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### Einstein’s Special Theory of Relativity

The theory of special relativity lays down two basic principles. First, that the laws of physics are identical in all constant frames of reference. Secondly, that the speed of light is constant no matter what your point of reference.

The first statement is easily explained. Suppose there is an individual standing by a train track holding a ping pong ball. He holds the ball out at arms length and drops. We all know that the ball will drop towards the ground. Now, what would happen if this person’s twin were to come by in a train traveling at a constant 40 km/hr and the twin dropped a ping pong ball? From the twin's point of view exactly the same thing would happen as if he had dropped it when he was stationary.

Now the second postulate is more difficult. It means that no matter what your state of reference, at rest or at constant velocity, the speed of light will appear the same to you. This can be illustrated with an example. You and a friend go outside to toss around a football. Suppose your friend throws the ball at a speed of 50 km/hr. If you were playing dodge ball and attempted to run away from the ball at 40km/hr it would appear the ball is moving at only 10km/hr. If you ran towards it at the same speed it will appear to be moving towards you at a speed of 90km/hr. This appeals to our intuition. What Einstein said is that this doesn’t happen with light. Say if your friend threw a light beam at you. Even if you attempted to run away at 99% the speed of light (quite impossible!) you would still see it approaching at the speed it was thrown, not 1% of that. Also, it will not appear to go faster if you ran towards it.

Now if a stranger were to walk by while you were playing this game of catch and was to see your friend throw the beam and then run after it at 99% the speed of light the stranger would see your friend almost keeping up with the beam. However, when asked, your friend would say that the beam sped away as if he was standing still. This is possible by getting rid of the idea of absolute distance and time. Time and distance change so that a second and a foot are not the same at great speed as they are when at rest.

The application of special relativity is important in GPS systems. The satellites orbiting earth have clocks that tick at a different speed then those on Earth because of their differing velocities. The technology needs to have clocks in perfect synch with those on Earth to allow it to nail down position to a precise degree. The slight difference between them would cause errors of several parts per billion. This would be enough to make them useless for all modern purposes including aerial and submersible navigation and military operations. Having your plan land twelve meters east of the runway is a major problem.

Problem: Special relativity led to several formulas for measuring the change in time, mass, and length of an object when seen by an observer at rest, as follows:

1) Time

2) Mass

3) Length

Now these formulas are needed for the application we proposed above. To figure out the difference in the perception of time between a satellite and the surface of the Earth we apply the first formula. Seeing as the average speed of a GPS satellite is about 20 000m/s we plug this in as v, the velocity. The speed of light is c, t0 is the proper time, and t is the time interval. So our equation is the proper time over the square root of one subtract 20 000 squared over the speed of light squared. This is equal to a correction factor of one. And so our satellite is aligned.

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By: Andrew Macdonald