Electromagnetic Theory Prof. Ruiz, UNC Asheville, doctorphys on YouTube Chapter B Notes. Special Relativity. B1. The Rotation Matrix

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1 Eletromagneti Theory Prof. Ruiz, UNC Asheille, dotorphys on YouTube Chapter B Notes. Speial Relatiity B1. The Rotation Matrix There are two pairs of axes below. The prime axes are rotated with respet to the original x-y system. The seret in relating ( x ', y ') to ( x, y ) is to onstrut the ute retangle you see in the figure. + y sinθ and y ' = y osθ x sinθ. + y sinθ y ' = x sinθ + y osθ Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense

2 We an define the following matrix R( θ ) osθ sinθ = sinθ osθ. In matrix notation we an write B. Trig Identities x ' x osθ sinθ x R( θ ) y ' = y = sinθ osθ y. Now the fun. We will proeed to derie trig formulas in one step. Remember those days, perhaps in high shool, when you first enountered ompliated trig identities inoling the sines and osines of sums and differenes of angles. Here you an derie these quikly. The ombined rotation osα sinα os β sin β R( α + β ) = sinα osα sin β os β has to be equal to os( α + β ) sin( α + β ) R( α + β ) = sin( α + β ) os( α + β ). Multiply the matries and your a 11 matrix element is your osine identity for os( α β ) +. The element 1 a takes are of sin( α β ) +. The results are: os( α + β ) = osα os β sinα sin β sin( α + β ) = osα sin β + sinα os β. Replae β with β and you arrie at the formulas inoling the differenes. Remember that the osine is an een funtion, i.e., os( β ) = os β, and the sine is an odd funtion suh that:sin( β ) = sin β. os( α β ) = osα os β + sinα sin β sin( α β ) = osα sin β + sinα os β. Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense

3 This is an example of the "magi" of the rotation matrix. You might say these are Feynmanesque deriations. We get the result in a ouple of lines, while the high shool proof goes on and on with intriate diagrams and multiple algebrai steps that an take oer a page. In fat, we are not een afraid of the triple-angle sum, os( α + β + γ ). Just multiply another matrix and pik off the appropriate part. This is another harateristi of Feynman - using theoretial tehniques to do een more general and more diffiult proofs with relatie ease. PB1 (Pratie Problem). Use the aboe formulas to derie the result for tan( α β ) β to arrie at the identity for tan( α β ) Then replae β with to look like the standard forms: B3. Galilean Transformation We onsider inertial frames in this hapter. An inertial frame is one either at rest or moing in a straight line with onstant speed. At the right are two suh frames. The K frame an be onsidered at rest and the K' frame moing at speed along the x-axis relatie to the K frame. tanα tan β tan( α β ) = 1 + tanα tan β and tanα + tan β tan( α + β ) =. 1 tanα tan β. +. Arrange your results We an introdue the time oordinate for eah frame, t and t', respetiely for the K and K' frame. We synhronize the loks so that t = t' = 0 when the origins oerlap. The ommon-sense lassial relationship between the oordinates (x,t) in the K frame an readily be round sine for any speifi point x' we hae x = t + x'. We obtain x ' = x t and t ' = t (absolute time for eeryone). Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense

4 The seond equation means absolute time. The two loks run at the same rate. This is known as the Galilean transformation. Howeer, this transformation is alid only for small speeds. When the relatie speed between the frames is great, i.e., appreiable ompared to the speed of light, it is the Lorentz transformation that is alid. B4. The Lorentz Transformation Hendrik Antoon Lorentz ( ) Courtesy Shool of Mathematis and Statistis Uniersity of St. Andrews, Sotland The Lorentz transformation between the frames K and K': x ' = x t and t ' = x t Lorentz arried at this to explain the Mihelson-Morley experiment (1887), whih indiated light appeared to trael at the same speed independent of the obserer. The Duth physiist Lorentz by the way shared the Nobel Prize in Physis in 190 with Zeeman for the disoery and explanation of the Zeeman effet. B5. Speial Relatiity Albert Einstein ( ) Courtesy Shool of Mathematis and Statistis Uniersity of St. Andrews, Sotland Albert Einstein put forth the Theory of Speial Relatiity in The postulates are: 1. First Postulate. The Laws of Physis are the same in all inertial frames.. Seond Postulate: The speed of light (in auum) is the same in all inertial frames. From the seond postulate we an derie the Lorentz transformation. Einstein also published in 1905 his famous paper on the Photoeletri Effet, whih won him the 191 Nobel Prize in Physis. And this is not all he published that year.. Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense

5 We will derie the Lorentz transformation in a Feynmanesque manner, i.e., in a ery elegant way. Consider a light beam emitted at the origin when the loks are synhronized at t = t'. For the two frames we hae x = t and x ' t ' =, where the speed of light is taken to be the same aording to Einstein's Seond Postulate. To allow for light traeling in the positie x and negatie x diretions we square both sides of eah equation, arriing at x = t and ( x ') ( t ') =. We an then state the following elegant result. x t = ( x ') ( t ') This arrangement is the same in eah frame. We say that the quantity x t is inariant. Einstein was always looking for that whih does not hange from frame to frame suh as the speed of light and relations like this one. What is inariant in our rotation sheme below? The length of the OP line. Coordinates may ary but eah frame agrees on the distane between O and P. The mathematiian Hermann Minkowski introdued the trik of letting y = it. Then, the inariant is x + y = x t. WARNING: This y is NOT our spatial y dimension in the referene frame aboe. The y = it is related to our time ariable. We still hae our regular y in our room. Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense

6 Our oordinate transformation + y sinθ y ' = x sinθ + y osθ with the Minkowski trik: y = it and y ' = it ', gies us the Lorentz transformation - well, almost. We still hae to deal with the angle. + it sinθ it ' = xsinθ + it osθ We hae an analogy here. The rotations with angles are analogous to our referene frames with arious speeds. All we hae to do is determine how the angle theta relates to the speed. We do this by looking at a point that stays put in the K' frame so that x ' = 0. Then, the obsered hange in that position measured by the K frame is the K frame wathes the K' frame moing on. We an write x ' = x osθ + i t sinθ = 0 x osθ = i t sinθ x = as t x t sinθ = i osθ = i tanθ tanθ = = i i Now this would freak out most folks. The tangent is a slope. You an't hae an imaginary slope. Thus we meet a harateristi of Feynman magi with math. Suh an Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense

7 imaginary slope wood not phase Feynman in the least and he would ontinue on as follows. In summary, we hae so far + it sinθ it ' = xsinθ + it osθ tanθ = i Proeeding as if nothing is strange, we set up our right triangle so that the tangent of the angle is orret. We then determine the hypotenuse with Pythagorean's Theorem, not phased in the least with the imaginary number. 1 1 hypotenuse = + i = This is amazing. Filling in for the trig we hae i 1 x ' = x + it x ' = x t, whih gies ; i 1 it ' = x + it, whih gies t ' = t x. I first saw this ool deriation in L. D. Landau and E. M. Lifshitz, The Classial Theory of Fields (Pergamum Press, Oxford, 196). Mihael J. Ruiz, Creatie Commons Attribution-NonCommerial-ShareAlike 3.0 Unported Liense