A new relativity paradox

 Recently, there was an article in PRL called "Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation".  I admit I didn't read the article, because I felt that the summary in Science was sufficient.

Figure from Science, credit P. Huey

The basic paradox is as follows.  You have a charge and a magnetic dipole, motionless with respect to each other.  In the reference frame where they are motionless, they exert no forces on each other.  In the reference frame where they are both moving, there will be an electric dipole in the magnet.  If you think of the magnetic dipole as a loop of current, the electric dipole can be understood as the result of Lorentz contraction (as discussed in an old post).

The positive end of the electric dipole will be repelled by the charge, while the negative end will be attracted.  Therefore, the charge applies a torque on the magnet, causing it to spin.  (The moving charge also applies a magnetic force, but this does not counteract the torque.)  But how can the magnet spin in one frame, but not spin in another?  Therein lies the paradox.

The author of the paper concludes that the Lorentz force law needs to be revised in the presence of polarized or magnetic materials.  However, I do not think this is the correct resolution for the following two reasons:

1. It is easy to construct the same paradox without any polarized or magnetic material.  Replace the magnet with a loop of current, which functions as a magnet.
2. It is easy (but tedious) to prove that the Lorentz force law is the same in all reference frames.  Showing that it behaves differently in two frames is akin to proving 1 = 2; the correct conclusion is that you made a mistake somewhere.

But I must admit that I could not figure out where the mistake was.  It's a good paradox!  Instead, I solved the problem using the research method.  I read two responses by Daniel Vanzella and Kirk McDonald (both of which said the same thing).  The resolution to the paradox is tricky, and I will attempt to summarize it for a lay audience.

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We need to talk about the momentum of the electromagnetic field itself.  It's well-known that light, a fluctuation in the electromagnetic field, can carry momentum.  But even constant fields can have momentum.  The momentum is proportional to the cross product of the electric and magnetic fields.  When you have an electric charge and magnetic dipole near each other, the momentum of the electromagnetic field is not zero.  But the total momentum of the magnetic dipole must be zero, since it is motionless.  Therefore, there must be some hidden mechanical momentum in the opposite direction.

In this figure, the magnetic dipole is represented by a loop of current.  The magnetic field produced by the current goes through the loop out of the screen.  The electromagnetic momentum is downwards.  The hidden mechanical momentum is upwards.

I claim that there's hidden mechanical momentum, but where does that come from?  I believe that it depends on the precise nature of the magnetic dipole, but we can at least consider a simple example.  In the above figure, we imagine that the magnetic dipole is actually a square loop of counter-clockwise current (with no dissipation).  Charged particles in section A are being slowed down by repulsion from the charge on the left.  Similarly, charged particles in section C are being sped up.  Therefore, the particles in section D are moving faster than the particles in section B.  And recall that in relativity, faster particles are heavier.  Therefore, there is more momentum in D than in B, and the total momentum is upwards

And that's how a loop that is motionless can still have momentum.

It turns out that even when we use a moving reference frame, there is some mechanical momentum upwards.  Torque, which normally causes an object to spin, is really the change in angular momentum around a single pivot point.  The angular momentum is proportional to the cross product of the momentum and the distance from the pivot point.  In the moving reference frame, there is torque on the loop, but this does not cause the momentum to change, or cause the object to spin.  It causes the distance from the pivot point to change (because the loop is moving, while the pivot point is motionless).  Thus the paradox is resolved.

Upon reading this resolution, I was puzzled, because there must be an equal and opposite torque applied somewhere else in the system.  I figured out that an opposite torque is applied to the momentum of the electromagnetic field.

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The responses from Vanzella and McDonald appeared before the paper was even officially published.  That might be kind of embarrassing for PRL.  Or maybe not?  Presumably, PRL felt that even if the paper was shot down, it would still generate productive discussion.

Note that there is precedence for revising old textbook equations.  I'm thinking of the Aharanov-Bohm effect (which shows that magnetic field has an effect on particles which go around the field even if they don't go through the field) and Berry's curvature (which modifies the laws of motion for particles within a crystal structure).  In both cases, the fundamental equations were known, but physicists overlooked a few of their strange consequences.

Electricity, magnetism, space, and time

[One of my most popular essays is "Relativity + Electrostatics =  Magnetism", which is about how the electric force, combined with Special Relativity, explain the existence of the magnetic force.  I think this is a topic worth revisiting, because I am a better writer than I used to be, and can make better pictures.]

The magnetic force is weird.  You're all familiar with bar magnets, which have a north and south pole.  Opposite poles attract and like poles repel.  But even though bar magnets are very familiar to us, they're relatively complicated, so we'll start with a more basic example.

A simple model of two wires.  In each wire, the electrons (in blue) are moving to the left, while the atomic nuclei (in red) remain stationary.

Two wires.  Wires conduct electric current, meaning that the negatively charged electrons move to the left or the right.*  The positively charged atomic nuclei remain stationary.  If the two wires have current going in the same direction, then they attract.  If they go in opposite directions, then they repel.

*I am making a simplification, since this is not true for all materials that the wire could be made of.  For reasons that I won't get into, some materials conduct electricity by things other than electrons.

Bar magnets are sort of like circular currents.  If you have a north and south pole together, these currents are going in the same direction, and they attract each other.  If you have a north and a north pole together, these currents are going in opposite directions, and they repel.

