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,
[math]
\left[ {\begin{array}{*{20}c}
{\cos \theta } & { – \sin \theta } \\
{\sin \theta } & {\cos \theta } \\
\end{array}} \right]
[/math]
So, the matrix equation expands to
[math]
x’ = x \cos \theta – y \sin \theta \\
y’ = x \cos \theta + y \sin \theta
[/math]
where [math]\theta[/math] is the angle of rotation. This is how a point [math]X=(x,y)[/math] in the original frame transforms to [math]X’=(x’,y’)[/math] in the rotated frame.
Similarly, LT in Minkowski space is X’ = L X. Lorentz Transformation matrix (in our 2-D case) is,
[math]
\left[ {\begin{array}{*{20}c}
\gamma & { – \beta \gamma } \\
{ – \beta \gamma } & \gamma \\
\end{array}} \right]
[/math]
where [math] \beta = v/c [/math] and [math] \gamma = 1/\sqrt {1 – \beta ^2 } [/math]
This expands to
[math]
t’ = (t – vx/c^2)/\sqrt{1-(v/c)^2} \\
x’ = (x – vt)/\sqrt{1-(v/c)^2}
[/math]
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.