also matter - the components of your H2 molecule are charged and moving). The potentials themselves are therefore no longer isotropic, and solving for a moving H2 molecule correctly will give you a different shape than for a stationary molecule.

Very interesting. So the laws of electromagnetism themselves dictate the answer. I assume this is all a result of Maxwell's equations, or is there more?

Also, do you know if this solution does store energy as I speculated?

What about other objects, such as black holes? I can only assume that the same basic thing is true--that for a moving singularity, general relativity no longer predicts a spherically symmetric 1/r^2 gravitational field, and that instead we end up with an ellipsoidal event horizon as the Lorentz contraction predicts?

It seems that in some sense, the Lorentz contraction isn't needed, and that it can be predicted from the other laws--it just happens that it's easier to find stationary solutions and transform them to a moving frame than it is to find the moving frame solutions directly.

Thanks for the great answers!

- Dr. Trintignant

 

[jsfisher]

Here's a question I've wondered about:

A one-foot ruler is traveling across a table at a high "relativistic speed" towards a one-foot hole in the table. From the vantage point of the ruler, the hole is contracted to an inch or two, so the ruler is expected to traverse the hole easily. On the other hand, from the vantage point of th table, the ruler is contracted to an inch or two, and so should drop into the hole.

Ok, since the ruler cannot fall and not fall, where's the flaw(s) in the analysis?

 

[Dr. Trintignant]

Ok, since the ruler cannot fall and not fall, where's the flaw(s) in the analysis?

The answer I've always heard is that the ruler will fall--and it can do so because from its perspective, there will be a tilt relative to the hole, and therefore there is enough room to pass through.

Essentially, the answer lies in the the fact that the two ends of the ruler will have different timelines depending on the observer. If you have a clock at each end of the ruler, and they appear synchronized to someone at the center of the ruler and moving with it, the clocks will not appear synchronized to someone at the stationary hole.

- Dr. Trintignant

 

[Ziggurat]

Very interesting. So the laws of electromagnetism themselves dictate the answer. I assume this is all a result of Maxwell's equations, or is there more?

It's classical electrodynamics, so yeah, Maxwell's equations.

Also, do you know if this solution does store energy as I speculated?

No, there's no additional energy in that respect.

What about other objects, such as black holes? I can only assume that the same basic thing is true--that for a moving singularity, general relativity no longer predicts a spherically symmetric 1/r^2 gravitational field, and that instead we end up with an ellipsoidal event horizon as the Lorentz contraction predicts?

Correct.

 

[Macoy]

H

To get our bearings, let's start with Newtonian mechanics; specifically, the classic bouncing ball on a train. To the moving observer, the ball simply bounces up and down. Let's say once per second. The observer can confirm that all the basic laws of physics hold: conservation of energy, conservation of momentum, gravity, the equations governing potential and kinetic energy, etc.

A stationary observer disagrees about what is going on--from his perspective, the ball is bouncing in long parabolas

- Dr. Trintignant

surely not? the bounces are spikes on the train's momentum, not the observer's

 

[Dr. Trintignant]

surely not? the bounces are spikes on the train's momentum, not the observer's

I'm not sure what you're saying here. The point is that even though the ball has a horizontal velocity component relative to the observer, the laws of physics do not change. The observers will disagree about the momentum of objects, but the conservation laws still hold.

- Dr. Trintignant

 

[Schneibster]

Ok, an extremely important principle in science is "the equivalence principle", which roughly states that the laws of physics are the same for all observers. The observers may disagree on what exactly is happening in a given experiment, but they will all agree that the laws of physics are the same and not being violated.

Zig's correct, the equivalence principle is the statement that the effects of being in a gravity field are indistinguishable from those of undergoing an acceleration. There are some caveats having to do with the fact that to be precisely equivalent, it would have to be a planar gravity field rather than the spherical one generated in the real world, but that's essentially the idea. The term for the laws of physics being the same for all observers is a series of statements of symmetry, or invariance, over rotation, translation (change of location), time, and state of motion; it's generally referred to as "Lorentz invariance" or "Lorentz symmetry."

