Question about Lorentz contraction

[69dodge]

"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.

I don't think it has anything to do with hyperbolic geometry. What matters is the number of dimensions. It works in three dimensions but not otherwise, because only in three dimensions do all the vectors orthogonal to a given one form a plane. (In general, they form a subspace that has one dimension less than the full space.)

 

[69dodge]

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]

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.

Don't know about the math. Just fixed the LaTex. (Inside the [latex] and [/latex] tags, you need to surround stuff by \[ and \] for a displayed equation or by $ and $ for inline math.)

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

Yes, I agree with this.

 

[Schneibster]

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

Sure. It's interesting.

"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.

It kinda depends on your definition of "plane," doesn't it? After all, when we look at it, it looks flat. We can't see time.

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.

Yes, and that's why I like the -+++ formulation of Minkowski's interval equation:
s² = x² + y² + z² - c²t²

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

I do too; that's why I like that way of looking at it. Enjoy.

 

[MortFurd]

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?

How high is the gravity? Assuming 1G, the ruler won't have much of a chance to fall through anything.

Ignoring the problem of accelerating a ruler to c, ignoring the friction that would slow it down and cause lots of heat, and ignoring what might happen if such an experiment were to go bad (mass of the ruler smacking into the table at c):
Given c=186283 miles per second=11802890880 inches per second.
The ruler would cross a 12 inch hole in 1.0167000713642114091967272343367e-9 seconds.
At 1G, the ruler would drop (9.8m/(s^2)) 1.013005454409752720865069310528e-14 millimeters in crossing the hole.

That "fall" is smaller than the diameter of a hydrogen atom (which Wikipedia gives as 1.06×10-10 m.) That difference would presumably be covered by whatever mechanism you were using to keep the ruler from burning up from friction while sliding across the table.

Assuming a table 2 millimeters thick and a ruler 2mm thick, then gravity would have to cause an acceleration of 3,869,673,142,366,310 meters per second^2 (approximately 394,864,606,363,909 times the gravity we experience here on the Earth's surface) in order for the ruler to drop through the hole in the time it takes to traverse it.

 

[Ziggurat]

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.

Actually, it's not as bad as I thought, and it demonstrates that the total energy works out to be the same. We've already solved for the electric fields, and found

[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]

The magnetic field has a similar (though not quite the same) angular dependence. Again dropping the force constants and the unit vector indicating field direction, we have

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

And for the two cases of interest:
[latex]\[ B(\theta=0) = 0[/latex]
[latex]\[ B(\theta=90) = \frac{qv}{R^2} \frac{1}{[1 - (v^2/c^2)]^{1/2}} \][/latex]

You are correct that this is an attractive force for the theta=90 case (for like charges). So what is the force? Well, it's q vxB for magnetic case, and qE for the electric case. So the total force ends up being (ignoring force constants)
[latex]\[ F(\theta=0) = \frac{q^2}{R^2} (1 - v^2/c^2) \][/latex]
[latex]\[ F(\theta=90) = \frac{q^2}{R^2} \frac{1 - v^2/c^2}{[1 - (v^2/c^2)]^{1/2}} = \frac{q^2}{R^2} (1 - v^2/c^2)^{1/2}\][/latex]
where the -v² on top in the second equation comes from the magnetic field contribution, and the 1/c² comes from converting the force constants (which I left out otherwise). Now these forces aren't equal, and they aren't even equal if we consider the length-contracted R for the theta=0 case. But we're not interested just in the force, we're interested in the potential. And for that, we need to integrate over R. The integral turns 1/R² into 1/R, and when you evaluate this at the length contracted position, you add in the missing (1-v²/c²)^1/2, which will make the potential for co-moving charges at both locations exactly equal. As expected from conservation of energy arguments. So there's nothing funky going on with quantum mechanics and relativity: the conservation of energy arguments I made works perfectly, the actual electrodynamics calculations back them up, and the potentials at the Lorentz-contracted positions all end up exactly the same, modified by the same (1-v²/c²)^1/2 factor you'd expect from just looking at relativistic equation for total energy of a system.

