Lecture12 - Lecture 12 Notes 95.658 Spring 2012...

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Lecture 12 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Velocities in Special Relativity - As was done in Galilean relativity, we can use the coordinate transformation to find out how velocities add. - If frame K ' is moving in the x 1 direction at a velocity v relative to frame K , then we found the transformation to be: x 0 = x 0 ' v c x 1 ' where x 0 = ct and x 1 = x 1 ' v c x 0 ' - If an object in frame K ' moves at some constant velocity u ' in the x 1 ' direction relative to its frame, e.g. a passenger on a moving rail car walks down the aisle, what we are saying is that its coordinate x 1 ' changes with respect to time t ': u ' = d x 1 ' d t ' or u ' = c d x 1 ' d x 0 ' and similarly u = c d x 1 d x 0 - An incremental displacement of a coordinate dx 1 transforms just like a regular coordinate: u = c  d x 1 ' v c d x 0 ' d x 0 ' v c d x 1 ' u = c d x 1 ' d x 0 ' v c 1 v c d x 1 ' d x 0 ' u = c u ' c v c 1 v u ' c 2 u = u ' v 1 v u ' c 2 - If the speed of the frame v or the speed of the object u ' is slow enough, the second term in the denominator approaches zero and the Galilean velocity addition equation is recovered: u = u ' + v. - As the speed of the frame v increases, the speed of the object as seen in frame K seems to be = 1 1 v 2 c 2
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slower than what one would expect from Galilean relativity. - Note that this equation tells us that if an object is traveling at the speed of light in one frame, it is traveling at the speed in all frames. This behavior is not unique to light, but applies to all objects traveling at the speed of light. - We must remember that a velocity is defined as a change in position over a change in time. Both position and time are not universal now, so the measured velocity experiences both length contraction and time dilation. - There is a certain symmetry to this equation that we should expect from the fact that we required that nothing goes faster than the speed of light. - If the frame moves at the speed of light, v = c , the velocity addition formula reduces to u = c . - A baseball thrown on a train traveling at c looks from the ground as if it is also traveling at c , i.e. it is stationary, stuck in mid-air, with respect to the train. This makes sense if we remember that looking at a frame traveling at c , its time has completely stopped. - Now consider if the object is moving diagonally in frame K ' so that the object has a component of its velocity which is perpendicular to the frame's velocity, u ' = u 1 ' x 1 ' u 2 ' x 2 ' . For instance, consider a baseball in the moving train that is thrown diagonally up and forward relative to the train (neglect gravity). - We have already derived the parallel component, and can now generalize. u par = u par ' + v 1 + v u par ' c 2 Parallel Velocity addition Formula for Special Relativity - Here “par” denotes the velocity component parallel to the frame's velocity v .
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