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Name: Simon Quellen Field
Location: Los Gatos, California, United States

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Wednesday, May 02, 2007

Einstein points the way

A good friend of mine is fond of describing the Global Positioning System as a device that depends on special relativity to work. He is correct of course — without the corrections for time dilation the system could not function. But the implication that it is the only such device he knows of is wrong.

A compass also depends on relativity to help us navigate.

Most people are aware of the strange effects of relativity, if only from exposure to science fiction. They know that time goes more slowly the faster you travel, and that rockets traveling near the speed of light appear shorter, due to an effect called Lorentz contraction. The general impression is that these effects are not significant at lower speeds. That impression is wrong.

The reason an educated person might have that impression is because of a little thing known as relativistic gamma, a function of speed that becomes larger as you approach the speed of light. Gamma tells us how much time dilation will occur, how much heavier you will get, and how much thinner you will become as you increase your speed.

At low velocities, gamma is very close to one. As the speed gets close to the speed of light, gamma gets very large:

The horizontal scale on the graph is β, the ratio of your speed to the speed of light. The curve explains why you can't accelerate to the speed of light. If you did, gamma would go to infinity, so your mass would go to infinity, and your width would go to zero. Since it takes an infinite amount of energy to move an infinite amount of mass, you just don't have the fuel to get there.

Gamma is usually written in one of these three ways:

As your speed v gets close to the speed of light c, the denominator gets very small, and thus gamma gets very large.

But look at the graph — until you get half the speed of light, gamma is still very close to the value one, meaning that your mass hasn't changed much, nor has your width, or your perception of time. Half the speed of light is still 335,308,315 miles per hour. That's really fast. At normal human speeds, such as walking, surely gamma is so close to one that we can ignore it, right?

Well, no.

It turns out that if you have something that is very small, it can be important if you have a whole lot of them. An atom might not weigh much, but an elephant is made of a lot of them. And there are a lot of electrons in a compass needle.

Magnetism seems like magic sometimes, but it is really just a whole bunch of electrons moving slowly, and showing us how relativity can happen at a walking pace.

To picture electrons moving, consider a wire connected to the terminals of a battery. The battery makes electrons in the wire move. They don't move very fast. If there is 10 amperes of current in the wire, the electrons are moving at 0.00053686471 miles per hour. Snails go 50 times faster than that. The value of gamma for that speed is so close to the value of one that the Google calculator can't tell the difference.

But there are 6,241,509,630,000,000,000 electrons moving past us in the wire every second. The electrons are negatively charged, moving past positively charged nuclei. Since the wire is neutral, there must be as many positive charges as negative charges.

But what happens if we walk along the wire at the speed of the electrons? The positive nuclei appear to be closer together due to gamma. A very very tiny bit closer together. The electrons are still the same distance apart to us, because we are moving at the same speed. But when we multiply a very tiny amount by a huge number of electrons, the effect of the positive charges getting bunched up together makes it appear that there are more positive charges than negative ones in the wire.

That will have an effect on any charged particle moving near the wire. The charged particle sees the wire as having a charge, but only when the particle is moving.

We call this magnetism.

It is special relativity happening at a pace that makes a snail look like a race car.