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Why can’t I go faster than the speed of light? Hints from electrodynamics

April 8, 2009

I should state from the very beginning that the title of this post is a bit of a tease.  I don’t know why nothing can go faster than the speed of light.  No one does.  It is a fundamental postulate of the theory of relativity: “The speed of light in a vaccum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.”  It is a weird assumption, and it leads to even weirder consequences.  It also happens to stand up to every conceivable test we have thought to put it through.  But it seems so strange and arbitrary; how did Einstein think of it?

It is a little-known fact that Einstein’s ground-breaking paper about relativity was called “On the Electrodynamics of Moving Bodies.”  It is even less well-known that the rule “nothing moves faster than the speed of light” is a consequence of the laws of electricity and magnetism, which had been known since the mid 1800’s.  Einstein was just the first scientist with enough imagination to realize how important this consequence was, and how generally it could be applied.

In this post, I’ll try to explain how simple rules for electric and magnetic fields demonstrate that all velocities must be less than the speed of light.  It can be done by imagining the consequences of a simple experiment, where a charged particle is watched by two different observers.  The demonstration done here doesn’t really explain why the speed of light is an absolute “speed limit”; it just shows that the question “how come nothing can go faster than the speed of light?” is equivalent to the question “how come the laws of electricity and magnetism look the way they do?”.  And even if this isn’t a real answer, there’s something profoundly satisfying about realizing how two hard questions are just different ways of looking at the same thing.

Imagine a very long, thin, rod that has been loaded up with positive charge.  The rod creates an electric field which pushes other positive charges away from it.  So, if you were watching as someone placed a small object with positive charge q next to the rod, you would see that it was repelled with some force F_E.  You can solve directly for the strength of the force, if you want, using Gauss’s Law: F_E = q (Q/L) / (2 \pi \epsilon_0 r), where Q is the amount of charge on the rod, L is its length, r is the distance between the rod and the object, and \epsilon_0 is a fundamental constant of nature called the “vacuum permittivity”.

If the rod is moving, however, it also creates a magnetic field.  This is also a well-known property of electricity and magnetism: moving charges create magnetic fields.  The shape of the magnetic field is circular, curling around the rod as it moves.  If some charged object is moving next to the rod, it will respond to the rod’s magnetic field.  If you assume that the rod and the object are moving parallel to each other with the same speed v, you can work out the strength of this “magnetic force” using Faraday’s Law: F_B = -\mu_o q (Q/L) v^2 / (2 \pi r), where \mu_0 is another fundamental constant called the “vacuum permeability”.  The negative sign is there because the magnetic force works in the opposite direction as the electric force.  It pulls the object toward the rod.

Here we see the beginnings of a paradox.  If the rod is standing still next to a stationary charge, there is an electric force repelling the two.  If the rod and the charge are both moving, however, there are two forces: an electric force repelling them and a magnetic force pulling them together.  How is that possible?

To make the paradox more apparent, consider the picture below.  One observer, named A, watches as the rod and the charged object whisk by him at speed v.  Another observer, named B, moves alongside the rod and the object with the same speed v.  Because A sees a moving rod, he observes an electric and magnetic field emanating from it, and consequently sees the object being pulled by two competing forces F_E and F_B.  From B’s perspective, however, nothing is moving at all.  Neither the rod nor the charged object are getting any closer or farther from him.  Therefore, he sees that the rod creates an electric field pushing the object away, but no magnetic field.  For him, then, there is only one force F_E.

A thought experiment with moving charges

A thought experiment with moving charges. Observer A sees electric field (black lines) and magnetic field (red lines) emanating from the rod. B sees only electric field.

The situation seems strange.  How can two different things be happening depending on how fast you’re moving when you look at it?  Maybe something subtle is happening, but one thing is clear: there is no way that A should see the object move upwards while B sees it move down.  If A sees that the total force on the object is upwards (F_B > F_E), then he will also see the object move upward, closer to the rod.  But B sees only one force, pushing the object downward.  This is a major problem.  One universe cannot accomodate both outcomes.

