Discussion in 'Physics & Math' started by Prosoothus, Mar 27, 2006.
At such moments, I wish I'd never chosen archaeology.
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Please Register or Log in to view the hidden image! Well, I had vowed never to post here again except in homework threads, but this is just too outrageous--even for sciforums--to let slide.
It's not a strong challenge, it is just as much a grammar school attempt as any other on this website. The fact that not a single character of mathematics was written down should immediately tip you off to that.
Click the following link. This problem is worked out exactly, and it is shown mathematically that the ratio of the magnetic force to the electric force is (v/c)<sup>2</sup>. So the magnetic force can only overpower the electric force if v>c, which is against relativity. So this isn't an internal contradiction, it's just a contradiction between relativity and some half-assed interpretation of relativity.
Good grief, talk about ironic. You make nothing but "vague statements" after not doing your own homework, and you call this nonsense "your position".
You don't have a position.
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I see in the preparation of my long response that Tom has already pointed out the complete silliness of this assertion. He and Dale are, of course, both completely correct. Since it doesn't look like CANGAS is going to do anything except talk, I have provided the complete relativistic calculation for your enjoyment.
I've divided this post into sections for convenience and ease of understanding.
We've got two coordinate systems, S1 and S2. In S1, the electrons are at rest and sit at the two points (D,0,0) and (-D,0,0). S2 is moving relative to S1 with speed v along the z axis of S1. The electrons are therefore moving in the negative z direction with speed v in S2. Everything with a big 1 refers to S1, and everything with a big 2 refers to S2.
Fields in S1:
The charges are at rest, so there is no magnetic field. The electric field in S1 is given by E1 = e/((x-D)^2 + y^2 + z^2)^3/2 [(x-D)i + yj + zk] + e/((x+D)^2 + y^2 + z^2)^3/2 [(x+D)i + yj + zk]. The magnetic field in S1 is B1 = 0. Notational note: i , j, k are unit vectors in the x, y, and z directions respectively.
Force in S1:
Remembering to drop the self force, the force on the electron at (D,0,0) is K1 = e^2 / 4 D^2 i. This force is clearly repulsive which we all expect. The force on the electron at (-D,0,0) is just the negative of the above.
These next few sections are inessential to main argument, I just thought I would show how one might carry out such a calculation using all the big powerful tools. Feel free to skip this part if you only want the result.
Transformation to S2:
The Lorentz transformation matrix L<SUP>μ</SUP><SUB>ν</SUB> has the following nontrivial entries. L<SUP>0</SUP><SUB>0</SUB> = γ, L<SUP>3</SUP><SUB>3</SUB> = γ, L<SUP>3</SUP><SUB>0</SUB> = - γ v, and L<SUP>0</SUP><SUB>3</SUB> = γ v. The other diagonal elements L<SUP>1</SUP><SUB>1</SUB> and L<SUP>2</SUP><SUB>2</SUB> are both 1. All other elements are zero. Notational note: 0 stands for time, 1 for x, 2 for y, and 3 for z, all in standard notation.
Description of the fields:
The electric and magnetic fields form together the 2nd rank tensor field F, often called the Faraday tensor. The components of F in a particular coordinate system are given as follows. F<SUP>0 i</SUP> = E<SUP>i</SUP> and F<SUP>i j</SUP> = ε<SUP>i j k</SUP> B<SUP>k</SUP>. A note of explanation for those only vaguely familiar with the notation. When you evaluate the component of the field strength tensor with one time index 0 and one space index i (i can be 1, 2, or 3), you get the appropriate component of the electric field. When you evaluate the component corresponding to two different spacelike indices i and j, you get a component of the magnetic field. The ε symbol is called the Levi-Civita symbol and simply means you have to scramble things a bit. Explicitly, F<SUP>1 2</SUP> = B<SUP>3</SUP>, F<SUP>2 3</SUP> = B<SUP>1</SUP>, and F<SUP>1 3</SUP> = -B<SUP>2</SUP> (there is the scrambled sign which arises because 132 is an odd permutation of 123). F is antisymmetric so all its diagonal elements vanish, and all the other elements can be obtained from the ones I've listed here.
