Einstein's Relativity has been around for 100 years and Physicists have got to know and like it. But is it not time to really look at alternatives.

Quaternion Relativity differs from Einstein's Relativity, Here is the Quaternion Formulation in general:

Here is a basic operation in Relativity.

Transform (Rotate) Point 2 around Point 1

(order counts in Quaternions, this is called non-commutativity, AB is not BA).

P1P2=(ct1 + ix1 + jy1 + kz1)(ct2 + ix2 + jy2 + kz2)=

(c^2t1t2 - (x1x2 + y1y2 + z1z2) ) + ct1(ix2 + jy2 + kz2) + ct2(ix1 + jy1 + kz2) + (ix1 + jy1 + kz1)X(ix2 + jy2 + kz2)

(whereXis the cross product of vectors that I will not multiply out here.)

Now we see the the first parentheses is like the 4-vector Interval:

(c^2t1t2 - (x1x2 + y1y2 + z1z2))

Note that there are other terms in Quaternion Relativity. Note that the cross product term can be zero if the vectors are parallel! Thus Einstein's Relativity to be for parallel vectors only. However, the points ct1(ix2 + jy2 + kz2) and ct2(ix1 + jy1 + kz1) are still missing from Einstein's Relativity Theory.

What happens if the the vectors are perpendicular? Then x1x2 + y1y2 + z1z2 =0 and the Interval reduces to "c^2t1t2" and the cross product term that is missing in Einstein's Relativity comes back. The cross product term is non-commutative like Pauli's Spinors and introduces non-commutativity into Relativity.

Finally the Interval can reduce to zero if c^2t1t2 = x1x2 + y1y2 + z1z2, however, ct1(ix2 + jy2 + kz2) and ct2(ix1 + jy1 + kz1) and the cross product non-commutative term is not zero. I hold as a point of physics that the Interval is never negative, namely that the speed of light is the limit and c^2>=(x1x2 + y1y2 + z1z2)/t1t2= v1v2.

Now you see the difference between Quaternion Relativity and Einstein's Relativity.

Here is another view of the Quaternion Relativity that might be more familiar:

(cosA + asinA)(cosB + bsinB) = cos(A + B) + csin(A+ B) =

(cosAcosB - sinAsinBcos(ab)) + asinAcosB + bsinBcosA + sinAsinBsin(ab)axb

This last formulation of Quaternion Relativity is quite familiar to those familiar with "Complex Number Theory". However, the last cross product term"sinAsinB sin(ab)axb" is not in complex numbers because complex numbers are commutative while quaternions are non-commutative. Non-commutativity is key to Quantum Theory and is also important in physics in general, whirlpool, handedness, etc.

Relativity is about Transformations aka Rotations in four space. Quaternions are the premier mathematical tool for rotations. There are differences in Quaternion Relativity and Einstein's Relativity Theory.

Einstein among his many achievements brought the idea of 4 dimensions into physics. This was revolutionary. However, there is only one 4 dimensional Associative Division Algebra, Quaternions. Associative means A(BC)=(AB)C.

In fact quaternions is uniquely the only Associative Division Algebra. The two other associative division algebras are Real Number Algebra and Complex Number Algebra, and they are commutative subsets of Quaternions.

The only other Division Algebra is the non-Associative Octonions, a 8 Dimensional Division Algebra.

There is more to Quaternions but this should give one reason to re-examine Einstein's Relativity Theory, there are things missing. Einstein's Relativity is related to situations where the angles A+B are multiples of pi (180 degrees) where the result is a real number. When angle A+B=multiples of 90 degrees, the result is vectors.