So, it must mean that the magnetic force F_B is always weaker than the electric force F_E, regardless of the velocity.  Since we have the expressions for F_E and F_B, we can check when this is true:

F_B < F_E

\Rightarrow v < 1/\sqrt{\epsilon_o \mu_0}.

The quantity 1/\sqrt{\epsilon_o \mu_0} = 3 \times 10^8 meters per second is exactly the speed of light.  So why can’t these two charged objects move faster than the speed of light?  Because if they did, then there would be two completely different realities: one for everyone who wasn’t moving, where the object got pulled up, and another for everyone who moved with the object, where it was pushed down.

So, by imposing the “speed limit” of 3 \times 10^8 meters per second, we have avoided a universe-splitting catastrophe.  But you might notice that there is still a problem.  Even if v is less than the speed of light, B still sees a stronger downward force than A sees, which means that B will watch the object move downward more quickly.  How is that possible?  Einstein realized this problem too, and to solve it he came up with a drastic fix: time dilation.  The only way to resolve the paradox is if time is moving at different rates for the two observers.  A must observe B to be “living in slow motion”.  That way, A sees the object falling slowly next to a person living in slow motion, which leads him to conclude “B must see that object falling faster than I do”.  Which he does, although from his perspective it is for very different reasons.

24 Comments leave one →
  1. ROGER ANDRES permalink
    April 15, 2009 4:09 pm

    IF LIGHT IS COMPOSED OF PHOTONS WHICH MOVE AT THE SPEED OF LIGHT ..
    THEN ENERGY (INERTIA OR POTENTIAL) COULD BE CONVERTED TO PHOTONS AND TRAVEL AT THE SPEED OF LIGHT.THE PROBLEM WOULD BE IN RE CONFIGURING THE PHOTONS BACK TO THEIR ORIGINAL FORM OF MATTER . I GUESS THIS COULD BE CALLED TELEPORTATION.MOVING FROM POINT A TO POINT B AT LIGHT SPEED.

  2. gravityandlevity permalink*
    April 15, 2009 6:38 pm

    Hi Roger, thanks for the comment.

    What you’re saying is true: light is another form of energy, which happens to move at the speed of light. The problem of “reconfiguring the energy back to its original form” can be a pretty enormous one, though!

    Think of it this way: when you drop a rock into a pond, some of its initial (gravitational) energy becomes “wave energy”, which moves as ripples across the water. These waves move at a particular speed, and in a sense they transfer the rock’s energy from one place to another. But using those waves to throw another rock back up into the air would be pretty hard.

    Note also that even though I derived the absoluteness of the speed of light in this post, nowhere did I talk about actual light. What we call the “speed of light” is a universal constant that has much more significance than “the speed at which light happens to travel.”

  3. HD Crowther permalink
    August 2, 2009 9:49 am

    (What an interesting website! Where’s the ‘least action’ explanation?)

    If the string theorists are correct, and/or every spatial dimension is
    connected to another 3 extra tiny dimensions (how odd an idea),
    then under special circumstances, ‘c’ is easily exceeded. Special
    circumstances being ‘entanglement’, wherein entangled particles
    are much closer together than they seem, through the ‘tiny’
    dimensions.

    • gravityandlevity permalink*
      August 2, 2009 9:56 am

      Thanks HD!

      The “principal of least action” is definitely on my list of big ideas to tackle. Stay tuned; I’ll probably try to write up something in the next month.

  4. Goldeigh permalink
    August 9, 2009 12:36 am

    GL! [“Gravity&Levity”, not “Green Lantern”] That was awesome! That’s exactly the kind of explanation I’m looking for. I’ve read explanations of relativity for us masses, but even if I can follow the relativity explanation and understand the concept of c as a limit, I could never for the life of me figure out what the hell one had to do with the other and why Albert would get from “motion is relative” to “speed of electromagnetism”. I have read that a moving electric field creates a magnetic field while a static one doesn’t — which still baffles me but, OK. But I wasn’t smart enough to connect that motion to the concept of relative motion (which is one example of why I couldn’t make it through college physics despite my fondness for it). But now it seems so obvious! Well, maybe not obvious, but at least I see how the mental connection was made.

