Review of Special Relativity,At the end of the 19th century it became clear that Maxwells formulation of electrodynamics was hugely successful. The theory predicted the existence of electromagnetic waves, which were eventually discovered by Hertz. In Lecture 5, eqns 17 and 18, we see that in source free regions of space the scalar and vector potential obey a wave equation. Wave equations were already known to the classical physicists in, for example, sound waves. These classical wave equations could be understood on the basis of Newtonian mechanics. Some medium was disturbed from equilibrium and the resulting disturbance propagates at a speed characteristic of the medium. If the medium was in motion relative to an observer, then the apparent speed of the disturbance to the observer was simply the vector sum of the velocity of the medium plus the inherent velocity of propagation in the medium. The speed of a sound wave relative to an observer, for example, depends on the speed of sound in air and the wind velocity. The Michelson-Morley experiment was an attempt to measure the motion of the earth through the aether, a substance hypothesized to be the disturbed medium for electromagnetic waves. The null result of the Michelson-Morley experiment, and all its successors, forced physicists to come to terms with the non-invariance of electromagnetic theory with Galilean Relativity. 1,http:/www.blgpt.com/,Galilean Relativity,Newtonian mechanics is invariant with respect to Galilean transformations. These are transformations between reference frames O and O given by eqns (1).,O,O,P,v,x,x,(x,y,z,t) , (x,y,z,t),x = x vt (1a) t = t (1b) z = z (1c) y = y (1d),Time is assumed to be a universal parameter, independent of the reference frame.,The coordinates of point P transform according to equations 1. O moves to the right with a velocity v with respect to O. Invariance of physical laws with respect to transformations of inertial reference frames was a long held and justifiable assumption. We assume that this invariance is a property of space and time. Observations by all competent observers are equally valid. In the case of sound waves we could say that a reference frame moving with the wind velocity is a preferred frame, for in this frame the equations are the most simple. 2,http:/www.blgpt.com/,In the absence of the aether there is no natural preferred reference frame for electromagnetic theory. We still conclude that all inertial reference frames are equally valid and hence the wave equations must have the same form in all inertial reference frames. However, it is straightforward to show that the wave equation does not satisfy Galilean relativity. Consider the transformation of the wave equation for a one dimensional wave V(x,t). In the O system,We will use eqns 1 to transform this into the O system.,In general the transformation from one coordinate system to another is given by,3,http:/www.blgpt.com/,(4a) , 4(b),Applying eqns 4 a second time gives,(5),So we see from eqn (5) that the wave equation is not a Galilean invariant. Equations 1 must be modified so that the wave equation is invariant in transforming from one inertial frame to another. The coordinates (y,z) perpendicular to v do not change. We must consider a more general transformation for the x and t coordinates. It makes sense to try a symmetrical representation of the transformation.,In eqns 6 we choose coefficients as and bs to be dimensionless. We now use equations 6 in equations 3 and derive for the wave equation,4,http:/www.blgpt.com/,(7),In order to ensure invariance w.r.t. coordinate transformation we need,We can try to find a solution to 7 that is symmetric, namely try a1=b1 , and a0=b0. Both 7a and 7b give the same result.,5,http:/www.blgty.net/,The counterpart to the Galilean transformations ( eqns 1), which makes the wave equation invariant is called the Lorentz transformation.,Inherent in this derivation are two assumptions. The first is that the speed of light, c, is the same in the O and O reference frames. This is actually an experimental fact. The second is that the laws of physics have the same form in all inertial reference frames.,There is, in fact, nothing special about electromagnetism other than in the vacuum the waves propagate at a universal speed, c. Any wave disturbance that travels at this speed will also require the Lorentz transformation. One point of view is that the Lorentz transformation says something about how space-time is constructed. 6,The inverse transformation from x to x simply requires changing the sign of v.,http:/www.blgty.net/,We would also have discovered the inadequacy of the Galilean transformation if physicists had had access to high speeds before the discovery of electromagnetism.,Lorentz Invariants,If we can frame our laws in such a way that they are Lorentz invariant then we have satisfied the re