Part 2: Lorentz transformations in matrix form
The fact that the Lorentz transformations are linear means that we can represent them with a matrix, just as we did for Galilean transformations last week.
2.1. Write the matrix equation transforming the spacetime vector (t, ) in S to the corresponding vector (t, ) in S.
2.2. We can make the Lorentz transformations a lot more symmetrical by writing all quantities in units of the speed of light. Lets define 3 = v/c (note< 1 because < c), and write ct for the time coordinate (in other words, if t is 1 second, then ct is 1 light-second, which has units of distance). Show that the Lorentz transformations can be written in matrix form as
^YYp
(5)
2.3. Solve the linear equations for (ct,) in terms of (ct,) to find the inverse Lorentz transfor- mation, in other words the matrix which transforms S to S. Once you have it, explain how you could have written down this answer immediately without doing any algebra.
2.4. Weve been neglecting the roles of the coordinates y and in the transverse directions, because they dont change when the relative motion between S and S' happens only in the direction: y = y and = z. Restore these coordinates to find the 4 4 matrix which transforms ct,x,y,toct,x,y,z.
2.5. Without doing any additional algebra, find the matrices which transform (ct,,y,) to (ct, , y, z) in the cases where S is moving in the positive y-direction or the positive 2-direction instead of the positive -direction.
We will close this part by introducing some notation which you will use for the remainder of the course. The object (ct, , y,z) is called a spacetime 4-uector (because it combines space and time, and has 4 components). It is often given the name , where the superscript is a Greek letter, and the indices start from zero: x = ct, = , 2 = y, x3 = z. The Lorentz transformation matrix is traditionally called A, and in index notation (using the Einstein summation convention) the Lorentz transformations are written as
# = x
(6)
Lorentz transformations are often referred to as boosts, and is also called the boost factor.