    Thanks so much!

  5. Goldeigh permalink
    August 9, 2009 1:13 am

    One small question: the rod’s moving positive electric charges “electrically” repel the moving positive charge object. Got it. You say the magnetic field created by the rod’s moving positive electric charge will attract the positively charged object. My first thought was, how does he know the magnetic field will attract and not repel the object? My simple experience with bar magnets says only if the magnetic charges are opposite will they attract. It seems like a positive electric charge would create a positive magnetic charge that would repel a positive object; but I’m sure that’s wrong. What decides whether the magnetic field is positive or negative, and why wouldn’t the rod’s moving charges create the same kind of field as the object’s moving charge, thus repelling each other?

    Thanks,

    MG

    • gravityandlevity permalink*
      August 17, 2009 8:44 am

      Hi Goldeigh,

      It is a surprising truth that there are really no such things as “magnetic charges”. All magnetic fields are created by moving electric charges. So somehow, when you have a bar magnet, there are electric currents moving around through the iron that create the magnetic fields. These current might just be the movement of individual electrons in their orbits around positive nuclei, but still every magnetic field must result from moving electric charges.

      • Largo permalink
        August 18, 2009 6:11 am

        Does this depend on the non-existence of magnetic monopoles? (Disclosure: I know only enough physics to make me dangerous — dangerous to making an ass out of myself by asking possibly inane questions!)

    • Goldeigh permalink
      August 17, 2009 12:40 pm

      Well I’ll be. Somehow I never knew that. I guess my next task is to better understand interactions between the magnetic fields and other electric particles. Does a magnetic field always work in the opposite direction as the electric field that created it? And does the word “direction” literally refer to a flow of . . . “something” that is the magnetic field around the wire in a spatial direction (i.e., from the perspective in your drawing, clockwise) or does it simply refer it’s effect on other electric charges?

      Thanks GL. Great writing.

  6. August 17, 2009 8:01 am

    Was this what James Clark-Maxwell worked out shortly before he died? I remember something about it from second-year physics, but that was 15 years ago. He didn’t get as far as Einstein, at least in writing, but was searching for a consistent solution if I recall correctly.

    Is there any reason that an uncharged fundamental particle (i.e. it has no charge at any level of being) cannot travel faster than the speed of light?

    Thanks for the great insight from this post, and the memories of pleasant times in the Maxwell Lecture Theatre!

    • gravityandlevity permalink*
      August 17, 2009 8:47 am

      Thanks, Richard.

      As far as I know, the statement that “nothing can go faster than the speed of light” is just an assumption of the theory of relativity. Einstein had some evidence (from Maxwell’s equations) that it must be true for charged particles, and he made the assumption that it must be true for all particles. Remarkably, it turned out to be true. But I don’t know of any good, simple thought experiment that will lead you to the same conclusion for an uncharged particle.

      Probably there is one. Anyone?

      • August 21, 2009 5:21 pm

        I don’t see how there could be. Tachyon theory is self-consistent (well, provided you are willing to deal with tachyon instability), and involves objects with going faster than light, sort of, whether they are charged or not. I don’t think there’s any good evidence for the measurable existence of tachyons, but the fact that the theory is self-consistent suggests that there’s no simple thought experiment that would rule them out.

  7. John Trumpener permalink
    August 17, 2009 8:06 am

    Great blog. However, I believe in your thought experiment that you should be thinking of length contraction, not time dilation.

    Allow me to make two changes to your thought experiment to illustrate: 1) Change the sign of the charge on the object so that the electric field provides an upward force, and the magnetic field (if there is one) is a downward force. 2) Let there be a downward gravitational field and let the object have a mass such that the gravitational force is exactly equal to the electric force (from B’s perspective).

