Commutative Hypercomplex Special Relativity

© 2000, 2007
Clyde M. Davenport

2-D Formulation 4-D Formulation Elementary Properties
Einstein's Postulates Relativistic Effects Unorthodox Effects Conclusions


In order to understand the following, the reader should first review the Hypercomplex Math page. It describes a system of commutative 4-D numbers of the form Z=1x+iy+jz+kct [not quaternions] that behaves in all ways like the 2-D classical complex variables. It is based upon elementary group, ring, and matrix principles.

We are motivated to attempt a formulation of special relativity in terms of commutative hypercomplex mathematics by one highly suggestive aspect of the mathematics, alone. The overall four-space is made up of the script D-space and a separate, closed subspace union of O+,O-composed of all the 4-D elements that are noninvertible in script D. That is not so remarkable until we realize that to be in script O, an element must have some coordinate value x= ±ct. This means that we are describing motion at the speed of light. It is also a description of the light cone of special relativity. Physics tells us that no material body can reach the speed of light. The script D-space mathematics tell us that script O is a closed subspace. For any material body to enter script O, it would have to reach the speed of light, reduce to two dimensions, and enter either the or plane. Similarly, physics tells us that as a material body moves, it foreshortens in the direction of motion, and in the limit as its speed approaches c, it flattens into a plane. We conclude that a key part of special relativity theory is intrinsic to the commutative hypercomplex mathematics.

In the following, we shall present a brief sketch of the hypercomplex formulation for the Internet reader. Refer to Einstein(1), 1923 or Bergmann, 1942 or any current treatment of elementary special relativity for more details on the basic subject, and the Hypercomplex Math page and Davenport(9), 1991 for more details on the hypercomplex formulation. Top

A 2-D Formulation

A script D-space formulation is readily constructable. In one space dimension and time, the following expressions give the traditional view of special relativity:

2-D special relativity

The relative motion v is always assumed to be along the +x direction, and any observer is assumed to be at the origin of coordinates. This means that only radially inward or outward motion is considered. In hypercomplex notation, this is

2-D hypercomplex relativity

The object T is a coordinate transform, as indicated in the last line. Here, it is applied in the vector notation, but it could be written in the 4 X 4 real matrix notation and applied in a traditional vector-matrix equation. However, the 4-D hypercomplex formulation is far more computationally efficient. Top

Notice that we applied the transform to only 2-D (x,ct) vectors. Nothing is said about what might be happening to the y and z coordinates that lie transverse to the line of motion. One may review Einstein's original paper on special relativity [ Einstein(2), 1905] to see that he assumed that the transverse coordinates might be affected in magnitude but not direction, and went on to deduce under those circumstances that magnitude, also, was not affected. We could do as he did and set y'=y, z'=z and declare that our hypercomplex formulation is complete. It would be correct according to the current physics paradigm.

Note also that the above formulations do not say how a given situation would actually appear to the aided or unaided vision of an observer, but state only how any measurements of objects and events would be affected due to the relative motion. In particular, nothing is said about the distance from observer to object or about the geometry and orientation of the object. Moreover, in these restricted-case formulations the observer is looking end-on at the object, so will "see" very little. Top

A 4-D Formulation

The above 2-D formulation is not completely general because it is not given in terms of a full 4-D motion vector whose line of action can be in any direction and may or may not go through the observer's origin of coordinates. We would like a concise hypercomplex formulation that allows the moving inertial frame (constant, linear motion) to move in any direction with respect to the observer's coordinate frame. It can be constructed as follows. Let

4-D position vector

be the 4-D position vector of the origin of the moving coordinate frame. Let the following dot notation indicate the ordinary derivative with respect to the observer's time. Then, the 4-D relative motion vector as measured by the observer (we can't explain how, here; it is a nontrivial problem) is Top

4-D relative velocity

The moving coordinate frame is considered to move in a parallel fashion with respect to the stationary frame, but their origins may or may not ever be coincident. The line of action may stand off at some distance from the origin of the stationary coordinates. Now, hold that notation while we do the following. The relativity transform T, above, can be rearranged by use of script D-space algebra and function rules to get it into a function of the relative motion vector, only:

T function of 2-D V

Note that, in the last step, we divided both the numerator and the denominator under the radical by the constant, k. This last form visually emphasizes that the relative motion v is always in the 1 (x-coordinate) direction in the traditional treatment. We want to allow it to be in any arbitrary direction. Looking at the above form, we notice that the signs are such that if we wrote both vectors in the canonical form, then the one in the numerator would be the same as the one in the denominator, except with the eigenvalues interchanged. We continue this property while adding in the two missing components for an arbitrary relative motion vector, as follows: Top

