Let's consider the space \(\mathbb{M}^4\), that is, the manifold \(\mathbb{R}^4\) with coordinates \((x,y,z,t)\) but with the pseudo-Riemannian metric given in each tangent space by \[ ds^2=dx^2+dy^2+dz^2-dt^2. \] This metric is left invariant by the group \(O(3,1)\) when acting on \(\mathbb{M}^4\). It also leaves invariant the cone \[ t^2-x^2-y^2-z^2=0 \] so is logical that we pay attention to it. Now, following TRTR from Penrose page 423, we can consider the family of hyperplanes inside \(\mathbb{M}^4\) given by \[ z+t+\lambda(t-z)=2 \] intersecting our cone in different 2-dimensional submanifolds \(P_{\lambda}\) of \(\mathbb{M}^4\). Here we have a picture from that book, where we consider only \((x,z,t)\) for purposes of drawability. If we analyse the case \(\lambda=0\), we observe that we obtain the submanifold \(P_0\), \(E\) in the picture with a shape of a parabola, given by the embedding \begin{align*} \phi \colon \mathbb{R}^2 &\longrightarrow \mathbb{M}^4\\ (u_1,u_2
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