 

[Dr. Trintignant]

I don't think it has anything to do with hyperbolic geometry. What matters is the number of dimensions.

<slaps forehead> Yes, I see you're right. For N dimensions, there must be N(N-1)/2 planes of rotation, if a pair of axes defines a plane. Only for N=3 does the number of planes equal the number of dimensions. Obviously it's easy to get trapped with the idea that the two must be equal, but clearly it's not for dimensions other than 3.

- Dr. Trintignant

 

[Dr. Trintignant]

So there's nothing funky going on with quantum mechanics and relativity: the conservation of energy arguments I made works perfectly, the actual electrodynamics calculations back them up, and the potentials at the Lorentz-contracted positions all end up exactly the same, modified by the same (1-v2/c2)1/2 factor you'd expect from just looking at relativistic equation for total energy of a system.

Wow--I think I'm convinced. I'm happy that you're not afraid to solve a few equations to answer the problem. Although verbal arguments have their place, sometimes at the end of the day you just have to run the numbers.

Sometimes I find the self-consistency of the universe amazing. No matter how complicated the situation, or how you rearrange things, or change reference frames, or whatever, the universe holds to its laws. I can understand why the perpetual motion types keep at it--the feeling that if you just keep trying, eventually you'll find a loophole. But at the same time I think if they really had an understanding of physical law, they would give up before starting: the universe is just too clever. It's already accounted for far more subtle problems than a silly machine can try to exploit.

- Dr. Trintignant

 

[Schneibster]

Look, simple argument: potential energy has mass. We all know the one about the stretched rubber band having more mass than the relaxed one.

OK, now how about squashed orbitals? Remember, it all still has to make sense, so do you REALLY want to argue that an atom that is speeding past has LESS ENERGY? Excuse me, I guess I learned relativity somewhere else.

 

[Ziggurat]

Look, simple argument: potential energy has mass. We all know the one about the stretched rubber band having more mass than the relaxed one.

Of course it's got more energy when it's moving. But it's got more energy whether it's squished lengthwise or from the side, and the additional energy is the constant factor (1-v²/c²)^-1/2, regardless of orientation (as predicted by simple relativistic arguments). And the electrodynamics calculations I did above back that up.

OK, now how about squashed orbitals? Remember, it all still has to make sense, so do you REALLY want to argue that an atom that is speeding past has LESS ENERGY?

What's got less energy is a hydrogen molecule compared to two free hydrogen atoms (the orientation of the hydrogen atom with respect to motion has no effect on the energy). And yes, of COURSE it's got less energy: it's a stable bound state, not a metastable state. The bond energy is negative, not positive. That's true at rest, and so it must be true in motion as well.

Edit to add: that means comparing a hydrogen molecule at rest to two free hydrogen atoms at rest, and a hydrogen molecule in motion compared to two free hydrogen atoms in motion, NOT comparing a moving hydrogen molecule to two free hydrogen atoms at rest.

Excuse me, I guess I learned relativity somewhere else.

The orientation of the orbital with respect to the direction of motion has no effect on the energy. The only argument you've got that it should is that it seems like it should. But I already went through the math and proved that the electric potentials are NOT different for positions on the surface of a lorentz-contracted sphere. And as I pointed out (and you still haven't dealt with), if the energy of an orbital changed by rotating it with respect to the direction of motion, then you'd be violating conservation of energy (since that operation has no energy cost in the object's rest frame). So why do you persist in believing this when both the electrodynamic calculations I showed above and conservation of energy indicate otherwise? Where's your actual evidence, besides hand-waving, that a moving hydrogen molecule will have an orientation dependence to its energy? You say you must have learned relativity somewhere else. But where, pray tell, did anyone ever teach you that the energy of atomic orbitals of moving atoms DID change depending upon their orientation with respect to motion?

 

[ponderingturtle]

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?

Length contraction is in the dirrection of motion, not perpendicular to it. So if the ruler was moving so that it would exactly fit into the whole its cross section would not change only length(in the dirrection of motion)

 

[ponderingturtle]

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?

With the reletivity of simulitnaity you if the rule fits in depends on the reference frame of the observer. It might in some but not in others.