On the left, the circular currents (dark red arrows) are going in the same direction, making the magnets attract.  On the right, the currents are going in opposite directions, making the magnets repel.  Alternatively, we can think of the magnets as having north and south poles, where opposite poles attract and like poles repel.

But let's return to the wires, because they're simpler.  The reason the wires attract is specifically because the electrons attract.  One way we know this is the electrons tend to gather towards each other at the closer edges of the wire (this is called the Hall Effect).  The electrons drag the rest of the wire with them, thus causing the wires to attract.

The reason why the electrons attract and the atomic nuclei don't is because that's just how the magnetic force works.  The magnetic force is proportional to the speed of the particle.  The electrons are moving, so they attract each other.  The atomic nuclei are not moving, so they don't attract each other.

But hold on!  Doesn't the speed of a particle depend on which way you look at it?  What if you put these wires on a moving truck, won't all the electrons and nuclei be moving faster than before?  For that matter, what if you put these wires on a moving Earth, doesn't that affect the speed?  Why is it that we can use motors and generators (which both require magnetic fields) without worrying about the earth's motion?

This question was the motivation for Special Relativity Theory.  You may have heard about Einstein's Theory of Relativity as something that radically alters our notions of space and time.  But that wasn't really the point of the theory.  The point was to explain something about magnetic fields.  The radical view of space and time was just an added bonus.

Einstein's Special Relativity is sort of like an expansion of rotations.  If you rotate your head 90 degrees, and look around you, several things change.  Ceilings become walls and walls become floors.  Right-left becomes up-down, and up-down becomes left-right.  Everything that was pointing in one direction (like gravity, which originally pointed down) is now pointing in a different direction.  But physics behaves the same way, just rotated.

The same is true if you're moving at constant velocity.  If I'm on a moving truck, things that were previously stationary are now zipping behind us.  If the air was previously still, it now becomes wind at our faces.  But physics behaves the same way, it's just that directions have changed.  In this case, we don't call it a rotation, we call it a boost.

According to Special Relativity, a boost does not just affect the motion of objects.  It also affects the electric and magnetic fields.  Just as right-left became up-down, and up-down became left-right when we rotated, electric fields can become magnetic fields and magnetic fields can become electric fields.  It's not quite like rotation (the math is harder), but the idea is the same.  A boost causes electric and magnetic fields to mix into each other.

And of course, it's not just electric and magnetic fields that mix into each other, it's space and time too.  But at everyday speeds this is difficult to observe, because they're such small boosts.  It's like rotating your head a fraction of a degree.  What was once horizontal is still mostly horizontal (but with a slight vertical component).  Similarly, at everyday speeds, what was once a distance is still mostly a distance (but with a small time component).

The mixing of electric and magnetic fields is deeply intertwined with the mixing of space and time.  But this is very difficult to demonstrate without getting into the mathematics of boosting.  There is one example where it is easy to demonstrate, and that is the example I started with.  Two wires.

In a real wire, the electrons are not really moving at the same speed.  But let's simplify and imagine that they are.  And then let's boost to a perspective where the electrons are not moving at all.

Two wires, boosted.

After we've boosted, it is no longer the case that the electrons are moving left while the atomic nuclei are stationary.  Now it is the electrons that are stationary while the atomic nuclei are moving right.

Since the magnetic force is proportional to speed, it can no longer affect the motionless electrons.  But this is easy to explain in terms of Relativity: what was once magnetic fields have now changed into electric fields.  The electric force causes the electrons in each wire to attract.  The same electric force also cause the atomic nuclei to repel, but that's okay!  The atomic nuclei have some speed, and are thus attracted by the magnetic force.  The magnetic force and electric force on the nuclei cancel out.

So we really have the same picture as before.  The electrons attract (but now due to the electric force, not the magnetic force), and they drag the entire wire along with them.

Now I will connect this to time and space.

If you look at my diagram, you'll notice I did something sneaky.  After the boost, there are more nuclei than there are electrons.  This is due to the mixing of time and space.  Previously, the protons had some spacing, some distance between them.  But as time and space mix, some of the distance changes into a time component, and the distance becomes shorter.  It's called Lorentz contraction.  When we boost from a frame where nuclei are stationary to a frame where they are moving, they become closer together.

Similarly, when we boost from a frame where electrons are moving to a frame where they are stationary, they become further apart.  It's backwards Lorentz contraction.

The end result is that each wire now has a positive charge because the nuclei are more dense than the electrons.  Electrons are attracted to positive charge by the electric force!

And so, there are two methods of solving the problem.  The first method is to mix the electric and magnetic fields through the mathematics of boosting.  The second method is to mix space and time, and then look at the resulting electric and magnetic forces.  Both of these methods are equivalent.

Hold on!  I know you have some questions.

Does that mean that the magnetic force isn't real?  No.  I'm not sure that it's meaningful to talk about whether these things are "real" or not.  What it does tell you is that the electric force, magnetic force, and Special Relativity are all connected.  Physics wouldn't make sense if you only had two out of the three.

What happened to the missing electrons?  The number of electrons does not change when we boost, so they must have gone somewhere.  The reason it seems they disappeared is because I only drew part of the wires.  For a wire to conduct electricity, you need to have the wire in a loop to complete the circuit.  The electrons will gather in the parts of the loop where they were going the opposite direction. There, they will be doubly Lorentz contracted.

