Proper orthochronous Lorentz transformation ($\Lambda \in S{O}^{+}(1,3)$)

A Lorentz transformation with determinant $+1$ that preserves future-directed timelike four-vectors. Equivalently, it belongs to the identity component of the Lorentz group. For the Minkowski metric $(-,+,+,+)$, its matrix satisfies ${\Lambda }^T\eta \Lambda =\eta$ and ${\Lambda }^0{}_0\ge 1$. Such transformations preserve the future light cone, so the retarded Green function of the d'Alembert operator transforms as a Lorentz scalar.