 

[jsfisher]

Length contraction is in the dirrection of motion, not perpendicular to it. So if the ruler was moving so that it would exactly fit into the whole its cross section would not change only length(in the dirrection of motion)

Maybe it was unclear, but the question was about the 1-foot ruler traveling length-wise across a table, much like a train. Would it fall down into a 1-foot gap in the table under the influence of gravity. Cross section doesn't come into the question as posed, only the length of the ruler and of the gap.

 

[ponderingturtle]

Maybe it was unclear, but the question was about the 1-foot ruler traveling length-wise across a table, much like a train. Would it fall down into a 1-foot gap in the table under the influence of gravity. Cross section doesn't come into the question as posed, only the length of the ruler and of the gap.

If the ruler and table are in the same reference frame then it is not an issue they would be contracted in the same fashion.

IF they are not in the same reference frame, what you are doing is making it needlessly complex by tilting the direction of motion of the ruler. If you are considering having it being accelerated by gravity that is still a different issue. But in almost any real gravitational field the gravitational acceleration is not going to make a meaningful change in the motion of the ruler.

If you are putting a ruler through a pipe and both have the same rest length, then if the ruler ends up entirely in the pipe is reference frame dependent as you are dealing with the relativity of simultaneity in this.

 

[MortFurd]

Maybe it was unclear, but the question was about the 1-foot ruler traveling length-wise across a table, much like a train. Would it fall down into a 1-foot gap in the table under the influence of gravity. Cross section doesn't come into the question as posed, only the length of the ruler and of the gap.

I think I answered that in an earlier post.
To make the answer clear:
The ruler won't drop into the hole, regardless of Lorentz contractions. At the speed of light (and assuming a 1G field,) the ruler will drop less than the diameter of a hydrogen atom while crossing a twelve inch hole. The surface roughness of the table presents hills and valleys that are hundreds if not thousands of times bigger.

In order for a (realistically thick) ruler to fall through an (unrealistacally) thin table, the gravity pulling the ruler through the hole would have to be a bazillion times higher (see the numbers in my earlier post) than 1G.

Try slapping a ruler across a table sometime. It shouldn't be too hard to make the ruler jump a twelve inch hole without falling through, just using speeds that yu can generate by hand. No need to call in relativity and the speed of light.

 

[drkitten]

If the ruler and table are in the same reference frame then it is not an issue they would be contracted in the same fashion.

IF they are not in the same reference frame, what you are doing is making it needlessly complex by tilting the direction of motion of the ruler. If you are considering having it being accelerated by gravity that is still a different issue. But in almost any real gravitational field the gravitational acceleration is not going to make a meaningful change in the motion of the ruler.

If you are putting a ruler through a pipe and both have the same rest length, then if the ruler ends up entirely in the pipe is reference frame dependent as you are dealing with the relativity of simultaneity in this.

I don't think you're quite understanding his scenario.

You are right, the rod-in-the-pipe scenario is well understood and the accepted answer is that whether or not the rod fits entirely within the pipe is frame-dependent. However, jsfisher is trying to establish a scenario where the results are not just quantitatively different, but where the object takes two different paths in space depending upon the events.

In particular, it's not physically possible for a 1 ft ruler to fit through a 1/2 ft gap, so if the ruler fit through the gap, the gap must have been larger than the ruler. This appears to be an ojbective (and frame-independent) way to compare two sizes -- but if the size is frame dependent, how can there be a frame-independent comparison?

The answer, of course, is that it can't. You can't physically set up such a comparison, because other (physical) factors such as the rotation of the rod interfere. He assumes that the rod remains horizontal throughout, when of course, that's frame-dependent. If I drop a rod "horizontally," someone zipping past at near the speed of light will see the rod dropping "tilted," with the amount and direction of the tilt being frame-dependent.

 

[drkitten]

In order for a (realistically thick) ruler to fall through an (unrealistacally) thin table, the gravity pulling the ruler through the hole would have to be a bazillion times higher (see the numbers in my earlier post) than 1G.

And to expand on this, if for some reason you decide to do this experiment on the surface of a black hole (and in that case, I want to meet your funding agency), then the rod will simply tilt because the front of the ruler will be unsupported while the back is still supported. With that degree of tilt, there will be no problem fitting through a relativity-foreshortened hole.

