Some insights
Hyperbolic Rotations
In elementary calculus they breezed past the hyperbolic functions with hardly a glance -- they seemed to be just a curiosity used by bridge builders.  In advanced math classes they were never mentioned at all (too elementary, I guess).  And then in relativity, suddenly they're just taken for granted; you should have picked them up by osmosis somewhere along the line.

That's how it was for me, anyway.  So after digging around in textbooks and working some things out for myself, I put this page together in case anyone else has encountered the same problem.  On this page, I'm going to consider just 2 dimensions (1 space and 1 time dimension).  My goal is to provide a little background on hyperbolic functions and rotations, and make them seem reasonably natural when they're used in relativity.

Definitions of the functions

The standard math class definition is that cosh and sinh are, respectively, the "even" and "odd" parts of the exponential function.  As such, cosh is even, sinh is odd, and they sum to the exponential function.  The formulas for the most important hyperbolic functions are:

A moment's calculation shows that

which is the equation of the unit hyperbola.  (The fact that u = 2 * the area shown in the graph is shown here, by simple integration):


The inverse hyperbolic functions are named with an ar prefix, as arcosh(x), to indicate that they return the area associated with that value of the function: it's short for "area of the cosh".  You may also occasionally see "arcsinh" rather than "arsinh".  As far as I know, that usage is just erroneous, and results from confusion with the "arc" in the inverse circular functions.

Hyperbolic Rotations

A hyperbolic rotation is what we get when we slide all the points on the hyperbola along by some angle.  To rotate a hyperbola by v, for example, we'd map each point on the unit hyperbola (cosh(u), sinh(u)) to (cosh(u+v), sinh(u+v)).  This is exactly analogous to a "circular rotation", in which we slide all the points on a circle around by some number of radians.

A hyperbolic rotation by φ is:

as we can easily verify.  Given a point (cosh(θ), sinh(θ)) which lies on the unit hyperbola, when we apply the rotation to it we obtain:

Writing out the two terms in the product in exponential form using (1a) and (1b), we see cosh(θ) transforms to:

and sinh(θ) transforms to:

Since hyperbolas which intercept the X axis at locations other than 1 are just a constant times the unit hyperbola, they will also be "rotated" by a hyperbolic rotation, as will hyperbolas about the Y axis.

The Lorentz Transform

The Lorentz transform, with c=1, is



Look at the first row of matrix (9), and note that:


which means the point (γ, -vγ) must lie on the unit hyperbola.  But in that case, if we define φ as


then we find that

And then the Lorentz transform can be represented as

Now matrices (9) and (13) are equal.  If we take the sum of the terms in the top row of (9) and equate it with the sum of the terms in the top row of (13), we can find the hyperbolic angle of the rotation, φ, in terms of the velocity:

Composition of Velocities in 1 Spatial Dimension

Suppose an object is moving at velocity u in frame F' and frame F' is moving at velocity v relative to frame F.  Then the velocity of the object in the F frame is just the composition of the Lorentz transforms from the object's frame to frame F' and from frame F' to frame F, applied to its rest-frame velocity of (1, 0, ...).  In terms of hyperbolic rotations, this is






We can work out sum of the angles in terms of u and v using equation (14c):

Comparing (18e) and (14c) we can see that the velocity viewed from frame F must be


which is special relativity's "composition of velocities" law.

Preservation of the Metric

In 1 (spatial) dimension, the metric is

or  -dt2 + dx2.  All points the same distance δ from the origin form a hyperbola, with the equation

Since a hyperbolic rotation maps each hyperbola onto itself, it will therefore map points of a particular distance from the origin to other points the same distance from the origin -- it preserves the Lorentz metric

Note that I haven't shown that hyperbolic rotations preserve the Lorentz distance between any arbitrary pair of points, which is also true.  Nor have I said anything about the behavior of hyperboloids in 4-space; I've just looked at 1 spatial dimension and 1 time dimension.  The goal, however, was just to provide some background on hyperbolic rotations and make them seem natural in the context of relativity.

Page created in 2004, and last updated on 11/15/06