I have been doing questions recently involving Lorentz boosts. However I was wondering if the Lorentz boost matrix $Λ$ is orthogonal.
$$ \left[\begin{array}{cccc}\hat {ct} \\ \hat x\end{array}\right] = \left[\begin{array}{cccc}{\cosh \varphi} & {-\sinh \varphi} \\ {-\sinh \varphi} & {\cosh \varphi}\end{array}\right] \left[\begin{array}{cccc}{ct} \\ x\end{array}\right] =Λ(\varphi)\left[\begin{array}{cccc}{ct} \\ x\end{array}\right] $$
My understanding: For a matrix to be orthogonal $ΛΛ^T=Λ^TΛ=I$
That is that $Λ^T=Λ^{-1}$, however this is not the case with the given matrix here. So instead of using that definition could I prove it is orthogonal in terms of an invarient?
My attempt: If I denote $\eta $ to be a Minkowsi metric which is an invariant.
The matrix representing a Lorentz boost is orthogonal with respect to this Minkowski metric $$ \Lambda \eta \Lambda^T = \eta \text{ or } \Lambda^{-1} = \eta \Lambda^T\eta.$$
Is this a correct statement?