Let $b:\mathbb{R}^3 \longrightarrow \mathbb{R} $ , $b(x,y)=x_1y_1+x_2y_2-x_3y_3$ be the Minkowski innter product and let $\mathbb{H}^2=\{x \in \mathbb{R}^3:b(x,x)=-1, x_3>0\}$. So $\mathbb{H}^2$, as a subset of $\mathbb{R}^3$, is the upper half of a two sheeted hyperboloid. $\mathbb{H^2}$ is a model of a hyperbolic plane (when we define what we mean by a line in our model and we do that by setting $l=\{x\in \mathbb{H^2}:b(n,x)=0\}$ where $n$ is a unit space-like vector in Minkowski space (i.e. $b(n,n)=1$)).
I'm interested in calculating the Gaussian curvature of our model. Book which I'm following says that every hyperbolic plane has constant negative Gaussian curvature. Just by looking at the $\mathbb{H^2}\subset\mathbb{R^3}$ I see it's Gaussian curvature should be positive.
But ok, let's do some calculations. I parametrized $\mathbb{H^2}$ as follows:
$\varphi:\mathbb{R^2}\longrightarrow\mathbb{H^2}\varphi(u,v)=(\sinh(u)\cosh(v),\sinh(v),\cosh(u)\cosh(v)$
Using the formula for Gaussian curvature $K=\frac{eg-f^2}{EG-F^2}$, where $E,G,F$ and $e,g,f$ are coefficients of, respectively, first and second fundamental form, I got $K=\frac{1}{({1-2z^2})^2}$
But than I thought I shouldn't be using the standard inner product when calculating $E,G,F,e,g,f$. I should use the Minkowski inner product.
So, using the same parametrization but the Minkowski inner product I got:
$E=\cosh^2(v)$, $F=0$, $G=1$.
$N=\frac{\partial_1 \times \partial_2}{\vert\vert{\partial_1 \times \partial_2}\vert\vert}=(-\sinh(u)\cosh(v),-\sinh(v),-\cosh(u)\cosh(v))$
$e=\cosh^2(v)$, $f=0$, $g=1$
So, $K=1$.
What am I doing wrong? I know I should get $K=-1$ but how? And if my idea of using Minkowski inner product is correct could someone provide intuition why?