T function of 4-D V

We have used the notation

T eigenvalues

Note that the eigenvalues of T (the radical quantities) are mutual inverses. This will be important, below. Top

By use of standard hypercomplex math methods, T can be expanded into 4-D vector form, as follows:

T vector form

Elementary Properties

We do not know at this point whether the above additions are acceptable and will produce the correct consequences. We do know that the modified transform contains the traditional 2-D formulation, because if we simply set the y,z components of the 3-D velocity to zero, then we are back to the 2-D formulation. We shall proceed by examining the properties and consequences and seeing if they agree with what is known from experimental physics. Top

We begin with some very minor and elementary, but nevertheless necessary, properties of the transform T. We cannot take the space to prove the following assertions, here, see Davenport(10), 1991 for the details.

(a) The transform is defined for all values of the 4-D relative motion vector for which det(V) is nonzero. This follows from the properties of the hypercomplex algebra, and is true everywhere the three-space speed (the magnitude of the 1,i,j components) is less than c (i.e., there is no possibility of rotating the spatial coordinate frame so that some component of velocity is c or greater; this is required by the known laws of physics).

(b) As mentioned above, the transform reduces to the traditional formulation if the i,j components of 4-D relative motion are set to zero. For a given relative motion V along a given line of action, we can always reorient the stationary frame to get the i,j components to be zero, as follows: Recall that the orientation of V is defined with respect to the stationary frame. Suppose there is a standoff between the stationary origin of coordinates and the line of action. Rotate the stationary frame so that the +y-axis is perpendicular to the line of action (along the line of standoff) and the +x-axis is pointed in the direction of motion. Then, with that orientation, the i,j components of relative velocity are always zero. Of course, if there is a standoff, then there is additional geometry that must be accounted for in any measurements.

(c) The set of all transforms T of the form given above and with det(V) nonzero make up a group. We shall denote it by script T. Top

(d) The transform T can be stated in the form of a 4 X 4 real matrix. Its determinant is always +1, yet T is not explicitly orthogonal; but, see below.

(e) Reversing the direction of the 4-D relative motion vector V leaves the associated transform T unchanged. (But, note that this is not the same as reversing only the three 1,i,j spatial components; time is also reversed.) Also, see next.

(f) Time reversal causes no effect. The position vector of the origin of the constant-motion coordinate frame can be written in the form X = X0+1at+ibt+ jet+kct, where X0 is a constant offset, and a negative sign on t yields a negative (reversed) 4-D relative motion vector V, which leaves the transform unchanged.

(g) Zero spatial velocity (the 1,i,j components of V all zeroed) produces a unit (no effect) transform. Top

Einstein's Postulates

All of the above properties are good, and promising, but now we come to the two most important properties. The transforms T must produce results in accord with Einstein's two postulates of special relativity.

Einstein's First Postulate

The First Postulate is that the laws of physics are the same in all inertial systems. An inertial system is one that is free of acceleration and external forces, but can have constant, straight-line motion. Agreement with this postulate is proved by showing that the equations for the laws of physics remain unchanged in form under coordinate transforms T. We shall do that here. We know that an orthogonal transformation of coordinates leaves the equations of physics unchanged in form, and we intend to show that our 4-D formulation T is a form of orthogonal transformation. The argument might appear disconnected, but it will make sense in the end.

To achieve our objective, we must do some look-ahead, back-and-forth, longitudinal thinking. We recall that the Lorentz transformation leaves the form x2+y2+z2 -(ct)2 invariant, and if we first perform the unitary transformation x'=x, y'=y, z'=z, ct'=ict, where "i" is the classical imaginary, then the form x2+y2+z2+(ct)2 is invariant. In the commutative hypercomplex formulation, we note that the form x2-y2+z2 -(ct)2 is invariant, and we take the clue from the above and apply the unitary transformation x'=x, y'= -iy, z'=z, ct'=ict. The result is that the form x2+y2+z2 +(ct)2 is invariant. Top

How does that help for our present purposes? Well, if we apply the latter unitary transformation (designate it by A) to the coordinate transformation equation X'=TX, we get:

AX' = ATA-1 AX

As shown above, the element T has a vector representation. Additionally, it also has a matrix representation, as do all elements of the script D-space. We shall use the following notation, with U,R,W,S as given above:

