Velocity Addition in Space/Time

4/23/2003

http://members.triton.net/daveb

Abstract

Using the space/time geometry developed in the previous paper (“Non-orthogonal Space-Time geometry, the Dirac Matrices and relativity.” ), I shall develop a velocity addition formula. I will show that the velocity addition formula proposed by Einstein can be developed using components of the true space/time velocity.

Space/Time geometry and velocity addition

We start by considering the space/time representation of the two objects B and C moving relative to A. They are both moving away from A and C is moving faster then B. Refer to Figure 1 below.

In Figure 1, the axes of motion for each moving object are shown. is the motion axis for object B and is the motion axis for object C. The dashed lines are reference lines for the purpose of determining the transformation equations for B. To avoid confusion the reference lines for C are not shown. The lines for C would simply be orientated relative to . We know from the previous paper and /or the above diagram that:

1.40

1.50

Now if we analysis the motion of C relative to B we get the diagram of Figure 2 below:

In Figure 2, the details were left out for clarity. The velocity of C relative to B is u and the axis of motion is labeled since the motion is the result of the projection of cT’. The transformation equation for the point is:

1.60

A vector velocity Addition formula

All the axes of motion in space/time have direction and therefore any velocity addition formula must be developed from vector considerations. We shall proceed with the simplest which is shown below in Figure 3.

In Figure 3, the motion of both B and C relative to A is shown. A vector labeled connects and . When equals , is zero and when equals c ( lies on the cT axis) equals . From B’s point of view the velocity of C in units of c, is . This follows from the definition of velocity in the previous paper. It clearly ranges from zero to one as viewed by B.

It is easy to develop an expression for . We start by considering the diagram in Figure 4 below.

In Figure 4 a perpendicular is dropped from the apex of the triangle to the line. From basic geometry we quickly arrive at the following equation.

1.70

From the previous paper we know that . Letting c equal 1 and defining (the velocity of C relative to B) we finally arrive at:

1.80

At this point it is instructive to calculate some actual velocities and see how they compare to the Einstein formula. The chart below show several combinations of velocity. Vb is the velocity of B relative to A. Uc is the velocity of C relative to B

A program for calculating values is available at http://members.triton.net/daveb/velocty_addition.htm

From the chart a number of things are clear.

For very small velocities both eq 1.80 and Einstein’s formula give identical results.

The largest variations between 1.80 and Einstein occur when one or both velocities are in the mid range.

One dimensional representation

If the velocities vectors above are converted to a momentum/energy representation, each would have a component along the line perpendicular to the momentum axis of their vector sum (the motion axis of C relative to A). These two components cancel when the vectors are added and therefore a velocity representation using only the components of the vectors along the C axis of motion is instructive. This is illustrated in Figure 5 below.

In Figure 5, The components of the velocity vectors are labeled and . It is assumed that point P, observer A and observer B are all at the same point in space/time at the intersection of ct and ct’, and are at the points shown in Figure 5 after a time T on A’s clock. The velocity A would use for the point P relative to B is given by the distance between P and B:

1.90

The time P is at that distance according to A is T’’.

From the previous paper and/or Figure 5 we know that:

2.00

The component of velocity of C relative to B from A’s view is then:

2.10

Now dividing the numerator and denominator by T and noting that P/T is the actual space/time velocity of the point P relative to A (), we arrive at:

2.30

It is interesting to note that and where

And hence taking X’ over T’ yields equation 2.30.

Of course X’/T’ is the conventional method of determining relativistic velocity composition.

Conclusion

The space/time geometry developed in my previous paper lends itself to a simple velocity addition formula. The similarity between the results of the formula and the Einstein formula would explain why there is no substantial experimental evidence contrary to the Einstein formula results. The Einstein formula is the result of considering only the components of the space/time velocities along the final axis of motion. Experiments with high velocity particles do not directly measure velocity but rather the results of momentum/energy.

Using the actual space/time velocities to calculate energy/momentum would require space/time vector considerations.

One should be able to test my theory with a properly devised experiment.