This post uses Easy LaTeX Pro to display equations.
Lorentz transformation is a rotation in the Minkowski space. In order to see it, let’s first look at rotation in Euclidean space, which can be written as X’ = R X. In the 2-D case, the matrix of rotation R is,
So, the matrix equation expands to
where is the angle of rotation. This is how a point in the original frame transforms to in the rotated frame.
Similarly, LT in Minkowski space is X’ = L X. Lorentz Transformation matrix (in our 2-D case) is,
This expands to
Note that rotation (and so LT) is a linear transformation, which means that the matrix R (or L) has to be independent of the vector it transforms. What happens when the matrix is a function of x, y or t? The geometry becomes non-flat and the metric tensor we defined doesn’t define the invariant distance any longer. The geometry requires a different metric tensor. Therefore, rotation or LT as we defined it and the associated single component equations is not valid any more. I will illustrate it further using 2-D rotation in the next post and show what they mean when they say that space-time is curved.