Lorentz transformations: new actual notation for a $4$-vector

Question

For the Lorentz trasformations I use this notation

\begin{equation*} \left\{\begin{aligned} x&=\gamma (x'+\beta ct')\\ y&=y'\\ z&=z'\\ ct&=\gamma (ct'+\beta x')\\ \end{aligned}\right. \end{equation*}

with this matrix

$$L^*=\begin{pmatrix}\gamma & 0 & 0 & \beta\gamma\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \beta \gamma & 0 & 0 & \gamma\end{pmatrix}$$ Introducing the imaginary unit $i=\sqrt{-1}$, the Lorentz transformations will allow you to switch from an orthogonal Cartesian coordinate system to an orthogonal one. Hence I, actually, use $L$ that is an orthogonal matrix. $$L=L(\beta)=\begin{pmatrix}\gamma & 0 & 0 & -i\beta\gamma\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ i\beta \gamma & 0 & 0 & \gamma \end{pmatrix}$$

My usual notation that I use is the following to define a quadrivector $\boldsymbol{\mathcal{X}}=(x,y,z,ict)$, or even better is:

$$\boldsymbol{\mathcal{X}}^\intercal=\begin{pmatrix} x \\ y \\ z \\ ict \end{pmatrix}$$ Why most physicists now use $(ct,x,y,z)$ instead of $(x,y,z,ict)$ (or $(ict, x,y,z)$) and let the electromagnetic field tensor have real components?

Answer 1

I had the same question before. I think there's a paragraph in Kip Thorne's Modern Classical Physics specifically pointed out that the imaginary number could not capture the total spacial/geometrical aspects of the GR (could not recall the detail), therefore I guess, people slowly used the real number and differential metric instead of the imaginary number (you don't actually need to write it in metric form if you use imaginary number, not necessarily). It's useful, though, to notice that imaginary number was a very “cheap”/neat solution, which was used in many old books. Like @bolbteppa has mentioned.