It's especially curious if one asks what keeps such a moving object in that compressed state. Consider a simple H2 molecule, moving along the bond direction. The two atoms "want" to be a certain distance apart; a distance dictated by electromagnetic and other forces. And yet when moving relativistically, that bond distance appears shorter. It seems to me that even the normally spherical electron orbital of a lone H atom will appear squished into an ellipsoid.

It seems clear that some mysterious forces are at work--but what are they? It seems that if the equivalence principle is to work, some kind of force--if if it's "just" a pseudoforce--must come into play to keep objects in this compressed state. Otherwise all kinds of other laws would break down. The laws of chemistry dictate what kind of shape molecules come in, and Lorentz contraction changes that shape. There must be a new force that holds molecules into their distorted shapes.

It's actually not a force, although Zig's overview is one way of looking at it, and you can consider it to be a pseudoforce. What's really happening is that velocity is a rotation in four-dimensional spacetime, sometimes referred to as "rapidity," and like all rotated objects, there is foreshortening. For example, imagine a stick, pointing transverse to your direction of observation; you look at it, and it has a certain length. If you now rotate the stick, its length appears to change- it gets shorter. The same thing happens to objects that have velocity, but the direction they rotate in involves the time dimension, so the only way to "revolve around" such objects and restore them to their original appearance is to "rotate yourself-" i.e., speed up to the same velocity as they have. This effect only occurs in the direction of the rotation, i.e. in the direction of the velocity vector, so you get the squashed atoms and molecules you've described.

Another thing comes to mind--there must be a tremendous amount of energy contained in that distortion. Electromagnetism is a very powerful force. Compressing any solid object along one axis must therefore require an enormous energy input. Is it possible that some--or even all--of the kinetic energy in an object is somehow "contained" in this distortion?

I am only guessing, but I'll bet if you worked it out, you'd find that this adds up to the relativistic apparent mass increase.

Well, that is all. As I said, it is quite likely that there is some fault in my reasoning process which makes the question irrelevant. If so, I'd like to hear where you think the fault lies. If there is no fault, am I right in thinking there must be a relativistic pseudoforce which results in Lorentz contraction? Thanks for the answers.

- Dr. Trintignant

I think there may be more than one correct answer to your question; Zig has, I believe, given one, and I have given another, and I suspect there may be a few more lurking out there.

 

[Schneibster]

Here's a question I've wondered about:

A one-foot ruler is traveling across a table at a high "relativistic speed" towards a one-foot hole in the table. From the vantage point of the ruler, the hole is contracted to an inch or two, so the ruler is expected to traverse the hole easily. On the other hand, from the vantage point of th table, the ruler is contracted to an inch or two, and so should drop into the hole.

Ok, since the ruler cannot fall and not fall, where's the flaw(s) in the analysis?

This is a brain teaser. I think the answer given is approximately correct, but my sense is, there's more to the story- and I have to point out that the ruler's front will drop, and as soon as that happens, it will strike the side of the hole. You can't have relativity in X and not in Y. But I think that's an over-simplified answer. Let me think about it; I've had a long day and I don't want to strain anything.

 

[jsfisher]

This is a brain teaser. I think the answer given is approximately correct, but my sense is, there's more to the story- and I have to point out that the ruler's front will drop, and as soon as that happens, it will strike the side of the hole. You can't have relativity in X and not in Y. But I think that's an over-simplified answer. Let me think about it; I've had a long day and I don't want to strain anything.

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.

 

[Dr. Trintignant]

The term for the laws of physics being the same for all observers is a series of statements of symmetry, or invariance, over rotation, translation (change of location), time, and state of motion; it's generally referred to as "Lorentz invariance" or "Lorentz symmetry."

Ahh, thank you. I recently read an interesting proof that the Lorentz correction is the only one possible given some very basic assumptions about symmetry. It could not predict the speed of light of course, and therefore Newtonian mechanics "fell out" of the Lorentz equation when assuming an infinite speed of light. With a finite speed of light, you get the Lorentz equation we know and love.

What's really happening is that velocity is a rotation in four-dimensional spacetime, sometimes referred to as "rapidity," and like all rotated objects, there is foreshortening.