So the answer depends upon the local gravity but is the same for all reference frames.

 

[ponderingturtle]

I don't think you're quite understanding his scenario.

You are right, the rod-in-the-pipe scenario is well understood and the accepted answer is that whether or not the rod fits entirely within the pipe is frame-dependent. However, jsfisher is trying to establish a scenario where the results are not just quantitatively different, but where the object takes two different paths in space depending upon the events.

In particular, it's not physically possible for a 1 ft ruler to fit through a 1/2 ft gap, so if the ruler fit through the gap, the gap must have been larger than the ruler. This appears to be an ojbective (and frame-independent) way to compare two sizes -- but if the size is frame dependent, how can there be a frame-independent comparison?

The answer, of course, is that it can't. You can't physically set up such a comparison, because other (physical) factors such as the rotation of the rod interfere. He assumes that the rod remains horizontal throughout, when of course, that's frame-dependent. If I drop a rod "horizontally," someone zipping past at near the speed of light will see the rod dropping "tilted," with the amount and direction of the tilt being frame-dependent.

Ah I see he is trying to establish an absolute reference frame. I still don't think it is possible. He is just hiding the effect in trying to make a overly complicated arangement.

He needs to learn to chose his coordinates better.

As for accelerating the rule, he is changing its reference frame.

 

[MortFurd]

And to expand on this, if for some reason you decide to do this experiment on the surface of a black hole (and in that case, I want to meet your funding agency), then the rod will simply tilt because the front of the ruler will be unsupported while the back is still supported. With that degree of tilt, there will be no problem fitting through a relativity-foreshortened hole.

So the answer depends upon the local gravity but is the same for all reference frames.

If G is high enough, then the ruler will tilt. The tilting will (I think) actually lessen the (apparent) shortening of the hole and make it easier for the ruler to pass through. If G were high enough to make the ruler do a 90 degree turn with in the length of the hole, then the contraction would disappear entirely, and you'd have the ruler dropping straight down through an aparent 12 inch hole. And I want some of whatever that ruler is made of. The G forces acting on it while sliding across the table would be staggering, and the forces involved in making a 90 degree turn in 12 inches while traveling at the speed of light would be even more fantastic. If the ruler can stand up to either of those, then that would be some truly far out stuff.

The point of all of this is that making that ruler fall through the hole while traveling at the speed of light requires a truly gargantuan acceleration perpendicular to the direction of travel.

 

[drkitten]

If G is high enough, then the ruler will tilt. The tilting will (I think) actually lessen the (apparent) shortening of the hole and make it easier for the ruler to pass through.

Half right. If the ruler tilts, then it will fit through the hole because it's now going through lengthwise. Put a pencil into a soft drink bottle. Put a meter stick through a milk ring. Hard to do crosswise, but easy enough if it's tilted any signficant amount. Nothing do do with decreased foreshortening and everything to do with the angle.

The point of all of this is that making that ruler fall through the hole while traveling at the speed of light requires a truly gargantuan acceleration perpendicular to the direction of travel.

No, it doesn't require that much acceleration.... just enough to miss the far edge of the hole and then go point-first into the gap.

 

[drkitten]

Ah I see he is trying to establish an absolute reference frame. I still don't think it is possible.

That's the ultimate goal of his argument, yes.

He is just hiding the effect in trying to make a overly complicated arangement.

I don't think it's overly complicated. And I don't think he's trying to "hide" anything; the absolute frame emerges as a consequence of his argument (or would if the argument were valid), not as a premise.

As I pointed out, the question "will this object physically fit through this gap" is not does not vary with reference frame. If one observer sees a collision, all will. If one observer sees no collision, none will. The presence or absence of a collision is frame-invariant. The exact details -- for example, the relative scales and timing -- may vary, but at the end of the day, the ruler will be either atop the table or below it, and all observers will agree.

As for accelerating the rule, he is changing its reference frame.

Not substantially. As with the twin paradox, the "paradox" emerges from the parts where the system is still operating in inertial frames.