What is the relationship between hyperbolic geometry and Einstein's special relativity?

Question

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic geometry, also known as the Minkowski model, takes place in Minkowski space (which I don't understand well) which is also the most convenient setting to formulate Einstein's theory of special relativity. Can someone clarify this for me? It has been pointed out to me that velocity in special relativity is a point in hyperbolic space, so I would like to use techniques and formulas in hyperbolic geometry to discuss things in special relativity.

Answer 1

The connection here is the Minkowski space, which can be used to describe both.

Hyperbolic geometry

For example, take hyperbolic 2-space in the hyperboloid model. You'd represent hyperbolic points as points on the hyperboloid, namely as

$$\left\{(x_1,x_2,x_3)^T\;\middle|\;x_1^2-x_2^2-x_3^2=1\right\}$$

This expression $x_1^2-x_2^2-x_3^2$ is the quadratic form which lies at the foundation of the Minkowski space $\mathbb R^{1,2}$. The corresponding bilinear form can be used to compute distances, as

$$d(x,y)=\operatorname{arcosh}\left(x_1y_1-x_2y_2-x_3y_3\right)$$

You can even think about this projectively: you may use a vector which does not lie on the hyperboloid, then use that vector to define a line which will intersect the hyperboloid in a given point, which is the point it specifies. This will work for every vector whose quadratic form is positive.

This idea is also very useful to define lines. A hyperbolic line (i.e. a geodesic) connecting two hyperbolic points is modeled by the intersection between the hyperboloid and a plane spanned by these two points and the origin. You can describe this plane by its normal vector, and you can compute that normal vector as the cross product of two vectors representing the two points. Conversely, you can obtain the intersection between two geodesics by computing the cross product between two normal vectors of such planes, although the quadratic form for that point likely won't be $1$ yet, but any other positive value instead. Therefore, this normal vector of the plane is a reasonable (and homogenous) representation of the line. Its quadratic form will be negative.

Common vocabulary

Now change the vocabulary to use terms which are common for Minkowski spaces. A vector whose quadratic form is positive is said to be time-like. So points of the hyperbolic plane correspond to time-like vectors, with scalar multiples of a vector representing the same point. Likewise a vector whose quadratic form is negative is called space-like. So a line in hyperbolic geometry corresponds to a space-like vector, and all its multiples. In between these two, there are those vectors for which the quadratic form is zero. These correspond to ideal points of your geometry. in a certain sense, an ideal point is as much a line as it is a point.

The set of hyperbolic isometries are those linear transformations of your vector space which preserve the set of ideal points, i.e. which preserve the light cone. These correspond roughly to Lorentz transformations in relativistic vocabulary (with some care because here we identify scalar multiples but there we don't, but the central idea of preserving the light cone remains).

Relativistic geometry

So where do these physically sounding terms come from? Imagining the whole vector space as some kind of space-time-diagram should be fairly simple. The first dimension (with the positive sign) would be time, the other two would be space. A vector would denote an event in this diagram. An event where all spatial coordinates are zero would happen at the same place as the origin, but at some different time. An event with zero time coordinate would happen at the same time but at some other place. The light cone would correspond to a cone of slope $1$, which is the speed of light in our coordinate system. Light travels from the origin along the light cone.

But time and space are relative, so the above choice of coordinate system is only valid for a given inertial system. To convert between inertial systems which meet at the origin, you'd again use a Lorentz transformation, i.e. a transformation which preserves the light cone. Using such a transformation, any event which is a time-like distance away from the origin can be made to happen at the same place but in the past or the future. You'd use an inertial system which travels to that event or came from it. Likewise, any event which is a space-like distance away can be made to happen at the same time, using the right movement to compensate.

Conclusion

So conversions between inertial systems correspond to isometric transformations of the hyperbolic space. And objects in hyperbolic space correspond to (equivalence classes of) events in space-time diagrams.

The above would generalize for higher dimensions, but the part about two points spanning a line would be more complicated to read, since you'd more likely talk about three points spanning a plane. $ $ $ $