I've heard this description before, though I haven't quite internalized it yet. Maybe it's not really possible, since I have only a 3-dimensional brain . I understand the analogy to foreshortening in the projection from 3D to 2D, though.

I am only guessing, but I'll bet if you worked it out, you'd find that this adds up to the relativistic apparent mass increase.

That would seem to differ with Zig's statement, but I suppose it might depend on your perspective and exactly how you define energy. If there's no way to extract energy from the distortion, does it really exist? I "like" your answer better, as it kinda bugs me that matter just happens to mass more in motion than at rest, but in the end the answer probably doesn't matter since the only way to tap that energy is to slow the object down.

I think there may be more than one correct answer to your question; Zig has, I believe, given one, and I have given another, and I suspect there may be a few more lurking out there.

If others have different interpretations, I'd enjoy hearing them. Thank you for yours.

- Dr. Trintignant

 

[Dr. Trintignant]

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.

There is no such thing as a perfectly rigid ruler. If there were, you could transmit information faster than light--simply tap one end, and instantaneously you have transmitted that signal to the other end. In other words, your thought experiment violates the laws of physics in its setup.

- Dr. Trintignant

 

[Ziggurat]

I am only guessing, but I'll bet if you worked it out, you'd find that this adds up to the relativistic apparent mass increase.

You don't have to guess, you can figure it out pretty simply. Now, the rest mass of two hydrogen atoms is changed slightly by bonding together in an H2 molecule. How much? Not much at all, and too little to measure. Now what happens if they're moving at relativistic speeds? Well, the energy of an H2 molecule moving at relativistic speeds is just given by E² = mc²/(1-v²/c²) (using rest mass for m - relativistic mass is redundant with relativistic energy). That doesn't depend upon bond direction, so whether the H2 molecule is aligned along the length of contraction or perpendicular to it doesn't matter. Note further that if instead of an H2 molecule, we try to apply this to two separate H atoms, the total mass is going to be almost the same, so the energy is going to be almost the same. There will be a difference caused by the bonding, but 1) it's the same fractional difference as for hydrogen at rest (meaning it's dwarfed by the energy of the atoms themselves), and 2) it will not depend upon the Lorentz contraction of the bond, and will apply even if the bond is uncontracted and perpendicular to the direction of motion.

 

[69dodge]

Here's a question I've wondered about:

A one-foot ruler is traveling across a table at a high "relativistic speed" towards a one-foot hole in the table. From the vantage point of the ruler, the hole is contracted to an inch or two, so the ruler is expected to traverse the hole easily. On the other hand, from the vantage point of th table, the ruler is contracted to an inch or two, and so should drop into the hole.

Ok, since the ruler cannot fall and not fall, where's the flaw(s) in the analysis?

Not really sure, but here's what my intuition tells me: If the ruler falls, what causes it to fall? Gravity. The Earth's gravity, to be precise. So, we should measure velocities relative to the Earth. So, the ruler is moving and the table isn't. So, the short ruler falls into the long hole.

Presumably, one could compute the gravitational effect of a moving Earth-and-table on a stationary ruler, and one would get the same answer. If one knew more about general relativity than I do, that is.

 

[Ziggurat]

The term for the laws of physics being the same for all observers is a series of statements of symmetry, or invariance, over rotation, translation (change of location), time, and state of motion; it's generally referred to as "Lorentz invariance" or "Lorentz symmetry."

Lorentz invariance is specific to the Minkowski metric (ie, special relativity applies when transforming between reference frames). The concept that the laws of physics should be the same in all inertial reference frames is actually more general than that, and is refered to as the principle of relativity, and it predates Einstein's theory of special relativity. The classical mechanics version of the principle of relativity is refered to as Galilean relativity). Galilean relativity happens to be wrong (and Lorentz invariance correct as far as we can tell), but the principle of relativity is the basis for both.

 

[King5555]

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).

What about a "stationary" jar vs a moving ruler in space? Is it even possible to label something as stationary in space?

