We have already given examples of quaternion functions in the introduction. Generally speaking we associate with each location qe in space time another quaternion Qf(qe). We restrict the domain qe to be Hermitian quaternions, because Hermitian quaternions correspond to space-time event locations(The Clifford algebra formation explains why). In relativity the important functions seem to be Hermitian or unimodular.
Consider a Quaternion function Qf(q). It takes on the value Qf0 at the point q0 The value of Qf at a neighboring point q0+ dq is Qf0+dQwhere dQ is given by the usual gradient formula:
We call this a quaternion one-form .Consider the general quaternion one-form
The exterior derivative of this one form is the two-form
We can also define exterior products of differential form expressions; however extra care is needed here because quaternion and differential form multiplication do not commute. To make things clear: consider a small rotation effected by the quaternion I+ Qr acting upon the general quaternion Qg.
(The equality is taken to be in the limit of Norm(Qr)--> 0.)
In General Relativity we will need to apply a differential quaternion operator representing a rotation to another differential quaternion representing a coordinate displacement. To achieve a meaningful result it is necessary to preserve the order of the differentials when applying the above formula.
Specifically, let H represent an Hermitian quaternion field. Values of H in the neighborhood of some location q0 are differentially defined by
Now consider another unimodular quaternion field R which is differentially defined by I+dR:
The effect of rotating H by the quaternion field R is differentially defined by the following two-form
Up to now we have dealt with the components of a quaternion. The components
are defined relative to a basis ie an orthonormal tetrad of unit
quaternions; but we have not specified how this basis is defined. It is
postulated (and experiment seems to confirm) that space time is a metric
geometry. Fundamental to a metric geometry is existence of a distance function
of any two points.(In ordinary geometry the distance function should be
positive,symmetric for distinct points, zero for coincident points, and
satisfy the triangle inequality, In space time geometry these requirements
are modified) Start with two dimensional geometry on a (curved) surface.
Take any two locations O and Ox; if they are close enough together, then
we can find a path connecting them such that the distance function is minimum.
We call path a geodetic; the resulting distance the geodetic distance.
We can mark off intervals of equal distance along this geodetic and construct
a ruler this way. Now take a third point Oy not on the geodetic. Find the
geodetic path of least length connecting Oy to the original geodetic. We
now have a orthogonal coordinate axis. Displacements of unit distance along
these axis constitute basis vectors. In the following diagram, the coordinates
of point P relative to point O in the basis O-Ox,O-Oy is (x,y)because the
geodetic P-x is the shortest geodetic from P to any point on O-Ox and likewise
for P-y. Also, for small x and y, the Pythagorean theorem applies: (P-O)2=(O-x)2+(O-y)2+..(terms
of order (x/R)4,(y/R)4) . What do we mean by small
x and y?: Small compared to the radius of curvature ,R of the surface.
The radius of curvature of the surface can be determined entirely from
measurements made on the surface so it is not necessary (but might be helpful
at first) to consider the surface as "embedded" in a a space of higher
dimensionality. It is also not necessary to define the basis vectors as
differential operators.
Refer to the two dimensional example
for
an introduction to moving frames on a curved surface.
In the world of space-time, distance is measured with a clock: take
any two events in space time transport a clock from one to the other along
some path; the measured time interval is the 'length' of that path. L length
is assigned to paths that are physically impossible (space like intervals
etc.) by a system of synchronized clocks and rulers. As outlined above
a system of moving orthonormal reference frames is established in region
of space-time. Consider a small displacement. dr from the origin of the
tetrad. Write dr as a sum of differential forms 
with
the basis Pauli quaternions s as coefficients
For example consider the schwarzchild geometry expressed in the traditional notation:
In quaternion differential form notation , we express this metric as
the actual displacement corresponding to a given coordinate change. We
call the scaling quaternion):
As we move in various directions the basis quaternion tetrad will rotate in accordance with the curvilinear coordinate system . The coefficients of sigma must also rotate contravariantly. The change in sigma can be described as a small rotation effected by another quaternion which we call Omega. Omega is a quaternion one-form determined from:
As already mentioned care must be taken to preserve the order of the differentials.
Write omega and sigma as:
Then the exterior product will be of the form:
The exterior derivative of Omega leads to the curvature quaternion 2-form theta:
Continuing with the Schwarzchild example , the exterior
derivative of the scaling quaternion
is:
The corresponding rotation quaternion is:
The curvature quaternion is:
Geometric interpretation of the curvature quaternion
Write the curvature quaternion Theta as as a sum of differential two forms with quaternion coefficients.
Then the
are
rotation quaternions that represents the amount of rotation a vector undergoes
when parallel translated about the rectangles defined by unit coordinate
displacements dx mdxn. These little rectangles will
change in size as we move about the space time manifold according to the
coordinate scaling--that complicates the formula. We gain better insight
into the symmetries of the manifold.by displacing along rectangles of uniform
size. We do this by scaling each of the differential two forms in the distance
units of the orthonormal basis; that is write the curvature quaternion
in the form::
In the case of the Schwarzchild metric we have:
Relation between the Riemann curvature tensor and the curvature quaternion
In actual distance units 
can be written as
Where the Rijmn are the components of the Riemann curvature tensor. Note that the symmetries Rijmn=-Rjimn and Rijmn=-Rijnm are immediately obvious: the first originates from the anti symmetry of an infinitesimal rotation matrix of which the Rijmn are components; the second from the anti symmetry of the exterior product.
Table of Contents BackTwo
Dimension Example Next
Rev 5/31/0 Copyright ©2000 George Raetz <bestwork.1@pcisys.net>
