What is a Lorentz reflection of $\mathbb R^3$? Is there a way to visualize it? Suppose I have a plane, P, what would (Lorentz) reflecting in it differ from (Euclid) reflecting in it?
I know that the Lorentzian metric is one where the inner product $(x,y)=x_1y_1+x_2y_2-x_3y_3$ but is the reflection along the (Euclidean) orthogonal to P but ending up at points with distance wrt the given metric?
In the Euclidean case, for $n\in \mathbb R^3$ such that $n$ is a unit vector normal to a plane P, the reflection in P is given by $R(x)=x-2(x,n)n$ , where $x\in \mathbb R^3$.
Is the equation for the Lorentzian case the same except the inner product $(.,.)$ is the Lorentzian one?