Part 1: Matrix multiplication 1.1. Consider the following 3 x 3 matrices: 0 cos 3 - sin 3 0 COS O - sin o R. = sin 3 cos B 0 (1) sin a COS CK 0 0 which (as you will see next week, or you can check for yourself now) represent active coun- terclockwise rotations by angles a about the r-axis and & about the z-axis, respectively. Compute the matrix product R, R, and also the matrix product in the reverse order, R, Rz. Do you get the same answer? Draw a sketch (or take pictures with your phone) of an object being rotated by these matrices to justify your reasoning. Feel free to choose whichever angles a and # make the visualization easiest. 1.2. Recall the Galilean transformation matrices from Discussion 3: M = (! " ). M' = (_: ") (2) Compute MM' and M'MY by matrix multiplication. You should find that both are equal to 1= (61) (3) the 2 x 2 identity matrix. 1.3. Multiply two Galilean matrices for different relative velocities, M. = (1 9). M. = ( " ) (4) Does matrix multiplication commute (i.e. is M, M,, equal to M,, M,) in this instance? Explain how this calculation yields the Galilean velocity addition rule. 1.4. Multiply the Lorentz transformation matrix and its inverse, A = (5) and show that AN' = A'A = 1.Part 2: Taylor series Taylor series are more than a neat calculus trick: they are perhaps the most important tool for com- Part 1: Matrix multiplication 1.1. Consider the following 3 x 3 matrices: 1 0 cos - sin 8 0 R. = 0 cosa - sin ax R . = sin / cos 8 0 (1) 0 sina COS CX 0 which (as you will see next week, or you can check for yourself now ) represent active coun- terclockwise rotations by angles a about the r-axis and & about the z-axis, respectively. Compute the matrix product RyRy, and also the matrix product in the reverse order, R, Ry. Do you get the same answer? Draw a sketch (or take pictures with your phone) of an object being rotated by these matrices to justify your reasoning. Feel free to choose whichever angles o and & make the visualization easiest. 1.2. Recall the Galilean transformation matrices from Discussion 3: M = ( ! " ) . A = ( _18 ) (2) Compute MM' and AI'M by matrix multiplication. You should find that both are equal to 1= (61) (3) the 2 x 2 identity matrix. 1.3. Multiply two Galilean matrices for different relative velocities, (4) Does matrix multiplication commute (i.e. is M, M., equal to M., M.) in this instance? Explain how this calculation yields the Galilean velocity addition rule. 1.4. Multiply the Lorentz transformation matrix and its inverse, A = (8 7 ), N= (3 78). (5) and show that AN' = A'A = 1. This formula was derived by Max von Laue in 1907, just two years after Einstein's 1905 paper on special relativity, and explained the results of numerous experiments from the 19th century. This result is a bit subtle. In fact, the Galilean transformations do not arise from the limit v -+ 0, but rather -+ 00. In a mathematical sense, these are actually quite different limits. But since c is not infinite in our world, your result (and not the Galilean one) is actually the correct transformation! See Morin 2.7 for more discussion.Part 3: Velocity addition revisited 3.1. The velocity addition formula is symmetric in vi and v2. Show that Ay Ay, gives the same matrix as AnAy, computed in class. In other words, longitudinal velocity addition (along the same direction) is symmetric because Lorentz transformations along the same direction commute. 3.2. Now consider transverse velocity addition: suppose that an object moves in the ay-plane with velocity (up, a;) with respect to S', which is moving in the +r-direction at speed v with respect to S. Find the velocity of the object in S by identifying its components , and wy. To do this, use the fact that the A's in the Lorentz transformations can be treated as infinitesimals, and u, = Ar'/At' and u, = Ay'/At'. You want to find up = Ar/At and uy = Ay/At. Express your answers in terms of up, uy, v, and % = 1/v1 -8/c', the Lorentz factor between $ and S'. (Feel free to refer to Morin Sec. 2.2.2). 3.3. For the special case u', = 0, we can imagine the velocity u, as arising from a Lorentz transfor- mation of a stationary object in a new frame S", which moves in the y-direction with respect to S' at velocity u'. The combined transformation from S" to S' and finally to S is given by the product of Lorentz matrices ArAy, with 0 0 Ar = (8) where % = 1/ v/1-v3/0', By = v/C, 71 = 1//1 - (u;)?/c2, and B. = /c. Perform the matrix multiplication and compute Ar/At and Ay/At for Ax" = Ay" = A:" =0 (since the object is at rest in S"). You should obtain up = v and by = u;/9, consistent with your results from 3.2 when u = 0. 3.4. Now perform the matrix multiplication from 3.3 in the opposite order: is AyA, the same matrix as ArAy? In other words, is transverse velocity addition symmetric? (Hint: if you find one element of the matrix that is not equal for the two orderings, you can stop; for the matrices to be equal, off entries must be the same.) 3.5. How could we verify whether or not ArAy is a Lorentz transformation? (Don't worry if you don't have a complete solution to this problem - the full story will come in the next two weeks!)