What about two electrons by themselves moving alongside each other?  In the wire example, electric and magnetic fields don't change over time.  In the two electron example, they do change over time.  This significantly complicates the way things work.  But suffice it to say that the faster the electrons move, the stronger the electric force repelling them.  The magnetic force between the electrons is counteracted by this stronger electric force. 

Aren't the electrons all moving at different speeds?  Yes.  But it works out the same.  It makes the math a lot more complicated.  The entire premise of this post is to simplify to an example where I don't have to show that.  If you would like to see it, you should be studying physics academically, not from blogs.

Is Lorentz contraction really that big?  No.  I exaggerated it so you can see it clearly.  In a typical wire, the electrons are moving with an average velocity of 10^-5 meters per second, which corresponds to a Lorentz contraction of 1 part in 10²⁷.  If it seems impressive that such a small Lorentz contraction can create a measurable force, that's because it is.

Near the speed of light

What happens when we travel near the speed of light?

Let's say you're traveling at 99.5% the speed of light. That's 299792458 meters per second, or 185351 miles per second. You're heading straight towards Proxima Centauri, the nearest star besides the Sun. Proxima Centauri is about 4 light years away, meaning that it takes a beam of light four years to get from here to there. 4 light years is about 2.3 x 10¹³ miles.

Large numbers, eh?

It might seem as if you would get to Proxima Centauri in about 4 years. After all, you're practically like a beam of light yourself now. But things are not so simple when we get close to the speed of light. To continue with our calculations, we will need something known as the Lorentz factor. The Lorentz factor is just some number, symbolized with (the letter gamma).

= 1/sqrt( 1 - (v/c)²)

v is the velocity, and c is the speed of light. In our example, is approximately equal to 10.

One of the things that happens when you travel near the speed of light, is that all distances in the direction of travel get shortened by a factor of . The Earth will seem like a squashed spheroid to you. It will also seem as if Proxima Centauri is not 4 lightyears away, but 0.4 lightyears away. So you will get there in 0.4 years without ever breaking the speed limit set by Relativity!

I would be on Earth, watching you. I would see you arrive at Proxima Centauri 8 years later. From my perspective, it took you 4 years to get there, and 4 more years for your signal to return to me. Let's say that you decided to return home, and sent a message to me saying so. It would take the message 4 years to get to me, and you will appear about one week later, since you travel at nearly the same speed as your own message.

By the time you returned, about 8 years would have passed on Earth. From your perspective, the entire trip only took 0.8 years. From your perspective, Earth has skipped ahead in time by about 7.2 years. From my perspective, it seems like you hardly aged over that long trip.

Let's say that you're about 100 kg (220 lb). Once you start moving near the speed of light, your mass increases by a factor of , so you will appear to be 1000 kg. Since your mass increased by 900 kg, guess how much energy it required to mobilize you? That's right, E=mc². In your case, this will be 8.1 x 10¹⁹ Joules, or about twenty thousand megatons. And that's just you by yourself. If you're in some kind a space ship, the space ship might carry about a million more megatons of energy. You would probably make quite an explosion if you hit one of Proxima Centauri's planets, if it has any. Not that we were planning to do anything like that.

You and your spaceship will appear to be flattened by a factor of . You will seem as flat as a pancake. You will probably look quite absurd, a massive, flying, pancaked person. What's more, your colors would be all shifted. When you're flying away, you will be redshifted by about a factor of 20. Basically, I will only see what little ultraviolet light you emit. But when you're flying towards me, you will be blueshifted by a factor of 20. Much of that infrared light which you normally radiate will now be blueshifted into visible light and x-rays. I'm not sure what color you would appear, but I estimate about the color of a star of 6,000 degrees, like the sun--a yellowish white.

You, while on your journey, will see something even more exciting. Everything behind you will be redshifted by a factor of 20. You would see the extreme ultraviolet range. In the ultraviolet range, most stars will seem much dimmer, except for the hotter ones, including white dwarfs. If you look in front of you, light is blueshifted by a factor of 20, so you will see part of the middle infrared spectrum. You can see giant dust clouds and nebula in this range. Some of these dust clouds are where stars will form. And as you swing your head around, looking from front to back, you will see all the ranges between middle infrared and extreme ultraviolet. I'm sure it would be quite an experience.

Great spiraling black holes!

Around this time is when you start to hear about everyone else's exciting plans for the summer. Hey, wait, I have one of those too! I got a research job at Caltech working with LIGO, the Laser Interferometer Gravitational Wave Observatory. It's probably not as glamorous as it sounds, but boy does it sound awesome.

Let's begin with the observatories. There is one observatory in Louisiana, and two in Washington state. The gravitational wave detector consists of two lasers which go in perpendicular directions. Each laser is 4 km (2.5 miles) long, and encased in a vacuum pipe. Once the lasers have bounced back and forth in their tubes many times, they recombine and interfere with each other. By looking at the interference pattern of the lasers, we can determine the difference in length of the two laser paths. And by that, I mean we can measure the difference very sensitively, down to 10^-18 meters. This is about a thousand times smaller than an a proton.

One of the Washington detectors. Credit: NASA

Why do we want to measure so sensitively the length of a laser path? It all goes back to Einstein.

