# Spacetime rotation matrix using mostly minus conventions

When trying to find the Lorentz transformation in matrix form in the $$x^2+x^3$$-direction, I tried simply mapping the Lorentz boost in the $$x^2$$-direction to the $$x+x^3$$-direction by rotating it $$45°$$ around the $$x^1$$ axis:

$$R_{x}^{-1}(\theta=45°)\left(\begin{array}{cccc}\gamma & 0 & -\gamma\beta &0\\0&1&0&0\\-\gamma\beta&0&\gamma&0\\0&0&0&1\end{array}\right)R_{x}(\theta=45°)$$ Where $$\beta=\dfrac vc$$ and $$\gamma = \dfrac{1}{\sqrt{1-\beta^2}}$$.

However, I am not sure of the proper form of the rotation matrix in 4-dimensional spacetime. Especially since we are using 'mostly minus' conventions. Using 'mostly plus' I would guess:

$$R_x=\left(\begin{array}{cccc}-1&0&0&0\\0&1&0&0\\0&0&\cos\theta&-\sin\theta\\0&0&\sin\theta&\cos\theta\end{array}\right).$$

How would this translate to mostly minus conventions?

• I believe the 00 component should be $+1$, as a rotation acts as the identity on $x^0$. – Dwagg Aug 11 '19 at 20:00
• What you are looking for is a Lorentz boost $B_{\beta}$ in the $x^2$ direction conjugated by a rotation $S_{\theta=\pi/4}$ in the $(x^2,x^3)$ plane, or $S_\theta^{-1} B_\beta S_\theta$, I believe. – Dwagg Aug 11 '19 at 20:27