Transformation of the fields:
The components of the field strength tensor in the new frame S2 are related to the components in the old frame by F'<SUP>μ ν</SUP>(x') = L<SUP>μ</SUP><SUB>α</SUB> L<SUP>μ</SUP><SUB>β</SUB> F<SUP>α β</SUP>(x). In words, the transformed field components at the transformed spacetime coordinates x' are simply the old components at the old spacetime coordinates x but scrambled with two L matrices. If you like pain, you can work out the transformations for yourself. The results for our case are as follows. E2<SUP>3</SUP> = E1<SUP>3</SUP>, E2<SUP>2</SUP> = γ E1<SUP>2</SUP>, E2<SUP>1</SUP> = γ E1<SUP>1</SUP>. And for the magnetic field we have B2<SUP>1</SUP> = v γ E1<SUP>2</SUP>, B2<SUP>2</SUP> = - v γE1<SUP>1</SUP>, B2<SUP>3</SUP> = 0. Remember also that the spacetime coordinates have changed. As a sanity check, the electric field along the z direction in S2 is E2<SUP>3</SUP>(x,y,z,t) = e/((x-D)^2+y^2+(z+vt)^2)^3/2 (z+vt) + e/((x+D)^2+y^2+(z+vt)^2)^3/2 (z+vt), and you can explicitly see the replacement of z with z + vt. If z and t are referring to coordinates in S2 then this is just what you would expect for an electron moving in the negative z direction. You can check also that the various magnetic field components have the right sign by using the right hand rule. Of course, it goes almost without saying that you could have obtained these fields by simply solving Maxwell's equations for the moving sources in S2.
Force in S2:
Let us now calculate the force on the electron at (D,0,-vt) in the frame S2. The Lorentz force law is K2 = e(E2 + vxB2) where all quantities refer to the frame S2 and are evaluated at (D,0,-vt). Since v = - v k, the magnetic part gives vxB2 = - v^2 γ e / 4 D^2 i. The electric part is E2 = γ e / 4 D^2 i . Now hopefully you see the magic. When you add the two together, you find the result that K2 = (1 - v^2) γ e^2 / 4 D^2 i = (1/γ ) e^2 / 4 D^2 i. The force is always repulsive and decreases to zero as v approaches c.
The easy way:
In general, the proper time derivative d/dτ of the four momentum p<SUP>μ</SUP> is the four force K<SUP>μ</SUP>: dp<SUP>μ</SUP>/dτ = K<SUP>μ</SUP>. The the proper time derivative of the momentum is related to the coordinate time derivative according to dp/dτ = γ dp/dt. In electromagnetism, the four force has the elegant expression K = e F(u), where F is the Faraday tensor and u is the four velocity. In coordinates, K<SUP>μ</SUP> = e F<SUP>μ</SUP><SUB>ν</SUB> u<SUP>ν</SUP>. Frame S1 is the initial rest frame of the electrons with v = 0, so the four velocity has components u<SUP>0</SUP> = 1, u<SUP>i</SUP> = 0. The 1 (or x) component of the four force is then just e^2 / 4 D^2. In frame S2, the electrons are moving. The components of the four force in S2 can be obtained by Lorentz transforming the components from S1, and the result is that K2<SUP>1</SUP> = K1<SUP>1</SUP> = e^2 / 4 D^2, the same thing! How does this relate to the previous answer? Well, the spatial components of the four momentum are the components of the three momentum, and upon dividing by the factor of γ that comes from d/dτ we find that dp<SUP>1</SUP>/dt = (1/γ ) e^2 / 4 D^2.
I made an effort to get all the signs right, but mistakes do happen. Oh yeah ... Argh! Why doesn't this forum have LaTeX?
Thanks to Physics Monkey - I knew I could count on you! (but it will take a little work /study on my part to follow you.)
It is clear from Tom2's reference (http://academic.mu.edu/phys/matthysd/web004/l0220.htm)
that "observer 3" can not exist because Fm/Fe = (v/c)^2.