Albert Einstein is most famous for his theories of Special Relativity and General Relativity. Special Relativity describes how physics behaves when things move near the speed of light. General Relativity is the theory which incorporates both Special Relativity and gravity. In fact, General Relativity is the theory which replaces the classical theory of gravity. The classical laws are very accurate under most conditions, but are decidedly incorrect nearby very massive objects and when things are moving near the speed of light.

In a way, it's rather surprising that General Relativity and classical gravity could possibly be describing the same thing. Classical gravity describes everything in terms of forces. General Relativity describes gravity as a distortion of the space-time topology. In other words, gravity influences the distances and time-intervals between different events. These small distortions cause a straight line through time appear to be curved, as if it were acted upon by some force.

One of the predictions of General Relativity is the existence of gravitational waves. Gravitational waves are analogous to electromagnetic waves (aka light). Electromagnetic waves are fluctuations in the electric and magnetic fields. Gravitational waves are fluctuations in the space-time topology. Electromagnetic waves are created whenever an electrically charged object accelerates. Gravitational waves are created whenever a massive object accelerates. Both kinds of waves are characterized by a frequency, which tells you how quickly the waves fluctuate. If a gravitational wave passes through the LIGO detector, it will cause the two laser arms to fluctuate in length. If the gravitational wave has a frequency of 40 Hz, then the lengths will fluctuate 40 times per second.

LIGO is only sensitive enough to detect gravitational waves with frequency 40 Hz or higher. At lower frequencies, it becomes too difficult to distinguish between gravitational waves and regular old earthquake activity.

What could possibly cause a gravitational wave of more than 40 Hz? Gravitational waves are caused by accelerating massive objects. For example, the earth is constantly accelerating towards the sun because it is in a circular orbit. But this should only cause gravitational waves with frequencies of about 1 per year. However, we might be able to detect orbiting objects if they are orbiting much faster than the earth. One of the objects we are interested in is the binary black hole* system. Black holes are very massive objects, and also very small. So two black holes could be orbiting very quickly and closely to each other. If a pair of black holes is what it takes, then let's look for black holes!

*It could also be any other type of massive astrophysical compact halo object (MACHO), like a neutron star.

One other thing about gravitational waves, is that they carry energy, just like light does. As two black holes orbit each other, they emit energy in the form of gravitational waves. This causes the black holes to slowly lose energy, falling slowly towards each other. Because they're closer together, the "force" of gravity is stronger, and they orbit faster and faster. The picture we have here is of two black holes, spiraling around each other, getting closer together and moving faster. Eventually, they collide, coalescing into a single black hole. When there is only one black hole left, it no longer emits gravitational waves, and its signal disappears.

This could really use some animation. So I found some animations on the net from the Numerical Relativity Group.

The detection of gravitational waves is not only a way to test Einstein's theory of General Relativity under new conditions, it is also a new way to do astronomy. It's much like how we build telescopes to detect electromagnetic waves from far away sources. We can use gravitational waves to detect objects like binary black holes, as well as exploding stars, and a certain kind of pulsar. Scientists are also trying to detect something analogous to the cosmic microwave background radiation, only it would be cosmic gravitational background radiation. It would be very difficult to detect, but it comes from a very early point in the universe's history, far earlier than even the microwave background radiation.

Experiments with relativistic asteroids

I've been playing around with this game: Relativistic Asteroids. It's the classic game of Asteroids, but it uses relativistic physics!* Light speed is set extra low so that you can see the effects of relativity. It is so awesome. This is the sort of game I dream of making. Thought experiments come to life!

Here's some relativistic weirdness that you can observe and experiment with in this game:

1. Lorentz contraction: If you are in the unmoving reference frame, your ship will appear to shorten when it moves near light speed. If you are in the ship's reference frame (press 'f' to switch between your own reference frame and the unmoving reference frame), the asteroids will shorten when near light speed.
   
2. Time dilation: In sandbox mode, you can create flashing squares by clicking and dragging on the screen. The faster the squares are moving, the more slowly they flash.
   
3. Tilting the plane of simultaneity: This one is a bit harder to observe. You must be in sandbox mode, and place two stationary flashing squares. If you time it right, you can make the squares flash in sync with each other. But if you change your reference frame, they go out of sync.
   
4. Make time go backwards: This is the effect I discussed in a previous post. While you are in an accelerating reference frame, clocks far behind you can tick backwards. The difficulty in observing this is that you have to build a clock. You may either use the flashing of a square as a clock, or you can set flashing squares to collide and then watch them reverse-collide.

But the game is not quite perfect. Here are a few "bugs" that I encountered, that are due to the programming, not due to relativity.

1. In sandbox mode, set light speed to 2 cm/s (use keys 'e' and 'd'). Make sure you are in the ship's reference frame. Place two squares, one directly above the other. Carefully navigate around the two squares, making sure to keep them away from the edges. If you go around the asteroids, you'll find that the one that used to be above will slowly move around to the side. I'm fairly sure this is impossible under Special Relativity. I suspect the programmer has incorrectly implemented the second dimension somehow.
   [ETA: I may have been mistaken about this point.  After all, the commutator of two boosts is a rotation *mutter mutter*.]
   
2. The paths of the bullets (which move at light speed) should be affected by an accelerating reference frame in the same way light is affected by gravity.
   