    Now, again from B’s perspective, the electric and gravitational forces exactly balance, and there is no magnetic force, so the object never rises and falls. However, from A’s perspective, the electric and gravitational forces cancel, and the magnetic field pushes the object down, so A sees the object fall.

    There is no way to use time dilation to reconcile A seeing the object fall, and B seeing the object stationary. (Unless you assume that the time is dilated to zero for B, for any velocity, which leads to a new and larger set of problems).

    The way to deal with this is to assume a contraction of length equal to (1-v^2/c^2)^1/2. This does not apply to B (since B is not moving relative to the rod), but it applies to A, so A sees a lineal charge density Q/L bigger than A sees (since the same charge is packed into a shorter length of rod). Therefore, A sees a bigger electric force, with the increase being exactly big enough to cancel out the magnetic force that A also sees.

    I believe this was postulated by Lorentz before Einstein’s 1905 paper, and that, in fact, the Lorentz transformation also includes time dilation. What Einstein did was to start with his postulate of the constancy of the speed of light for all observers, and from this simple assumption he could derive length contraction, time dilation, and eventually mass-energy equivalence and everything else.

    • gravityandlevity permalink*
      August 17, 2009 8:51 am

      Thanks Richard,

      You’re right, of course. The difference between A and B is that time is moving at different rates for the two, and that the charges in the rod appear to be at different densities. That is, the rod gets “dilated” for A and therefore the charge on it appears more tightly packed. In this particular case both phenomena work in the same direction. But I like your example, which illustrates that the two don’t always contribute in the same way.

  8. John Trumpener permalink
    August 17, 2009 9:48 am

    The example I remember from physics class (these many decades ago) is slightly different but with a big wow factor. The setup is the same, except that the rod is now electrically neutral, it’s a metal conductor and it carries a current of electrons.

    Observer A sees a magnetic field created by the rod and the moving, charged object in this field experiences a magnetic force, which will accelerate the object towards the rod.

    Observer B also sees a magnetic field, but from B’s perspective, the object is not moving, and since its velocity is zero, the magnetic force on the object is zero, so the object experiences no acceleration. How to reconcile A and B?

    The answer is that B sees the bulk of the rod, including all the positive charges, moving past with velocity v, but the moving electrons have a drift velocity d, so they move past at velocity v+d.

    Now, despite the fact that electrical signals move through a wire very fast, the electron drift is really slow, perhaps millimetres per second in a typical situation. However, it is enough that the length contraction of the electrons is a bit more than that of the positive charges (protons) in the wire, so the electron density looks a bit higher than the proton density, and the wire looks like it has a tiny net negative charge. This causes B to see an electric force on the object, which is equal to the magnetic force that A sees, so there’s no contradiction.

    I was always very impressed that the millimetres/second electron drift, in a factor that includes the speed of light squared, is big enough to make this effect.

  9. October 21, 2009 7:32 pm

    “I have read that a moving electric field creates a magnetic field while a static one doesn’t”

    Um my question is this. How do you create a static (pertaining to or characterized by a fixed or stationary condition) electric field on a planet that is moving?

    It doesn’t seem like it would be a truely static field.

    • gravityandlevity permalink*
      October 22, 2009 9:10 am

      The question you’re asking is pretty much the heart of the problem. Whether an electric charge is truly “static” depends on your point of view. For someone standing on the earth, a charge fixed to the earth doesn’t appear to move, so it generates an electric field but no magnetic field. For someone standing in outer space and watching the earth fly by, the charge generates both electric and magnetic fields. The point of this experiment was to say that we learn something about the speed of light by demanding that those two observers see consistent realities.

      Somehow, electric and magnetic fields must be two sides of the same thing. Looking at it from a particular reference frame, you might see an electric field, while from another you would see a magnetic field. In reality, there is only one “electromagnetic field”; it just appears different depending on your speed as you move through it.

  10. IgnoQuest permalink
    November 6, 2009 7:40 pm

    I’m sure this has been asked before, but I’ve never seen the answer and I’m curious.
    Thanks in advance!

