I am in general confused about what Minkowski space is. I'll write down what I know and what I believe Minkowski space is. I'd appreciate any corrections.
A Riemannian manifold is a manifold (so it locally looks like $\mathbb R^n$) equipped with a non-negative positive symmetric bilinear form (the metric).
Hyperbolic space is a type of Riemannian manifold, where it locally looks like $\mathbb R^n$, but globally the space has negative curvature. This gives it all of the weird properties we know and love [geodesics getting exponentially farther away, thin triangles, etc.]
'Minkowski space' naively speaking is some space $\mathbb M \equiv \mathbb (\mathbb R^4, d)$ equipped with the metric $d(p, q) = p_0 q_0 – p_1 q_1 – p_2 q_2 – p_3 q_3$. This looks exactly like the hyperboloid model of hyperbolic space. So it far to say that Minkowski space is literally the hyperboloid model of hyperbolic space?
Next, a 'Lorentzian manifold' is a pseudo-riemannian manifold which locally looks like Minkowski space $\mathbb M$ [contrast with the Riemannian manifold which locally looks like $\mathbb R^n$]. Globally, it is is given by a manifold which is equipped with a non-degenerate symmetric bilinear form: note that here, the metric can be negative definite.
When we talk about a 'flat Lorentzian manifold', we are talking about how the different 'local Minkowski spaces' fit together. A flat Lorentzian manifold is still hyperbolic, because minkowski space is hyperbolic. Rather, the flat here refers to the fact that there is no curvature across the local Minkowski spaces fitting together. So we are to imagine many copies of Minkowski space, each of which fit together 'perfectly', and hence there is no curvature. But locally, the manifold is Minkowski, and thus has constant negative curvature 'at each local point'. Wikipedia talks about the phrase [locally flat
Is this correct? am I completely off? I find this very confusing, because Wikipedia keeps talking about float Lorentzian manifolds. To quote:
Just as Euclidean space $\mathbb {R} ^{n}$can be thought of as the model Riemannian manifold, Minkowski space $\mathbb {R} ^{n-1,1}$ with the flat Minkowski metric is the model Lorentzian manifold.
My understanding of the situation is that because in a Pseudo-Riemannian manifold we can have the metric be negative, we can simply set the metric to $diag(1, -1, -1, -1)$ and get hyperbolic space. This is flat because the second derivatives vanish (indeed, the first derivatives vanish), and hence the space cannot have curvature. On the other hand, in the Riemannian case, we need to setup the hyperbolicity through curvature by assembling copies of $\mathbb R^n$.
Is what I have written sane, or am I completely off the mark? I'm looking for clarifications and spotting mistakes in my mental model of the physics I am studying with the math that I know.
Best Answer
The first thing to know is that the sign of the Lorentzian metric on Minkowski space, which is well adapted for applications to special relativity, is poorly adapted for applications to hyperbolic geometry. If you want to construct the hyperboloid model of hyperbolic space, start instead with the Lorentzian metric of opposite sign: $$d(p,q) = -p_0 q_0 + p_1 q_1 + p_2 q_2 + p_3 q_3 $$ In order to get around this confusion for the isolated purposes of this answer I am going to do something terrible and henceforth refer to this as "anti-Minkowski space".
The second thing is that anti-Minkowski space is not literally the same as the hyperboloid model. Instead, the hyperboloid model is one of the two components the subspace of anti-Minkowski space that is cut out by the equation $d(p,p)=-1$, namely the component containing the point $p = (1,0,0,0)$, and thus it is one of the sheets of the two-sheeted hyperboloid $-p_0^2 + p_1^2 + p_2^2 + p^3_3 = -1$, or equivalently $p_0^2 - p_1^2 - p_2^2 - p_3^2 = +1$. This leads to several important differences:
So, your paragraph "When we talk about..." is quite a bit off the mark.