3. One of the major obstacles in making Asteroids truly relativistic is that the playing field has a donut topology. That is, if you or an asteroid go off the edge of the screen on one side, you'll appear on the other. The problem here is that Lorentz contraction shouldn't simply affect the asteroids, but should also affect the entire screen. If you're going really fast, you should find that the distance required to wrap around the screen is much shorter than it was before. The percentage of total screen space taken up by asteroids should remain constant.
   
   But I understand the programmer ignoring this fact, because combining Relativity with a donut topology opens the door to much more weirdness than anyone's bargained for. All relativistic hell breaks loose. A donut topology implies a "special" reference frame in which the screen is largest. It makes for a more complicated twin paradox. You might see multiple copies of the same asteroid in different locations. Finally, you would be simultaneous with your future and past self.

Nitpicking aside, it's still so awesome.

Next, I want to see relativistic billiards! And then I want to see a game based on Maxwell's equations, where you guide a charged particle through an obstacle course by inducing electromagnetic fields. And then we can add some plasma physics to the mix too. Oh, the possibilities!

*For my introductory explanations of basic relativity concepts, refer to my earlier posts, Parts 0, 1, 2, and 3. (I think my writing has much improved since I've written those posts...)

The metaphysics of Relativity?

One of the things people like to ask about physics is, "What is the meaning of it all?" What does physics tell us about metaphysics--the ultimate reality behind all that is? Does Special Relativity support Moral Relativism? Does Quantum Mechanics provide a mechanism for consciousness or miracles? Does the Big Bang Theory prove God? No, no, and no.

In general, I feel the answer is that, no, physics does not tell us much about metaphysics. Furthermore, people who try to connect physics and metaphysics are often engaged in gross abuse of the science. Much more often, physics only helps us realize how much we don't know about metaphysics.

I've already tried to explain Special Relativity (Explained: 1, 2, 3), so I'm going to use it as an example to demonstrate my point.

First off, Relativity is not even related to Moral Relativism, or indeed any sort of subjectivist philosophy. They have nothing to do with each other. Yes, Relativity suggests that different observers can have different perceptions of time and space. But "observer" is a technical term that ultimately has no connection to humans, much less human ethical theory. Similarly, the details of time and space have little to do with ethics. If I say that time and space depend on your frame of reference, "frame of reference" is a technical term that depends on your velocity, not your point of view. People seem to think that Relativity tells us, "It all depends on your point of view," but the more accurate statement is, "It all depends on your direction and magnitude of motion." I guess that just doesn't have the same ring to it.

xkcd: "Moral Relativity"

Another problem with the idea is that Special Relativity doesn't say everything is relative. Previous to Special Relativity, we used what we retrospectively call Galilean Relativity. In Galilean Relativity, the distance between two objects, and the time between two events would always remain the same, no matter your frame of reference. Time (Δt) and distance (Δx) are what we call "invariants". In Special Relativity, there is an invariant that combines space and time: what we call the "interval".* That means that though space and time are relative, space-time, taken as a whole, is not. Changing reference frames is analogous to rotating a keyboard (only you're actually rotating space-time). You may have changed the positions of the keys, but the shape of the keyboard stays the same. You may have changed space and time, but some things about space-time never change. For example, the speed of light always remains the same.

What Relativity does tell us is that our intuition of time and space can be incorrect. Our common experiences bias us towards thinking that time is absolute and independent of space, but that's not necessarily the case. Therefore, if you want to philosophize, you should be careful about your assumptions about time and space. If you want to talk about the cause of the universe, you should be careful with ideas like "before" and "after", since these concepts may or may not exist independently of the universe. Furthermore, in these matters, I think it is irrelevant whether Relativity is in fact true. Relativity is just a heads-up to philosophers to be careful with hidden assumptions. Ideally, philosophers would realize this without science's help, but whatever it takes.

*For the record, the equation for the interval is (ΔS)² = (Δx)² - (cΔt)²

Relativity + Electrostatics = Magnetism

[Update: Given the popularity of this post, I made a clearer rewrite.]

There is another dramatic success of Special Relativity that is virtually unknown among laymen. Special Relativity is what causes magnetism.

First, what is the electrostatic force, and what is the magnetic force?

The electrostatic force is what causes opposite charges to attract, and like charges to repel. Electrons, negatively charged, tend to stick to protons, positively charged. Two protons would repel each other, as would two electrons. On a macroscopic scale, the electrostatic force is what causes static electricity, in which an object accumulates an excess or shortage of electrons. It also causes lightning, which is basically static electricity on an even larger scale.

The magnetic force acts upon moving charges. If I've got an electric current, where the electrons are moving forward in a wire, the current creates a magnetic field. If I place two of these wires next to each other with the currents going in the same direction, they will attract. If the currents go in opposite directions, the wires repel. What we call magnets are materials with permanent circular currents on an atomic scale. The north pole of a magnet has currents going counter-clockwise, and the south pole has currents going clockwise. The north and south poles attract because when they are placed together, the currents go in the same direction.

The magnetic and electric forces interact and affect each other, but it is not clear why. Why should currents in the same direction attract? The wires, after all, have no net charge. There are just as many electrons as protons in each wire. So it can't be that the electric force is somehow sneaking in, disguised, right?

