Solving problems in special relativity with quaternions

Subject: Solving problems in special relativity w/quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/03/27
Message-Id:
Newsgroups: sci.physics.research
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Hello:

Back in September, I said I was taking 8.033 at MIT, Classical and
Relativistic Mechanics. To make the introductory class more fun, I
tried to solve all the assigned homework using quaternions. In this
post, I will explain why I was able to succeed. To back up the claim,
there is now a web site with over 50 problems solved explicitly using
quaternions, hence the following caveat:

[Moderator's Note : Caveat emptor. The moderators of
sci.physics.research make no claim as to the content of this web
page.]

Consider two quaternions q and q':

q = t + x I + y J + z K

q' = gamma t - gamma beta x I + y J + z K .

where beta is the relativistic velocity and gamma is 1/(1- beta^2)^1/2
as is usually defined in special relativity. If q is an event, then q' is
that event seen by an observer boosted along the x axis using a
Lorentz transformation. Since quaternions are a skew field or
division ring, there exists a quaternion L such that

L q = q' .

L performs the work of a Lorentz transformation, but a quaternion is
not a member of the Lorentz group, because members of the Lorentz
group are not quaternions. At this point, it can be said that
any problem in special relativity can be solved using quaternions
because the problems are always some application of Lorentz
transformations. The details have been worked out explicitly for
over 50 problems at the web site:

http://world.std.com/~sweetser .

The problems include: kinematic effects of special relativity, the
Lorentz transformation, additions of velocities, the Doppler effect, 4-
vector invariants, the twin paradox, energy-momentum-mass
problems, the Compton effect and threshold collision problems. For
anyone bored with teaching special relativity the standard way,
there is an alternative approach which can be used for the typical
range of problems assigned to undergraduates.

One of the people teaching the class, Prof. Randall of MIT's Center for
Theoretical Physics, said, "Cute, but so what?" (tough audience!)

At the time, I could only appeal to Occam's razor. There are three
tools required of increasing complexity to solve problems in special
relativity: 4-dimensional vectors which can be added together or
multiplied by a scalar, a Minkowski metric so that any two 4- vectors
can be multiplied together, and the Lorentz group to transform the
4-vectors but preserve the inner product. Using quaternions, one
needs to explain the properties of a 4-dimensional mathematical
field which involves addition and multiplication of elements in the
field. The rest is an application - not a trivial application - but
nothing more is needed. If events in spacetime are described with
terms that can be added and multiplied together, then the
Minkowski metric and Lorentz group become unnecessary.

This approach is distinct from using Lorentz transformations.
Quaternions can be represented as Pauli matrices which are like the
generators of the Lorentz transformations to within a factor of i.
Complex-valued quaternions have been used as substitutes for the
Lorentz group. This is _not_ my proposal. Complex-valued
quaternions are not a mathematical field, and it is the power of a
mathematical field I am trying to harness. The Lorentz group,
however it is represented, takes a 4 D vector and transforms it into
another 4 D vector. It will not accomplish the task proposed: to take
a 4 D element of a field into another element of the 4 D field using
the same 4 D field.

To address Prof. Randall's complaint, I've added a few more powerful
tools. Consider the quaternion differential operator:

d/dt + d/dx I + d/dy J + d/dz K (all partial derivatives)

Using the properties of a mathematical field, it is possible to start
from a 4-dimensional number and get to any other 4- dimensional
number. It is also possible to start from this 4-dimensional operator
and get to any other 4-dimensional differential operator employed in
physics (the road may be very challenging). So far, I've built
operators that generate Newton's second law in an inertial and a
rotating reference frame, the four Maxwell equations as one
quaternion wave equation, and the Klein-Gordon equation as part of
a quaternion simple harmonic oscillator. There is even a thread to
the affine parameter of general relativity by calculating the interval
as was done for special relativity, but allowing the origin to move at
the same time.

For those who visit the site, you can expect to see Mathematica 3.0
notebooks that have been converted to HTML for easy web browsing.
I attempted to make the work look more like math than like
Mathematica, but there is a file about understanding the notebooks.
Mathematica was used to assure that the proposed solutions and
derivations work precisely as promised. The smallest error is very
evident using a computer package to check the math. The pages lack
an intellectual depth one might expect given these topics: there are
few references previous work in the area or to the literature. That is
due to the author's lack of formal training. The web pages are a
"how to use" quaternions to understand a broad range of physics, not
a "why this works compared to what we already know". If more
learned people read the pages and find it a valuable approach, this
weakness might be addressed someday.

In the mean time, I will continue to apply quaternions to a broad -
and broadening - class of problems in physics. It has been a
productive six months.

Doug Sweetser
http://world.std.com/~sweetser

4 D fields of numbers are fun
4 D fields of operators are dangerous