I still think that reduced to only observers (1) and (2) CANGAS's paradox is more effective than the "twin paradox", but perhaps that is just because it is new to me. Glad I said:
"I am confident that someone well versed in SR and Maxwell's equations can show that it is only a paradox, not a hammer smashing SR ..."
My delay in responding is a reflection of my schedule today, not a reflection of the difficulty of your challenge. However, in this case it is not my homework but yours; I always do my own homework and occasionally MacM's too (since he is admittedly severely math-deficient). But I shouldn't need to do yours since you are well-versed in the math. You are making a claim that SR predicts a specific result (which happens to be logically inconsistent). It is your claim about SR's prediction and you should be willing to back up your own claims with math that shows, according to SR, the results you claim.
This is the key point. What is that "exactly right" velocity? According to SR it is c. This has already been mentioned by przyk in the 4th post on this thread and derived in one of the links I provided on the 11th post of the thread. (by edit: and now also claimed and demonstrated by Tom2 and PM)
Are the reports contradictory? The observer moving at c relative to the charges will see (in his rest frame) that the electron's "clock" is stopped. So he calculates that the acceleration in the electron's frame should be time-dilated to zero in his frame. The observer moving faster than c relative to the charges will see (in his rest frame) that the electron's "clock" is running backwards. So he calculates that the repulsive acceleration in the electron's frame should be time-reversed to attraction in his frame. So, where is the inconsistency?
It is also not unnoticed that you have replied with words rather than math. You have made no blunder in your set-up other than assuming, without doing the math, that SR predicts anything paradoxical here.
The second observer can't exist either.
I do not doubt that either, but think that is harder to see, for one as lazy as me.
The second observer is the one who sees Fm/Fe=1. For that to happen he has to travel at v=c, which he can't do.
Ha ha. Dale, I sure wish I had seen that you posted those links back on the first page. I wouldn't have bothered spending 25 minutes of my life writing that monster up above. For some reason my first instinct is always to derive everything from scratch, and I always forget to ask google if someone has written it up for me already. Alas! It's just as well though since I don't mind showing off my mad tensor skills.
Please Register or Log in to view the hidden image! Well, I am glad you wrote it. If for no other reason the words "Faraday tensor" have made googling relevant information much easier for me!
By the way, what are the frame-invariant quantities for relativistic EM? Obviously the electric and the magnetic fields are frame-variant. The charge and current are also frame varying. I think that the Faraday tensor is also frame variant. I can't tell what is invariant.
OK. Let's summarize this issue. Everyone is saying something different. I want to know :
1) If there is a current flowing, in the same direction, through two parallel wires, will they attract each other? What is the relative reason behind this attraction in one of the electron's frame of reference?
2) If two parallel streams of electrons are moving through a vaccum, will they attract each other by the same amount as in #1 if the same number of electrons that are moving through the vacuum per second is equal to the number of electrons flowing through the conductors in #1? What is the relative reason behind this attraction in one of the electron's frame of reference?
3) At what relative speed does the attractive force negate the repulsive force in #2? Does it occur at a speed of c, or lower than c? James stated that two parallel streams of electrons would attract each other, which would indicate that it happens at a speed that's lower than c. But Dale indicated that it would happen at c? Which is it?
Physics Monkey: Thanks for spending the time for posting the math, but it just left me more confused than I was when I started this thread. I want to know the logical reasons for the attraction between the two streams and the two conductors according to relativity. I don't think that this should be too difficult considering that in the conductor example, there are only protons and electrons to consider, and in the stream example, there are only electrons. I also don't need the exact answers that the math provides, I only want the reasons for the attractions per relativity. And although you, and other scientists, have high regards for mathematical explanations, I don't. Over my four years at sciforums I've found that you can just about prove that anything is true, or untrue, using math. You can even prove that the speed of light is invariant. Please Register or Log in to view the hidden image!
