The curvature quaternion is a sum of two-forms with quaternion coefficients. Each quaternion measures the curvature in each of the six orthogonal planes of four dimension space-time. Define the following projection quaternions

Referring now to the decomposition of the quaternion as a sum of two-forms; define the Einstein quaternion three-forms:

Geometrically, the coefficient of each three form dx_idx_mdx_n is the sum of the normal components of in the respective three-dimensional subspace. The signs of the R_i components insure consistent application of the right hand rule. For example, in cartesian xyzt coordinates, the coefficient of dxdydz is Qxy_z+Qyz_x+Qzx_y.

The normal components of Qxy,Qzx,Qyz 

We can relate Q_einstein to the components of the Einstein tensor defined in traditional tensor notation:

Energy three form

In relativity, a distribution of energy is characterized by a tensor

We can express this energy momentum tensor as a sum of four three-forms with Hermitian quaternion coefficients:

To understand the various components of the energy momentum, first consider a uniform distribution of n partcles per unit volume, each having rest mass m and all at rest in some region. There is only one non-zero component of the energy momentum tensor: the energy density of the particles . That is is the total energy in the volume element dxdydz at any instant of time. Now transform to a reference frame moving at relative velocity v. let

then the new components of the energy momentum tensor are:

If we had started with a group of particles not at rest,but having randomly distributed velocities with root mean square value v_rms then T₁₁,T₂₂ and T₃₃ would have terms corresopondng to pressure p=½ n m v_rms² . One can further generalize the tensor to cover situations where the particles interact,other forms of energy such as electromagnetic, cases where the number of particles is not constant, etc. The only form of energy that we don't include is the gravitational field itself. Concerning this last point, attempts to define a locally conserved energy density remain controversial. At the risk of oversimplifying we can say that the energy of the gravitational field is a fictititious quantity. Just as a distribution of matter (or a partcile if you wish)can take on different values of kinetic energy depending on the velocity of an oberservor, matter in a gravitational field can take on different values of energy depending on the reference system; however the reference system itiself is determined by energy content. See the Usenet Relativity FAQ Is Energy Conserved in General Relativity-for more.

Einstein Law of Gravity

A distribution of energy specified by the energy density three form W induces space-time curvature described by Q_einstein according to

The constant k has the approximate value k=2.07 10 ^-48 cm/erg.

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_{Rev 5/20/0 Copyright ©2000 George Raetz <bestwork.1@pcisys.net>}