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The theory of relativity is a theory of space and time, proposed in two parts by Albert Einstein in the early twentieth century, and supported by a great deal of experimental evidence. The first part, the special theory of relativity, was published by Einstein in 1905 (a year sometimes referred to as his "annus mirabilis" (tr. "year of wonder") because of this and two other revolutionary papers he published that year). The second part, the general theory of relativity, was completed some ten years later.

The Special Theory

The special theory primarily describes the behavior of objects moving at speeds close to the speed of light. It is based on two assumptions, known as Einstein's postulates, which were inspired largely by Maxwell's equations of electrodynamics:

1. The fundamental laws of physics are the same in every inertial reference frame.
2. The speed of light in a vacuum is the same in every inertial reference frame.

A host of mind-boggling consequences follow from these two simple assumptions.

Length contraction -- The length of an object, in a reference frame in which that object is moving, is shorter than the length of that same object in a frame in which it is at rest. The amount by which it is shorter is given by a factor that turns up a lot in relativity:

                <math>\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}</math>,

where v is the speed of the object, and c is the speed of light. The contraction takes place only along the direction of motion. Notice that as the speed of the object approaches the speed of light, this factor approaches infinity, meaning that the object gets infinitely short at the speed of light.

Time dilation -- The time between two events, in a frame in which those events occur at different places, is longer than the time between those same two events, in a frame in which those events occur at the same place. A more succinct, but less general way of stating this is that a moving clock runs more slowly than a stationary clock. (By "clock," we're referring to anything that notes or experiences the passing of time. Plants will grow more slowly, people will age more slowly, unstable particles will decay more slowly, etc.) The amount by which time is dilated is given by the same factor that's shown above. So as the speed of a clock approaches the speed of light, the rate at which the clock runs slows to zero.

An example -- Suppose a spaceship flies by earth at 3/4 the speed of light, en route to a star that's 6 light years away. By our reckoning, it should take the ship 8 years to get there. However, since the people inside the ship are moving very rapidly, their clocks (and bodies and minds) should run slower, so that in their experience the trip doesn't take as long.

On the other hand, in their frame of reference, they are stationary and the space between us and the star is moving rapidly past them. In this point of view, their clocks run at normal speed. But the distance between earth and the star, because it's going past them so rapidly, is shorter due to length contraction. And so, again, the trip takes less than 8 years for them. Cool, huh?

The speed limit -- The theory of relativity gives us a limit to how fast something can travel. Specifically, nothing can be accelerated to a speed greater than that of light. There are a number of reasons for this prohibition. Mathematically, you can see that the gamma factor defined above becomes undefined for a speed at or above that of light. It can also be shown that it takes an infinite amount of energy and force to accelerate an object up to the speed of light. Finally, the logic of cause and effect breaks down if we allow speeds faster than light. That is, if objects can travel faster than light, then it's possible in some frames of reference for the cause of an event to occur after that event has taken place.

The mass-energy relation -- Einstein showed that mass is effectively a form of energy. This is illustrated in Einstein's famous equation for the energy of an object at rest:

                <math>E = mc^2</math>,

where m is the mass of the object, and c is the speed of light. Not only does this say that there's energy associated with the mass of an object, it says that it's a staggering amount of energy!

In most everyday occurrences, this energy is unimportant. That's because it's seldom converted into other forms of energy. However, where "rest energy," i.e. mass, is converted to other forms, one can get quite a lot of it. This is what happens in nuclear power and nuclear weapons, for example. Both these inventions rely on nuclear processes in which the particles in the final state have a total mass that's less than the particles in the initial state. This loss in rest energy is accompanied by a gain in kinetic energy.

Even in nuclear processes in which a small amount of radioactive material results in a huge amount of kinetic energy, only a tiny fraction of the mass is converted to kinetic energy. A full conversion from mass to kinetic energy is only possible in the case of a matter-antimatter annihilation involving fundamental particles. For example, in an electron-positron collision. The problem with this as an energy source, however, is that there are no great stashes of antimatter lying around waiting for us to collect. If we want antimatter, we have to make it ourselves, and this takes up just as much energy as we would get out of it. If starships ever use antimatter, it will be as a convenient way of storing vast amounts of energy, not as an actual source of energy.

The General Theory

Einstein published his general theory of relativity in 1916. The general theory is a theory of gravity, in which the traditional gravitational force described by Isaac Newton is replaced by curvature in four-dimensional spacetime. That is, in the general theory of relativity, four-dimensional spacetime is described by a non-Euclidean geometry whose curved spacetime gives us the effects of gravity. Here's what this means. When we throw an object into the air and watch it soar in a parabolic arc, we can interpret the motion two different ways.

1. The object moves in a curved, rather than straight path, because of the gravitational force that acts on it. (Newton)
2. The object moves along what is considered a straight path in spacetime, but the path does not appear straight to us because the geometry of spacetime is curved, not Euclidean. (Einstein)

Of course, Newtonian gravity gives a highly accurate description of gravity in almost all cases, which means that if the general theory is to have any validity, it must give the same result as Newtonian gravity in such situations. In fact, this is the case, and the two theories differ mainly in just the case of very strong gravitational fields. Two such cases are especially interesting.

The deflection of light -- One prediction of the general theory of relativity is that light should be deflected by a gravitational field. The scientist Sir Arthur Eddington set out to test this effect in 1919 by observing stars in the direction near the sun during a solar eclipse. The idea is that during an eclipse it's possible to see stars in the direction of the sun that would normally be unseeable due to the sun's brightness. And since the light from these stars must travel close by the sun in order to reach us, that light will be deflected by the sun's strong gravitational field.

What this means is that the light from these stars will be arriving at earth from a direction which is slightly different from the known positions of those stars. Eddington measured the direction of the light from these stars, and the deviation from the actual position of the stars was found to be approximately what was predicted by Einstein's theory. It was this test of the theory that made Einstein a celebrity.

The precession of the perihelion of Mercury -- According to Newton's theory of gravity, a single planet orbiting a spherical mass should have a "closed" orbit. That is, each time around it comes back to the exact same spot. However, when we take other objects in the solar system into account, we find that the perihelion, the point in the orbit where the planet is closest to the sun, shifts a little bit each time around. This shift is called a precession. Astronomers were able to calculate this precession of the perihelion for Mercury very precisely based on Newtonian gravity. And they were able to observe the precession very accurately as well.

It turns out, the calculated precession differed from the observed precession by about 43 arc seconds per century. (In angular measure, there are 60 minutes in one degree and 60 seconds in one minute.) However, when one includes corrections due to general relativity, the discrepancy disappears. This was another triumph for Einstein's theory.