I'm glad you got a little something out of it. Regarding frame invariant quantities, you are certainly correct that the electric and magnetic fields are frame variant, as are the charge and current densities. However, the Faraday tensor is invariant. The components of the tensor differ from frame to frame, but the tensor itself is invariant. It's kind of a subtle issue, but the point is that when you change frames your components change, but so do the basis tensors (just like basis vectors, but for tensors!). The Faraday tensor is a combination of these two and remains the same. It's just like ordinary rotations. If you just draw an arrow, the arrow doesn't change when you rotate coordinates. However, both the components you ascribe to the arrow and the basis vectors change.
It's easier to wrap your head around the scalar invariants. You can check for yourself that both the quantities E^2 - B^2 and E*B are relativistic invariants. You can actually write them in terms of contractions of the Faraday tensor. For example, the first invariant I listed is proportional to F<SUP>μ ν</SUP> F<SUB>μ ν</SUB>. The other is a little more complicated to write since it involves something called the dual of the Faraday tensor.
1) Two neutral conducting wires carrying parallel currents will attract. There is no electrostatic force in the frame where they are neutral, but there is an attractive magnetic force.
2) Two parallel streams of electrons in empty space will repel. The key difference here is that the system is no longer charge neutral. In the rest frame of the stream there is a huge repulsive electrostatic force and no magnetic force.
3) As Dale indicated, and I demonstrated above, the electromagnetic force approaches zero as the velocity of the electrons approaches c. It is always repulsive and never actually reaches zero.
Yes. The protons are length-contracted and the electrons are not (see cato's links) leading to a net positive charge and electrostatic attraction.
No (but this is a poorly worded scenario since there is no current in the electron's frame). There are no protons and no current in their rest frame so there is only electrostatic repulsion. In the frame where the current is the same as in #1 the repulsion will be reduced by the same amount as the attractive force in #1. This is due to the electron's time dilation.
c. But then you already knew I would claim that Please Register or Log in to view the hidden image!. James is wrong unless he is talking about the reduction of the repulsive force as an attractive force (the pseudo-force that you were complaining about earlier). The net force is repulsive for any c>v.
As you probably would guess I disagree most emphatically with this statement. If Bob has mathematically proved X and Bill has mathematically proved not-X then either Bob or Bill made a mistake and has not mathematically proven anything. You cannot "prove that anything is true, or untrue, using math". Your disdain for mathematical explanations is a disdain of logic because math is the language of logic.
Ironically, what I thought was a paradox challenging SR only a few hours ago, I now think may actually be able to support, if not be used to derive SRT!
My observer 2, who moves just fast enough to supposedly make the magnetic attraction balance the electrostatic repulsion, of course cannot exist, just as Tom2 told me, because he must be traveling at the speed of light. However, in my simple intuitive approach, I think it true that a fast traveling observer (2') does see some reduction in the net force acting on the charges, so for him, they move apart more slowly* than for observer 1, (Who is always stationary wrt the initial position of charges when they were restrained).
Let us modify the set up slightly: Imagine that when restrained, the charges are D apart and symmetrically located between two plane electrodes that are 2D apart. When the electrons reach these electrodes, a current pulse exists in circuit back to the (metal) posts that initially restrained them. This pulse advances the hands of a clock. (Not hard to repeat this so "clock" continues to "tick" but I will not bother - one "tick" seen in the two frames is enough)
I am too lazy, and perhaps not able to if I tried, but note that if observer 1 & 2' both have identical more conventional clocks and for observer 1, the tick rate of the "flying electron clock" (by choice of "D") is the same as one tick of the more conventional clock, then more than one tick of proper time for observer 2' on his identical conventional clock will pass before he see the "flying electron clock" "tick" once. I.e. this gedenken demands time dilation in a frame moving wrt to you. Or time must pass more slowly in the moving frame than in yours!
What I wonder, is can the quantitative magnitude of the time dilation, as a function of observer 2' speed, be derived from Maxwell's equations alone and simple corrdinate shifts (a new derivation of SR) or is SR required in the derivation of how much slower the "flying electron clock" is ticking for observer 2' ?
