Hyperbolic Geometry & Special Relativity

Ivar Thorson bio photo By Ivar Thorson

While studying An Introduction to Particle Physics on a recent flight to Japan – studying math that is way above my head is a nerdy hobby of mine – I came across a simple viewpoint on special relativity that really stunned me:

Consider a spatial rotation around, say, the z-axis (or, equivalently, mixing the x and y coordinates). Such a transformation is called an Euler Transformation, and takes the form

\[\begin{eqnarray*} t^{\prime} & = & t\\ x^{\prime} & = & x\cos\theta+y\sin\theta\\ y^{\prime} & = & -x\cos\theta+y\sin\theta\\ z^{\prime} & = & z \end{eqnarray*}\]

where \(\theta\) is the angle of rotation, called the Euler Angle. We can simultaneously express a Lorentz transformation as a sort of “rotation” that mixes a spatial dimension and a time dimension, as follows

\[\begin{eqnarray*} t^{\prime} & = & t\cosh\theta-x\sinh\theta\\ x^{\prime} & = & -t\sinh\theta+x\cosh\theta\\ y^{\prime} & = & y\\ z^{\prime} & = & z\end{eqnarray*}\]

where \(\theta \) is defined by the relationship \(\beta = \tan(\theta) \).

After staring at that for a while, I was struck by the analogy between the “mixing” between coordinates when we rotate something, and the relativistic “mixing” between time and space when something accelerates. Although this is just an analogy and perhaps squeezing and stretching is a better metaphor than rotation in this case, is very tempting to search for the geometry of what is happening rather than just the algebra.

Whether it was the in flight wine or just the jet lag, I found myself lost in thought about the connection between hyperbolic geometry and special relativity – after all, what does it mean to say that our space is curved in a way consistent with hyperbolic geometry? We all study flat, Euclidean spaces in school, and as a roboticist who worked a lot with rigid body dynamics I became pretty comfortable with the concepts surrounding rotation matrices, quaternions, and SO(3) groups, I but I don’t ever recall using the hyperbolic cosine even once in my life. This was surprising to me, as I use math on a fairly regular basis.

To try to clear the matter in my mind, I made a little table of some of the definitions of sine, cosine, and their hyperbolic equivalents:

\[\begin{eqnarray} \sin(x) & = & \frac{1}{2}(e^{ix}-e^{-ix}) & = & \sum_{k=0}^{\infty}\frac{(-1)^{k}x^{2k+1}}{(2k+1)!} \\ \cos(x) & = & \frac{1}{2}(e^{ix}+e^{-ix}) & = & \sum_{k=0}^{\infty}\frac{(-1)^{k}x^{2k}}{(2k)!} \\ \sinh(x) & = & \frac{1}{2}(e^{x}-e^{-x}) & = & \sum_{k=0}^{\infty}\frac{x^{2k+1}}{(2k+1)!} \\ \cosh(x) & = & \frac{1}{2}(e^{x}+e^{-x}) & = & \sum_{k=0}^{\infty}\frac{x^{2k}}{(2k)!} \end{eqnarray}\]

Look at how the hyperbolic trig functions have no need of the imaginary number \(i=\sqrt{-1}\) or the alternating -1’s in the infinite series. One of the most frustrating things about \(i\) is that it is so incredibly useful algebraically, but it also makes things more cumbersome because expressions in general become more – pardon the pun – complex. The geometric meaning of the imaginary number is often associated with rotations, but I wonder if there are more geometric ways of understanding the same operation that have no need for the imaginary number per se. I would prefer to see mathematical expressions that are as simple as possible, but no simpler.

Even now, as I stare at the infinite series for hyperbolic sine and cosine, and their definitions in terms of the all-important exponential function, I wonder if in fact that “things with the imaginary number are wrong” in the same way that “pi is wrong”. It is not that sines are incorrect in an algebraic or mathematical sense, because they are clearly not. Rather, I am wondering whether, due to the way mathematics developed historically, perhaps we are using algebra that is more cumbersome than necessary to describe two legs of the same geometric elephant.

I would welcome good book recommendations connecting the essential geometry of hyperbolas to special relativity!