 

[Ziggurat]

I've heard this description before, though I haven't quite internalized it yet. Maybe it's not really possible, since I have only a 3-dimensional brain . I understand the analogy to foreshortening in the projection from 3D to 2D, though.

Sometimes it helps to get the math definitions nailed down solid before trying to get an inuitive grasp of what they mean. For this you don't actually need to work in higher dimensions, even two dimensions will do.

OK, so what is a rotation? A rotation is any transformation which
1) maintains the origin and
2) maintains all distances between pairs of points
3) are continuous

Each of those criteria matter, so I'll give examples of transformations which violate them (using Euclidean 2D space).
1) the transformation x' = x + 2, y' = y does not maintain the origin. It is a translation.
2) the transformation x' = 1/2 x, y' = 1/2 y does not maintain distances between points.
3) the transformation x' = -x, y' = y is a reflection. It maintains the origin and all distances between points, as well as all distances between points. But it is not continuous: there's no way to construct that transformation as a series of infinitesimal transformations which also obey 1) and 2). So the only class of transformations which satisfy all three requirements in Euclidean space are the standard, well-known rotations in the plane. Rotations in higher Euclidean dimensions are much the same.

Here's the thing, though: criteria #2 depends upon what we mean by distance. If what we mean by distance is the standard Euclidean metric, then what we get are the standard, familiar rotations. But what if distance means something else? Then the class of transformations which satisfy those criteria will also have to be different. In special relativity, the way we measure distance in 4D space is quite different, though you can get the same results with just a single spatial dimension and a time dimension. The distance "s" between points in a 2D Euclidean space is:
s² = x² + y²which is the familiar Pythagorean theorem. In special relativity, its:
s² = x² - (ct)²Note the minus sign. That's what makes all the difference. The class of transformations which satisfy our requirements for rotation above are no longer the familiar rotations of Euclidean geometry, but the more complicated and less familiar Lorentz transformations. If you draw a spacetime graph with an x and t axis, and you apply one of these rotations, the time axis tilts (corresponding to the position of something moving), and so does the x axis. But instead of rotating in the same direction as the t axis did on our paper, it rotates in the opposite direction. Both the x and the t axis will squeeze together. So it's a very weird rotation compared to what we're used to. But it's still just a rotation, because it satisfies the requirements above.

 

[Schneibster]

Well, the energy of an H2 molecule moving at relativistic speeds is just given by E² = mc²/(1-v²/c²) (using rest mass for m - relativistic mass is redundant with relativistic energy). That doesn't depend upon bond direction, so whether the H2 molecule is aligned along the length of contraction or perpendicular to it doesn't matter.

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.

Note further that if instead of an H2 molecule, we try to apply this to two separate H atoms, the total mass is going to be almost the same, so the energy is going to be almost the same.

I think you've missed the point: the shapes of the orbitals will be compressed along the direction of motion. This will change the energy level of the orbital, and so forth as above. Like I said, it's some pretty gnarly calculation.

There will be a difference caused by the bonding, but 1) it's the same fractional difference as for hydrogen at rest (meaning it's dwarfed by the energy of the atoms themselves), and 2) it will not depend upon the Lorentz contraction of the bond, and will apply even if the bond is uncontracted and perpendicular to the direction of motion.

<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.

 

[Schneibster]

Lorentz invariance is specific to the Minkowski metric (ie, special relativity applies when transforming between reference frames). The concept that the laws of physics should be the same in all inertial reference frames is actually more general than that, and is refered to as the principle of relativity, and it predates Einstein's theory of special relativity. The classical mechanics version of the principle of relativity is refered to as Galilean relativity). Galilean relativity happens to be wrong (and Lorentz invariance correct as far as we can tell), but the principle of relativity is the basis for both.

Mmmmfff. I tend to think in terms of the underlying symmetries; and the symmetry of physical law over velocity is specifically the Lorentz symmetry. Both the Galilean and Einsteinian principles of relativity are composed of the symmetries over position and time location, and over rotation. Einsteinian relativity adds Lorentz symmetry, and my perception of the implicit question is, "what is the name of the principle that guarantees symmetry of physical law over velocity?" YMMV.