Some "paradox" this turned out to be! It supports SR‘s time dilation, and may even be a way to derive SR; but I will leave that to Tom2 and PM et al to think about. - I retired 12 years ago, and lost interest in physic 10 years before that, when I turned my attention to how the mind works, how “free will” can be consistent with physics (Yes, it can, but "you" must pay quite a price for tht "free will.") and related subjects, especially how vision / 3D perception works, foundation of illusions, etc.
* Some of this slowness may be due to their increased mass, but perhaps this can come out, without assuming SR? or at least only the SR mass increase needs be assumed? The SR mass increase is very easily brought in as an "experimental fact", (I.e. more easily demonstrated experimentally than “time dilation,” I think.)
A few notes and some wild ideas relative to the foot note of my last post:
In that post, which reading this paragraph can replace, I suggest that CANGAS's two "released electrons" initially separated by "D" and subsequently collected by two plane, symmetrically-placed, electrodes were a clock, that "ticked" at least once. ( 1 Tick = time of flight of the electrons.) Both the moving and stationary observers have identical conventional clocks and D is such that one tick of the stationary observer's conventional clock = one tick of the "flying electron" clock. However, the reduction in the net force of repulsion caused by the magnetic force of attraction present only in the moving frame, makes more than one tick of the moving conventional clock pass during one tick of the "flying electron" clock, as seen by the moving observer. Thus, instead challenging SR, CANGAS's "paradox" predicts "time dilation."
The footnote of prior post acknowledges that it may not be possible to derive SR's equations for time dilation from Maxwell's equations and coordinate transforms alone because, the mass of the electrons has increased for the moving observer and this also will make "electron clock" take longer to make one tick. However, "relativistic mass" (term most physicist more modern than me hate, but I still like and use) is not a "scalar" like ordinary mass. It increase only in the direction of motion (except possibly for some higher order effects in the SR equations, as I recall). Thus perhaps the mass of the "flying electrons" in the direction they are accelerating is unchanged (Good old F = ma or really a = F/m can still be twice integrated to find how long it takes for a “tick” in moving frame with moving frame's F' < F of the lab frame by the reduction of the magnetic force. Etc.)
Thinking about all this last night, as I was going to sleep, lead me to again wonder why inertial mass is same as gravitational mass, worry if I should make this post (embarrass my self again by speaking of relativistic mass etc.) So I began to think if there was some way I could state what I wanted to about extending the footnote of prior post in more modern mass terms (avoid ever mentioning "relativistic mass.")
The idea of marble on large stretched rubber sheet always comes to mind when I start to think about these things (I have actually seen demonstrations of how marble moves on such a sheet with some sections raised and lowered. - The dynamic equations are the same as an electron's flying thru a potential field. Sometimes in the "good old days" (before digital computers, made many experiments pointless. Model airplanes and wind tunnels useless for airplane design, etc.) rubber sheets were used (as fast, efficient, special-purpose, analogue computers) to help design electron guns, electron microscope lenses, etc.)
First thing I thought was: Mass is "warping space" just like the marble on the rubber sheet. Next: Obviously to move the mass, in a "virtual displacement," you must push it up the "hill" it has created. Well, mass or that "dent" and the steepness of the hill" are directly proportional - Hence "gravitational mass" = "inertial mass." But next thought was: That is nonsense, because inertial mass is found/acts in real, not virtual, displacements. I.e. as the marble is pushed on the rubber sheet, the depression it makes moves along with it. -It is not “climbing any hill.”
(I have been down this mental path before. I always tried to associate "inertial mass" with the effort needed to move the "dent" in the rubber sheet, but that would depend upon the characteristics of the rubber which might not be exactly in direct relationship to the characteristics of the rubber that determine the depth of the "dent" so I always dropped the idea, especially with its modern “stress tensors,” "space warping mass"could have a lot of freedom to make these two conceptually different “characteristic” totally unrelated. Why they should be critically related, would just be the “new question“ replacing old question: "Why is inertial mass = gravitational mass?". I would be in “way over my head in deep yogurt math“ with this "new question." Conculsion: I had better just go with Mach‘s answer, or none, to the original question.)
