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Hyperbolic Rotations
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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
(9) 
where 
Look at the first row of matrix (9), and note that:
(10) 
which means the point (γ, -vγ) must lie on the unit hyperbola. But in
that case, if we define φ as
(11)
then we find that
And then the
Lorentz transform can be represented as
(13) 
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
(15) 
or
(16) 
or
(17) 
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
(19) 
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