I am trying to obtain the embedding diagram of Minkowski space. I took the metric:
$$ d s^{2} = - d t^{2} + d r^{2} + r^{2} \left( d \theta^{2} + \sin^{2} \theta d \phi^{2}\right) $$
for $t =$ cte and $\theta = \pi /2$,
$$ d s^{2} = d r^{2} + r^{2} d \phi^{2} $$
I know this should give me a disk, like the one below:
But I can't understand how to graph or if I should do any parameterization before. I know it's something basic but I would really appreciate if someone could guide me.
Let us follow your assumptions and say you had Minkowski spacetime in rectangular coordinates:
$$ds^2=-dX_0^2+dX_1^2+dX_2^2$$
Now, using your transformation:
$$X_0=\cosh\rho,$$ $$X_1=\sinh\rho\cos\theta,$$ $$X_2=\sinh\rho\sin\theta.$$
Then, after solving, we get the following, for a unit hyperbolic radius:
$$ds^2=d\rho^2+\sinh^2\rho d\theta^2$$
You have everything; all you need is just to plug your transformation rules in Mathematica:
ParametricPlot3D[{Sinh[ρ] Cos[θ], Sinh[ρ] Sin[θ], Cosh[ρ]}, {ρ, 0, 0.01}, {θ, 0, 2 π}, PlotRange -> Automatic] which gives you the required embedding diagram,