    If a particle in the LHC can go 99.9999991% the speed of light and Earth is going around the Sun at ~30km/s and our solar system travels around the Milky Way at 250km/s and the Milky Way is traveling at 600km/s; if these speeds are added to the particle, why won’t it exceed the speed of light (299,792.458 km/s).

    • gravityandlevity permalink*
      November 7, 2009 9:39 am

      The answer is that there is a problem with simply adding all the velocities together, since length and time appear different for observers traveling at different velocities. In fact velocities add by a more complicated “relativistic rule”: http://en.wikipedia.org/wiki/Velocity-addition_formula

      You can think of it like this: Imagine a person is standing in a space shuttle that flies past the earth at 1/2 the speed of light. To an observer on Earth, two things will seem strange about the space shuttle and the things in it. 1) Everything inside the shuttle will seem shortened (or flattened) in the direction of motion. Kind of like this: http://www.iafe.uba.ar/e2e/phys230/Gamow/pic2.jpg . 2) Everything inside the shuttle will seem to be moving in slow motion.

      To the person standing inside the shuttle, though, everything seems normal. So imagine what will happen if the person on the shuttle throws a ball from the back of the space ship toward the front, at a speed of 1/2 the speed of light. To the person in the shuttle, the ball’s motion seems normal. But to the observer in earth it moves slower, and it covers a shorter distance. As a result, the ball doesn’t seem to move as fast as 1/2 + 1/2 = 1 * the speed of light, but somewhat slower. In this case, the apparent speed of the ball is (0.5 + 0.5)/(1 + 0.5*0.5) = 0.8 * the speed of light.

      The more general result is that nothing can move as fast as the speed of light relative to anyone else, even though a direct sum of their velocities makes it seem like they should.

  11. Ellison permalink
    September 5, 2010 2:34 pm

    This applies to charged particles, but how does this related to uncharged neutral masses? Every atom in ground state has a balanced number of protons and electrons, so shouldn’t the electric and magnetic forces all cancel each other out?

    • gravityandlevity permalink*
      September 7, 2010 12:26 pm

      This is just a simple thought experiment; it wasn’t meant to derive the constancy of the speed of light for all objects.

      In the case of a neutral atom, though, you can imagine that the individual electrons and protons still feel all the forces that I outlined above, so the same conclusion holds for them.
      I haven’t proved it, but it isn’t hard to believe that if c is a maximum speed for all of the individual charges that make up a neutral atom, then it must be true for the whole atom as well.

      • Ellison permalink
        September 7, 2010 4:58 pm

        Well, I’ve been doing some reading on the issue of moving faster than the speed of light. As you mentioned, they’ve postulated this since the late 1800’s. The earliest reference I know of was the Searle paper, “On the Steady Motion of an Electrified Ellipsoid”. Here’s the link for this article: “http://en.wikisource.org/wiki/On_the_Steady_Motion_of_an_Electrified_Ellipsoid”. This was about a charged elliptical shell. His conclusion was that it was not possible to move at the speed of light, due to the EM fields induced and that it would take infinite energy.

        However keep in mind that at this time they did not know what the construction of the atom was. Searle assumed a shell, of charges, which was rigid, and he did not know about neutrons or protons and had a vague idea of electrons. So the assumption was that the charge of the body cannot change nor can the electrons and protons rearrange themselves.

        Although your assumption that electrons and protons both feel the same forces is true. However, due to symmetry, they will react with opposite forces. Because of this, it may be possible that a neutral atom will not feel any net effect of EM interactions. I would further postulate that perhaps in a system of ground state atoms, they may reconfigure to maintain a neutral charge. The electron cloud is not rigid or uniform as the Searle paper assumes.

        In fact when particle physicists use particle accelerators to accelerate charged particles using magnetic fields, they will indeed never be able to get those particles to reach the speed of light. It’s like that old saying goes, they are pissing against the wind. However, more experiments should be done on systems of neutral atoms instead. Someone should re-visit the question of whether we can indeed travel beyond the speed of light. Remember how everyone thought we couldn’t exceed the sound barrier? I’ll need to do more reading!

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