Last night, half asleep, I had a new twist: What if all motion really is set of very small virtual displacement steps in zero time. Then, “after” each virtual step, some time is required for universe to adjust to the new configuration caused by these "Plankian small" moves. If cause and effect is preserved, then Plank moves, in non-zero time, cause changes in forces acting. These force changes cause slight changes in next "Plank move" of everything. Etc. as universe setps forward discretely first in space, then a time step.* (Why it does not "step backwards" in the next Plank move, is a question I now ignore as it is not relivant to the why inertial mass = gravitational mass question but to the why entropy increases question. Note I am admitting another "bad" characteristic of us old timers. We tend to ask pointless "why" questions.)
I.e. on the Plank scale, all moves are "virtual and Plankian" then universe adjusts forces for new configuration and this takes time. (Even for God to do all those calculations Please Register or Log in to view the hidden image! and make sure his "laws of physics" are not violated.). If all moves are virtual, then I need not abandon the idea that inertial mass = gravitational mass because the forces acting in the zero time duration move are still directly related to the last configuration of the universe. (In old “rubber sheet” model, the hill does not change with the move.) I.e. all motion and time is a set of very small steps. (Quantized time and space is certainly not a new idea. It is an old idea, discussed by many better informed than me.) What I am trying to say, that may be new, is perhaps this discrete stepping of time and configurations of the universe, is at the heart of WHY inertial mass = gravitational mass.
Hope this not too boring and may stimulate someone to either: (1)think more about it or (2) tell me I need to read reference XXXXX where this idea was shown to be at best a dead end, or more probably just nonsense.
* this note is already too long, but some may enjoy my rambling style so I note that the idea of first step in space then in time has had great success in explaining why when you shake a box full of various size round stone the big ones end up on top. Model was: start with random mix and center of mass locations, but of course no two stones "interpenetrate." Step 1: Let every stone move vertically, if it can, starting with the bottom most stone, until all stones are supported vertically and total gravitational potential is reduced, but lost (stones at rest at end of this step). Step 2: Shake stones to randomize them horizontally, but with no change in gravitational potential and no "inter penetrations." Step 3, test for "stop condition". Step 4: return to step 1. When stop condition (no stone can move vertically) you find the big stones on top and the little ones on the bottom.
I.e. some real processes can be modeled as if a sequence of independent steps with adjustments to new configuration produced after, instead of during, the step. Universe adjusts "after" plank step to make new forces and forces acting in Plank step are constants. (I may not have model exactly correct, but it is something like this. The little stones, on average, move more vertically down in each cycle more than the big ones.)
In typical fashion you have screwed up once more. If you had taken an actual look at the processes you would have noticed that time dilation; which is a function of (1 - v<sup>2</sup>/c<sup>2</sup>)<sup>1/2</sup> does not equal v<sup>2</sup>/c<sup>2</sup>.
They are very distinctly different and produce different curvatures. They are equal mathematical at two points v = 0 and v = c.
The process DOES NOT predict nor support time dilation.
v<sup>2</sup>/c<sup>2</sup> is the ratio of the magnetic force to the electric force, not the time dilation. This ratio equals 1 at v=c where the time dilation is infinite and it equals 0 at v=0 where the time dilation is 1. Infinity does not equal 1 nor does 1 equal 0, so they are not equal at those two points (or any other points).
In any case, BillyT is correct, Maxwell's equations imply time dilation etc. when analyzed in different coordinate systems using the principle of relativity (first postulate). This shouldn't be terribly surprising, even to a reflexive anti-relativityist like yourself, since c is also a result of Maxwell's equations.
Well for others information here is a table of the two terms:
They are not equal nor complementary. My comments were directed at the assertion that the magnet force/electric force ratio directly inferred time dilation. :bugeye:
The ratio does infer time-dilation (when the different frames are analyzed using only the first postulate).
You are confusing A infers B with A equals B.
Separate names with a comma.