There is, in fact, a paradox associated with magnetism. Magnetic forces only act upon moving charges. But if we consider a moving particle's reference frame, the particle always has zero speed relative to itself. Therefore, from the particle's reference frame, it cannot be affected by magnetic forces. These forces shouldn't be disappearing just because our reference frame is different!

Let's consider a specific case: two wires with current going in the same direction. Wires, along with most everyday objects, consist of equal numbers of protons and electrons. If a wire has electric current going through it, that means that the protons are remaining still while the electrons are moving in one direction along the wire. The electrons, in fact, are moving at a large range of speeds, but for simplicity's sake I will assume that they are all moving at one constant speed.

Let's consider the wires with Relativity in mind. Of course, from the protons' motionless frame of reference, the wire is electrically neutral. But what happens if we consider the frame of reference of a moving electron? From the electron's point of view, the other wire contains a bunch of motionless electrons and a bunch of backwards-moving protons. Since the electrons and protons are moving relative to each other, we must take into account Lorentz contraction. If you don't recall, Lorentz contraction makes all distances in the direction of motion smaller. Lorentz contraction causes the protons to be closer together, more densely packed. As a result, the other wire has an overall positive charge, creating an electrostatic force. The electron will be attracted by this force.

So from my point of view, standing still, the wires attract because of magnetic forces. From the electron's point of view, they attract because electric forces. Both of us are correct, much in the same way that we would both be correct in thinking the other's clock ticks slower than our own. The resolution to the paradox is that electrostatic and magnetic forces transform into each other as we change reference frames. It turns out that magnetism is necessary for Special Relativity and electrostatics to make any sense together.

What's interesting about this is that it occurs at extremely low velocities. I did a bit of math, and I found that if we have 10 amps (a quantity of current) going through a copper wire of diameter 1mm, then the average velocity of electrons is 9.4*10^-5 [edit: corrected math] 3.3*10^-5 meters per second. That doesn't even begin to approach the speed of light (3*10⁸ meters per second). And yet, if you place two of these wires next to each other, there will actually be a measurable magnetic force. Not a significant force (about the weight of a paperclip per foot of wire), but not negligible either. We rely on this force for electric motors, generators, and countless other applications.

Usually, when you learn Special Relativity, teachers are quick to say that it is entirely ignorable at everyday speeds. But it turns out that even at microscopic speeds, Relativity does no less than power the modern age.

Special Relativity Part 3: Mass-energy

Part 0: More historical background
Part 1: The problem
Part 2: Space-Time

One of the results of SR is that velocities do not add. If one rocket is going to my right at 3/4 the speed of light, and another is going to my left at 3/4 the speed of light, these two speeds to not sum to 3/2 the speed of light. Instead, each rocket observes the other to be going at 24/25 the speed of light. The equation for the relativistic "sum" of two speeds is as follows:

(v₁ + v_2)/(1 + v₁v₂/c²)
...where v₁ and v₂ are the two speeds, and c is the speed of light

The derivation of this equation unfortunately relies on math that I skipped, so you'll have to take my word for it. It basically arises from the equations that fully describe lorentz contraction and time dilation. One thing to note is that if either v₁ or v₂ is equal to c, then the relativistic sum is equal to c. That means that no matter how fast an observer is moving, light will still travel at speed c. Einstein originally set out to explain why light is always observed to move at speed c, and now we've done it.

Also note that if v₁ and v₂ are both less than c, their relativistic sum will be less than c. So no matter how much you try, you can never push an object above speed c. In fact, what happens, is that the closer its speed is to c, the harder and harder it becomes to increase its speed. That means that Newton's second law (Force equals mass times acceleration) is wrong. At high speeds, it takes greater force for a smaller amount of acceleration. In some sense, the mass of the object increases when it speeds up. Specifically, the "relativistic mass" of the object increases. Usually when physicists talk about mass, by default they refer to "rest mass" (the mass of an object when it isn't moving) rather than the relativistic mass. The rest mass stays constant during acceleration.

Whenever you apply force to a moving object, you are transferring energy. You are also changing the object's relativistic mass. With a bit of calculus, we can conclude that energy is relativistic mass multiplied by a factor of c². That's where we get the equation E=mc².

Technically, the famous equation is incorrect. The correct form is E=ɣmc², since these days, relativistic mass is written as ɣm. The letter m represents the rest mass of the object, though in Einstein's time, it was used to represent the relativistic mass, thus the source of this confusion. The number ɣ (represented by the letter gamma) is called the Lorentz factor, and it turns up in the equations for Lorentz contraction and time dilation. For the interested reader, ɣ=1/sqrt(1-v²/c²).

For a non-moving object, ɣ=1, but when the object's speed approaches light, ɣ approaches infinity. That means that E=mc² is only true for an object that is not moving. In order to push an object to the speed of light, it would require an infinite amount of energy, and the object would gain an infinite amount of mass. Along with time-travel, this is why the speed of light is the ultimate speed limit. (However, this does not disprove the existence of tachyons, which, in theory, already start out faster than light, so that there is no need to push them through the light-speed barrier.)

I don't think most people realize the implications of this equation, so I want to expound upon them. Most people have heard that this is how atom bombs work. When an atom bomb explodes, the nuclei of Uranium atoms split apart. The way they split is such that the total mass decreases. Because c² is such a large number, a small difference in mass results in a huge amount of energy. The same basic idea is behind hydrogen bombs, and nuclear power plants.

