Lecture 1 - Physics 325 Lecture 17 Special Relativity As...

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Physics 325 Lecture 17 Special Relativity As you no doubt know already, Newton’s laws are only an approximation to classical mechanics. A nearly perfect approximation for everyday velocities of objects, but an approximation nonetheless. The Newtonian world has a universal clock that ticks on at a constant rate independent of the motion of an observer. Vector components however, clearly depend on the reference frame, and the transformation of vectors is described by a matrix that preserves dot products. This puts time and space on different footings since time is independent of the observer while space has very definite transformation properties. The concept of an invariant dot product is expanded in special relativity in such a way that time and space are put on equal footing. The theory of special relativity (I will henceforth refer to it as just relativity , but I will mean special, not general relativity) is based on the following two postulates: I. The laws of physics are the same in all inertial reference frames. II. The velocity of light in vacuum is a universal constant and is independent of the relative motion between the source and the observer. The first postulate is a hold-over from Newtonian mechanics. The second postulate is revolutionary. In the Newtonian world all velocities add like simple vectors, so if a person is moving towards you at a velocity v G and shines a light at you that is traveling at velocity in her reference frame, then the light is traveling towards you at velocity (this is called a “Galilean transformation”). Not so says Einstein! The light is traveling at velocity in the running person’s frame, and it is also traveling toward you at in your reference frame. c G vc + GG c G c G This counter-intuitive postulate seems to be disproved by our everyday experience. There are many examples in which we observe the simple addition of velocities when moving from one frame to another. For instance, if you walk on a moving conveyor belt the speed with which you pass stationary objects is certainly the sum of your walking speed and the speed of the belt. Relativity avoids this contradiction by redefining the addition of velocities in such a way that they work “normally” at velocities small compared to the speed of light, but saturate at large velocities so that the result is never larger than the speed of light.
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We have seen that Newton’s laws are invariant under Galilean transformations between two reference frames that move with a constant velocity 0 v G relative to one another: ( ) ( ) () () 0 0 rt rt v t vt v = = G GG G (17.1) Here we have assumed that space and time are completely separable so that the t in the first equation above can be defined independent of the reference frame (when we invert the equation to write r in terms of G r G , the same t appears. Therefore tt = for a Galilean transformation.
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Lecture 1 - Physics 325 Lecture 17 Special Relativity As...

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