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# Special relativity

We have an intuitive notion of space and time that seems natural. But space and time have a structure just as objects that exist in space and time do. That structure is not quite as it appears. For example it seems obvious that the shortest distance between two points is a straight line. However we live not on a plane but on the surface of a sphere and the shortest distance is not necessarily a straight line. It is a great circle route. Special relativity is a theory about the structure of space and time that is a radical departure from our intuitive ideas. Relativistic effects are only noticeable at relative velocities much faster then we encounter in every day activities. The first relativistic effects were observed in trying to measure the earth's motion through space.

The waves that ripple out from a rock thrown in a pool are structurally similar to sound waves. When you speak the air vibrates with a pattern of pressure changes in the form of waves. Light was shown to have many of the properties of waves. Unlike sound light travels through empty space. Whatever contains the changing levels of pressure associated with a wave is called its medium of propagation. With sound this is air and with water waves it is water. Scientists naturally wondered what supports the propagation of light in empty space. They assumed there must exist such a medium and they called it the ether.

The earth circles the sun which in turn circles the galaxy which in turn moves away form neighboring galaxies. The absolute motion of the earth relative to the ether must be very fast. By measuring the speed of light in one direction and then the opposite direction one should be able to determine the absolute motion of the earth through the ether. When the earth is moving in the same direction as the light beam the speed relative to the earth will be slower because light has to travel not only the distance between two points on earth but also the distance the earth moved in the time between the measurements at the two locations. When moving in the opposite direction the speed is higher because the speed of the earth is added to, not subtracted from, the speed of light. Take the difference of the two speeds and divide by two and you have the absolute speed of the earth.

Michelson and Morley devised an ingenious experiment to make these measurements. It was difficult because light travels at approximately 300,000 kilometers a second and the motion of earth through space was expected to be only a tiny faction of that speed. The result was that one could detect no difference. The speed of light was the same in all directions.

Einstein explained this mystery by assuming the experiment results were correct and generalizing them. He assumed the speed of light is the same no matter how fast one is moving relative to a light beam. A system that is moving at uniform speed (not accelerating or decelerating) in a constant direction is called an inertial frame of reference. Einstein assumed that not just the speed of light but all physical measurements would be the same as long as the experiment was carried out in an inertial frame of reference. This implies that our measurements change depending on how fast we are traveling relative to the object being measured. Time itself slows down as we approach the speed of light. If we could travel as fast as light time would stop completely.

At first this seems absurd. We think of distance as being absolute and not relative to how we measure it. Mathematically topology is independent of distance. We can set up a topology or mathematical set of points and then impose on this any distance function we choose. To specify location one must describe the connectedness of the geometry. For example we can use pairs of real numbers to specify the ordering of points in a two dimensional space. The connectedness of the points is determined by the ordering of pairs of real numbers . We can then assign a distant function. The standard Euclidean distance function between and is as follows.

There are many other possible distance functions. For example if one lived in a world where diagonal motion was not possible and one could only move parallel to the or axis the distance function would be as follows.

Note denotes the absolute value of x. If x is negative otherwise .

The relativistic distance function is strange because the measurements two astronauts make of each others space ships depend on the relative speed of the two ships. In relativity there is no absolute speed and no absolute frame of reference against which absolute speed could be defined. This does not mean that relativity is inconsistent with assuming there is an absolute reference frame. On the contrary one can assume any inertial frame of reference is the absolute reference frame. Relativity treats all inertial frames the same.

In special relativity the Euclidean distance function works as long the object being measured is not at motion relative to the measuring apparatus. If it is in motion than the measured distance will be less than the measurement made of the same object at rest by a factor of where is the relative velocity and is the speed of light. This comes from the Lorentz transformation named for its inventor. Einstein did not create the equations of special relativity. He interpreted them. Different observers moving at different velocities (different frames of reference) will get different measurements of the same object and all of them will be correct. The Lorentz transformation shows how to translate the measurement in one frame of reference to a measurement of the same object in a different frame of reference.