  ## Theoretical Basis for Special Relativity

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### Theoretical Basis for Special Relativity

My theory of special relativity results from two statements -- the two basic postulates of special relativity:

The speed of light is the same for all observers, no matter what their relative speeds.
The laws of physics are the same in any inertial (that is, non-accelerated) frame of reference. This means that the laws of physics observed by a hypothetical observer traveling with a relativistic particle must be the same as those observed by an observer who is stationary in the laboratory.
Given these two statements, Einstein showed how definitions of momentum and energy must be refined and how quantities such as length and time must change from one observer to another in order to get consistent results for physical quantities such as particle half-life. To decide whether his postulates are a correct theory of nature, physicists test whether the predictions of Einstein's theory match observations. Indeed many such tests have been made -- and the answers Einstein gave are right every time!

The Speed of Light is the same for all observers.
The first postulate -- the speed of light will be seen to be the same relative to any observer, independent of the motion of the observer -- is the crucial idea that led Einstein to formulate his theory. It means we can define a quantity c, the speed of light, which is a fundamental constant of nature.

Note that this is quite different from the motion of ordinary, massive objects. If I am driving down the freeway at 50 miles per hour relative to the road, a car traveling in the same direction at 55 mph has a speed of only 5 mph relative to me, while a car coming in the opposite direction at 55 mph approaches me at a rate of 105 mph. Their speed relative to me depends on my motion as well as on theirs.

Physics is the same for all inertial observers.
This second postulate is really a basic though unspoken assumption in all of science -- the idea that we can formulate rules of nature which do not depend on our particular observing situation. This does not mean that things behave in the same way on the earth and in space, e.g. an observer at the surface of the earth is affected by the earth's gravity, but it does mean that the effect of a force on an object is the same independent of what causes the force and also of where the object is or what its speed is.

Einstein developed a theory of motion that could consistently contain both the same speed of light for any observer and the familiar addition of velocities described above for slow-moving objects. This is called the special theory of relativity, since it deals with the relative motions of objects.

Note that Einstein's General Theory of Relativity is a separate theory about a very different topic -- the effects of gravity.

Relativistic Definitions

Physicists call particles with v/c comparable to 1 "relativistic" particles. Particles with v/c << 1 (very much less than one) are "non-relativistic." At SLAC, we are almost always dealing with relativistic particles. Below we catalogue some essential differences between the relativistic quantities the more familiar non-relativistic or low-speed approximate definitions and behaviors.

Gamma ()
The measurable effects of relativity are based on gamma. Gamma depends only on the speed of a particle and is always larger than 1. By definition:

c is the speed of light
v is the speed of the object in question

For example, when an electron has traveled ten feet along the accelerator is has a speed of 0.99c, and the value of gamma at that speed is 7.09. When the electron reaches the end of the linac, its speed is 0.99999999995c where gamma equals 100,000.

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