T transform representations

Now, if we apply the similarity transform T ' = ATA-1, then we get Top

T similarity transformed

Recall that to find the inverse of any element of script D, one finds the inverse of each eigenvalue. Now, look back at the eigenvalue form of T. If one takes the inverse of each eigenvalue (the radical quantities), the effect is that the eigenvalues exchange places. That is fortuitous, because we know that if we have T in the vector form, then we can interchange the eigenvalues by merely changing the signs of the j,k components:

T inverse

If we use this observation to construct the inverse of T' (the similarity-transformed T), we get:

T' inverse

Finally, we are at our objective: Notice that the inverse of T' is merely its transpose! T' is complex orthogonal. That means that under the right viewpoint (the proper unitary transformation of coordinates) and by virtue of 4-D orthogonality arguments, the group script T leaves the equations of physics invariant. Top

Einstein's Second Postulate

The Second Postulate is that the speed of light in a vacuum is constant, the same in all inertial frames, and is independent of the source and any observer. In the traditional approach, it is built into the derivation of the Lorentz transform that the form x2+y2+z2 -(ct)2 is invariant from frame to frame (i.e., under the Lorentz transformation group). Therefore, if a spherical wavefront expands at a rate given by x2+y2+z2 =(ct)2 in the observer's own inertial frame of reference, with c constant, then it does so equally in all other inertial frames.

We remarked earlier that an invariant quantity of the group script T is x2-y2+z2 -(ct)2 . We can use the unitary transformation of coordinates x'=x, y'=iy, z'=z, ct'=ct, where "i" is the classical imaginary, to transform the latter into the desired form x2+y2+z2 -(ct)2 . We conclude that under the right viewpoint (the proper unitary transformation of coordinates), the group script T leaves the speed of light in a vacuum to be constant and the same in all inertial systems.

For those uncomfortable with the use of unitary transformations in our proofs, note that physicists accept as equivalent any two mathematical formulations that are linked by a unitary transformation that transforms one formulation into the other. It is a widely-used technique, especially in quantum mechanics. Top

Relativistic Effects

Compounding of Velocities

As in the traditional special relativity theory, one finds the effect of compounding (adding) two velocities (for example, the velocity of the origin of the moving coordinate frame, plus a second velocity measured with respect to the moving frame) by multiplying their two associated Lorentz transforms, then deducing the single equivalent velocity that would produce the same effect. In the script D-space case, the product of transforms is

T1 T2 product

What we are after is a single velocity expression that will replace the product of velocities in the denominator. We know how to form the products under the radical. They are Top

T1 T2 quotient under radical

Notice that the product numerator and denominator have the same form as the single-velocity original. The components of the numerator and the denominator correspond term by term, and the j,k signs are negative in the numerator, as in the original. This is a remarkable, fortuitous, and important property. Now that we have verified that the product numerator and denominator have the requisite form and sign patterns relative to each other, we can focus on the denominator to extract an equivalent relative motion vector. Top

These products have the units of velocity squared, but we want simply equivalent velocity expressions. We will have to find a common factor in both numerator and denominator with units of velocity and cancel it from both. Moreover, the k component for a 4-D relative motion vector is always c (in agreement with Einstein's second postulate), so our factoring/canceling should leave c as the k component in both the numerator and denominator. In effect, we are renormalizing to keep the speed of light constant. Looking ahead toward what we will need to be compatible with the Einstein law for compounding of velocities, we see that we will have to divide through by some quantity containing a term of the form . Accordingly, we find that if we divide both numerator and denominator by a quantity involving the 1-component of each: Top

T1 T2 reducing factor

then we get a promising-looking result for the denominator:

T1 T2 normalized denominator

We shall call this expression the generalized Einstein's addition law for relative velocities. Recall that before renormalization, this was the product of two 4-D vectors, V1 and V2. Now notice the remarkable symmetry of the renormalized expression: If one sets the subscripted-2 quantities to zero, then the expression reverts to V1, and if one sets the subscripted-1 quantities to zero, the expression reverts to V2. Similar statements apply to the renormalized numerator. Also, the compounded velocity expression reduces properly to the single velocity case when either 3-D spatial velocity is set to zero. Top

The acid test of this expression is: Does it reduce to the Einstein addition law for velocities when V1 and V2 are aligned in the same direction, along the +x-axis? It certainly does. If we set all y,z quantities to zero (forcing alignment with the x-axis), then we are left with