What most people don't realize is that mass-energy equivalence is true of all energy. If you use energy to heat up an object, it becomes a tiny bit heavier. This should not be surprising, since the temperature of an object tells you how fast the individual particles are moving, and I've already said that objects moving at higher speeds become more massive. But more surprising is that if you compress a spring, giving it potential energy, it becomes heavier. If you snap your fingers, you become slightly lighter, since a small amount of energy is released through the snapping sound.

Mind you, none of these mass differences come in large enough quantities that we would ever notice them, and they would be completely swamped by other effects. But mass-energy equivalence also gives protons and neutrons the vast majority of their mass. Without it, atoms would be too light, the force of gravity would be less effective, and we probably wouldn't exist. It is by the grace of Special Relativity (along with a ton of other cool physics) that we're all here.

This concludes my series on Special Relativity, though I may still talk about it, and go off on other tangents. If you've got any questions, ask, and I could write more to explain the answers.

Special Relativity tangent: Time travel

Part 0: More historical background
Part 1: The problem
Part 2: Space-Time

Ok, I'm going to go off on another tangent, but I promise it will be interesting.

One thing that SR necessarily implies is that if you move faster than light, you can travel backwards in time. In the diagram below (see part 2 for an explanation of space-time diagrams), we are considering the yellow world line, which represents a particle moving at 3 times the speed of light.

Note that the yellow particle is moving from the left side of the diagram to the right side. You might notice that I left one of the other world lines (dark blue) in there, along with its plane of simultaneity (light blue). This is to show that the yellow particle moves from after the plane of simultaneity to before the plane. Therefore, from the reference frame of the dark blue person, the yellow particle is moving backwards in time. This is true of anything that moves faster than light; there exists a reference frame in which that object is moving backwards in time.

In advanced physics, sometimes you can come up with theories in which there are particles that always move faster than light. These particles are called tachyons. Despite being intriguing little things, tachyons are usually a sign that the theory needs improvement.

The problem with tachyons is that time travel creates numerous paradoxes. What if you go back in time and kill your grandfather? This is called the grandfather paradox. This can actually be a rather compelling argument against tachyons (unless they never interact). However, there are ways around this, as we can see in countless sci-fi stories. Perhaps history is quite resilient, and it is difficult to change anything so as to create paradoxes. Perhaps at the moment of a paradox, the entire universe disappears. Perhaps the entire universe splits into separate possibilities. Perhaps the rules of the universe just don't make any sense.

Although those are all great premises for a work of fiction, they're kind of far fetched as real physical theories if you ask me. But I think there's another way that Special Relativity allows for a sort of time travel. Under special relativity, you can make time go backwards just by accelerating! This is just a natural consequence of the tilting of the plane of simultaneity. Whenever you accelerate, your plane of simultaneity tilts in the direction of your motion. At a certain distance behind you, the plane of simultaneity is tilting faster than time is moving forward. Therefore, whenever you accelerate, any clocks more than a certain distance behind you move backwards.

I did the math, and this distance is equal to c²/a, where a is your acceleration, and c is the speed of light. That means that if you're in a car, and you accelerate from 0 to 60 mi/hr in 12 seconds facing directly away from alpha centauri (the nearest star aside from the sun), then all clocks on alpha centauri will start moving backwards (if I've done my math right). No paradoxes attached, since once you accelerate towards alpha centauri, the clocks will more than make up for their lost time.

That said, it will be impossible to actually observe the clocks moving backwards because of the manner in which the light would reach us.

Special Relativity Part 2: Space-Time

This post will focus on the solution to the paradox in Part 1: The problem. Also see Part 0: More Historical Background. I should probably also cite this website, where I got the paradox. In fact, that website probably explains the resolution better than I, but I that hasn't stopped me before.

Before resolving this paradox, I think it is important to introduce the space-time diagram. (All diagrams were created by me in excel.)

This is the diagram for the paradox I explained last time. The vertical axis represents time, and the horizontal axis represents space. The vertical dark-green line represents something that is staying still in space while moving through time--that represents me. This line is called my "world line" because it shows my path through space and time. The two red lines represent the world lines of two beams of light that I shoot forwards and backwards. Naturally, they move at the speed of light, which makes a 45 degree angle with the axes. The blue line represents you on your longboard, coasting forwards at 60% of the speed of light. At the initial point in time, you and I are in the same location. The light green line represents the "plane of simultaneity". All points on the plane of simultaneity happen at the same time.

Of course, for our universe, which has three dimensions of space and one of time, a space-time diagram would actually have 4-dimensions. But that's too hard to draw, so I'm sticking to this. Another important note is that though everything on a plane of simultaneity really happens at the same time, it takes some time for the light to travel to my eyes before I see what's happening. Here, we do not talk about what I see, only what is.

In order to find if I am at the midpoint between the two beams of light, I first look to see where their world lines (red lines) intersect my plane of simultaneity (light green line). Then, I measure the distance between me (the green square) and the two intersections. The two distances are the same, so I must be at the midpoint.

But what of the paradox? How can you also be at the midpoint between the two beams of light? You're clearly closer to the one going forward than the one going backward. But once I draw your plane of simultaneity, all will become clear.