T einstein's addition law

and the 1-component is precisely the Einstein addition law for velocities, the resultant being aligned along the +x-axis, as required. Top

Length Contraction

We can readily provide a formula for this property. Any 4-D position vector X transforms according to X'=TX. The prime ( ' ) mark refers to a quantity at rest in the moving system, that is measured in the observer's (at rest) system. In all cases below, the unprimed coordinates refer to the (arbitrarily-assumed) stationary frame of reference, and the primed quantities refer to the (arbitrarily-selected) moving frame of reference. An interval vector transforms the same way, with U,R,W,S as given above: Top

T 4-D coordinate transform

From the traditional treatments of special relativity, when one is discussing length measurements one apparently does not consider how much time might be required to make the measurements. It is as if a measurement is instantaneous ( both primed and unprimed time intervals are zero). In the above, if we take that approach, then we are left with Top

T spatial coordinate transform

From this, we can construct the generalized Lorentz contraction law:

Length contraction formula

The first formula provides the means to calculate the at-rest length Lo of an object in a moving frame from observations made from a stationary frame. This is the reverse of the traditional Lorentz contraction formula, which states how an object with a known length and in constant, linear motion would appear to a stationary observer (foreshortened in the direction of motion). It might appear that we could go back to the 3-D vector-matrix equation above and solve for the length in terms of the primed-system measurements, but that would do no good - we would get the same result! That is because neither observer in either frame can tell who is moving and who is stationary; to each, the other appears to be moving. The T transform as constructed by the moving observer would be the same as for the stationary observer. The second formula, above, gives the apparent length as seen by an observer in the stationary frame. Top

Again, the acid test of this formula is: Does it give the same result as the traditional Lorentz contraction formula under the same conditions? The answer is yes, but it is messy and space-consuming to explain in detail because of the complicated definitions of U,R,W,S. Nevertheless, it is just a matter of algebraic manipulation. We shall simply give the results here. If the dimension to be measured is aligned along the +x-axis, then , and in the length contraction formula we are left with only

Length contraction on x axis

Moreover, in the traditional treatment the relative velocity is aligned along the +x-axis. That yields R=W=0 and

Length contraction U calculation

We are left with

Lorentz contraction formula

The delta-x quantity is the length measurement L made by a stationary observer of the moving object whose length is Lo. Therefore, the length as seen by a stationary observer is:

Observed Lorentz contraction

which is precisely the Lorentz contraction formula. Top

Time Dilation

We can dispense with this one rather easily. It is expedient to go back to the 4-D vector transform equation X'=TX and work with the inverse, X=T-1X':

Inverse interval transformation

As we pointed out above, because of its peculiar structure we can get the inverse of T by merely changing the signs on its W,S vector components. This is very helpful in calculations. Now, the above transformation indicates that the time component of a 4-D vector transforms according to:

Time dilation formula

In the second equation, we have merely factored out the delta-t quantity in order to compare it with the corresponding rest-frame quantity. The result is an odd expression at first glance. Recall that the relative motion vector V is incorporated into the U,R,W,S vector components of the relativistic transform T. Apparently, the derivatives in parentheses represent any additional components of velocity measured with respect to the moving (primed) coordinate frame. We are not interested in adding velocity in the primed frame. A comparison of time effects between the stationary and moving origins of coordinates will be sufficient. We therefore set the additional velocity components to zero, leaving Top

Time dilation formula

We shall call this the generalized time dilation formula. It readily reduces to that from the traditional special relativity analysis. As we saw earlier, in the restricted case of relative velocity along the +x-axis, U reduces to a division by the Lorentz contraction factor:

Restricted time dilation formula

The effect of this formula is that, from a stationary observer's perspective, time appears to run more slowy for a moving body. This has been tested in elementary particle physics experiments and is found to be true. Top

Unorthodox Effects

Finally - we can put it off no longer - we get to the nontraditional aspects of the present formulation. First, in all of the above formulas for relativistic effects, the observer's perceptions change dramatically if there is a standoff of the line of action and the relative velocity line of motion is rotated off of the observer's line of sight. For example, look at the definition of T:

T function of 4-D V

We have already shown that it reduces to the restricted case when the i,j components are set to zero. But look what happens when the 1,i components are set to zero (leaving the remaining velocity turned perpendicular to the line of sight): the transform reduces to unity, no effect! In the latter case, it is assumed that the line of action does not go through the observer's position; otherwise, we have the traditional restricted case, just oriented along a different direction in space. Of course, with a standoff, the nonlinearly-varying origin-to-origin distance between the frames will cause an apparent deceleration/acceleration effect as the moving frame passes the point of closest approach, with acceleration dropping to zero only at closest approach. It would be difficult to make measurements under those circumstances. However, the acceleration effects are minimized if the moving frame is standing well off, say at astronomical distances; then when it is near the closest approach, no relativistic effects will be observed. There are many cases of astronomical objects moving apart at apparently superluminal speeds [Hughes, 1990], which they couldn't do if a transverse view of the action still showed full relativistic effects. But, note: If we were on one of the objects and viewing the other along the line of motion, then we would see all of the traditional relativistic effects. Therefore, relativistic effects must shade out from the traditional ones along the line of motion to none when there is an offset of the line of motion and the moving object passes through the point of closest approach. Top

Secondly, all coordinates are affected equally in the 4-D commutative hypercomplex formulation. In Einstein's treatment, the transverse coordinates were more or less assumed to be unaffected. To see how the hypercomplex formulation differs, go back and look at our 2-D hypercomplex formulation. Recall that it addresses the restricted case of relative motion along the +x-axis. In it, we have a transform T with only 1,k components, which is applied to position vectors X, also with only 1,k components (the latter to maintain agreement with Einstein's formulation). However, even an element T with only 1,k components has a 4 X 4 real matrix representation. It is

T 2-D matrix

This has an attractive symmetry. If we apply it in a coordinate transformation X'=TX, but this time with 4-D position vectors, then we get Top

Transformed coordinates

In this view, all coordinates are treated qualitatively the same by the relativity transform. The result has a compelling symmetry. More importantly, it is in keeping with continuity of effects, in the sense of analytic functions, and it obeys Einstein's two postulates. Top

Observe that (x',ct') and (x,ct) are a transform pair [i.e., do not involve (y,z)], and similarly for (y',z'),(y,z). Each of these transformations is of the form

Hyperbolic transformation

with real. The determinant is always +1 due to a common hyperbolic identity:

Hyperbolic identity

Therefore, these transformations represent hyperbolic rotations. We are discussing the restricted case of motion along the +x-axis, but similar statements hold for the general transform T; it is just a matter of our perspective. A similar observation regarding the nature of the Lorentz transform is made by Aharoni, 1965, adding another note of agreement between the two separate formulations. However, an important note of disagreement is that the transverse coordinates will be affected both in direction and magnitude. Top


We have produced a direct, concise formulation of special relativity with an arbitrary relative motion vector, and have shown that it reduces properly for the restricted case of relative velocity along the +x-axis. We showed that it obeys Einstein's two postulates of special relativity, under the proper viewpoint (similarity transformation). Recall that Einstein conditioned his formulation on only the two postulates, so inasmuch as our present formulation obeys the two postulates, we can argue for a successful alternative formulation. Top

The group script T of relativistic transformations corresponds to the group of Lorentz transformations of the traditional treatments, but the two cannot be related on a fundamental level because the Lorentz group is noncommutative and the script T group is commutative. In both cases, this is due to their formulation in a particular mathematical system, the first in the noncommutative vector analysis schema and the second in the commutative hypercomplex notation. One may argue that vector analysis is not used directly in the development of special relativity, but one must admit that all of the field equations of physics, with which special relativity was designed to meld, are based on vector analysis. It has also been proved that the Lorentz group is the only invariance group for the equations of physics, but one must complete that statement with "when stated in vector analysis notation." We are wearing the rose-colored glasses of vector analysis and saying that everything looks rose-colored. Let me pose the following question: If long ago we had found ourselves with only some non-vectorial mathematics, would not today all of our physics equations look entirely different, yet would be describing the same reality? Top

One will also hear arguments that much of physics is inherently noncommutative. Examples will be given such as successive rotations of a 3-D object around orthogonal axes. One cannot get back to the original orientation by merely reversing the order of operations. However, operations that look noncommutative in three dimensions may well be commutative in four. Note that a rigid rotation in four dimensions does not necessarily result in a rigid rotation of the three-space and vice-versa.

For those who will say that they don't understand or don't believe the strange mathematics, we point out that everything can be stated in terms of 4 X 4 real matrices. One would not normally want to go that route, however, because the canonical form is so much more concise, computationally efficient, and theoretically illuminating. Its formulation upon matrix eigenvalues and with numerous rotational invariants should be attractive to physicists. Top

© 2000, 2007
Clyde M. Davenport

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