The solution to all of this is that your plane of simultaneity is not parallel to mine. If you see two events happening at the same time, I would disagree and say they're happening at different times.

In order see that you are at the midpoint of the two beams of light, you first find the intersections between your plane of simultaneity (light blue) and the world lines of the beams of light (red). Then you compare the distance between yourself (light blue square) and the two intersections. We really can be at two different midpoints!

From your point of view, of course, you'd be the one standing still, while the rest of the universe and I travel backwards. So the diagram below is from your point of view.
If you can imagine a smooth transition between this diagram to the previous one, you're in excellent shape. If you can't, maybe this animation will help.

I drew these diagrams out so that the two squares, the green and blue ones, correspond to exactly one clock tick after the initial moment. Notice that from my point of view, my clock ticks before yours does. I must conclude that your clock is slower than mine. From your point of view, your clock ticks before mine does, so you must conclude that my clock is slower than yours. Yes, each clock is slower than the other, depending on your point of view. This effect is called "time dilation", and it becomes greater at greater speeds.

Notice that the light green and light blue lines change length depending on your point of view. This is because of an effect called "Lorentz contraction". From your point of view, all distances would appear shorter than usual. I would appear to be skinnier than usual. From my point of view, you would appear to be skinnier than usual. This effect becomes greater at greater speeds.

But before you start concocting relativistic schemes to become skinnier, I'll add that you also become heavier than usual. The reasons why are to be explained in Part 3.

I will take questions.

Special Relativity Part 0: More historical background

For a long time in physics, there had been a debate about the nature of light. Is light a particle or a wave? Nowadays, we know that it's both, but that's another story. In Einstein's time, light was primarily thought of as a wave due to the discovery of Maxwell's equations.

Maxwell's equations are a bit advanced to go over for my purposes, but all you need to know is that they completely describe the behavior of electric and magnetic fields. Electric fields affect magnetic fields, which affect electric fields, and so forth. To gloss over all the math, this feedback loop creates an electromagnetic wave that moves at the speed 2.998 x 10⁸ meters per second. This number matched the speed of light so well that Maxwell correctly guessed that light is made of electromagnetic waves.

At the same time, there was something very odd about Maxwell's equations. In classical mechanics, all velocities are relative. We cannot feel the Earth moving beneath us because we're moving at the same speed. We cannot feel the Earth's orbit because we orbit along with it. We cannot feel our solar system's orbit around our galaxy, or our galaxy's motion through the cosmos. If the entire universe was moving in one direction, we could never know, since we move along with it. Maxwell's equations predict a specific velocity for light, but a velocity relative to what?

At first, scientists postulated that light was moving with respect to something called the luminiferous aether (nice sci-fi name there), also just called ether. The ether is supposed to be a medium for light waves the same way that the ocean is a medium for ocean waves. The ether would operate on light similarly to how wind operates on sound. Light would move faster when moving in the same direction as the ether, and slower when moving in the opposite direction. Since the earth cannot possibly be staying still with respect to the ether (since for one thing, it rotates), scientists predicted that light would move faster in some directions than in others. The famous experiment to test this was called the Michelson-Morley experiment, which involved putting two light sources on a rotating table. The experiment failed spectacularly, showing that light moves at the exact same speed in all directions.

In science, a failed experiment only provides new opportunities. To Einstein, the Michelson-Morley experiment showed that we must discard the concept of ether, along with many other things from classical physics. Einstein found that the speed derived from Maxwell's equations is very special in that there is no need to specify a reference frame. The speed of light is the same no matter which way you are going.

Special Relativity Part 1: The problem

So I'm an undergrad physics major, as my brief profile mentions on the left. But I don't know that there's a whole lot to write about. I actually love all my classes, and manage my time very well, so there isn't exactly a whole lot of drama going on. I'd love to explain physics concepts, but I have to stick to what I know, which is certainly less than, say, the physicists over at Cosmic Variance.

But there's still a whole lot, and it includes Special Relativity (abbreviated SR). Lay people seem to think that you need to be an Einstein to understand it. There's also the idea that not even physicists can get a true understanding of the theory; they only know how to do the math and output the numbers. These are all wrong, at least when it comes to SR, which is a relatively simple theory. I think SR is something that any interested reader can come to fully understand.

Oh, and don't confuse Special Relativity with General Relativity, which by contrast is a much more difficult theory to understand. If you're imagining the warping of space--that's General Relativity.

The first thing to understand is why SR is even needed. The problem arose from many experiments involving the speed of light, and most famously the Michelson-Morley Experiment. These experiments showed that light moves at the same speed in all directions, regardless of the motion of the source of the light, or the observer. That means that if I shoot a beam of light at you, you will measure the speed of light to be going at the exact same speed regardless of whether you are moving towards me or away from me.

If that seems only vaguely paradoxical at first, consider this thought experiment: you are on a longboard, and I am standing still. Right when you pass by me, I shoot two beams of light, one forwards and one backwards. Because both beams of light move at the same speed, I will always be right at the midpoint of the two beams of light. However, the same is true for you, even though you are moving on past me. So at any instant, each of us is at the midpoint of the two beams of light, even though we are at two different locations. How can that be?

The solution to this, as many people know, involves radically changing our view of time, space, and whatnot. I'll leave that explanation for future posts.