Velocity Addition in Space/Time David Barwacz 4/23/

Transcription

1 Veloity Addition in Spae/Time 003 David arwaz 4/3/003 Abstrat Using the spae/time geometry developed in the previous paper ( Non-orthogonal Spae- Time geometry, the Dira Matries and relativity. ), I shall develop a veloity addition formula. I will show that the veloity addition formula proposed by Einstein an be developed using omponents of the true spae/time veloity. Spae/Time geometry and veloity addition We start by onsidering the spae/time representation of the two objets and moving relative to A. They are both moving away from A and is moving faster then. Refer to Figure 1 below. In Figure 1, the axes of motion for eah moving objet are shown. P is the motion axis for objet and P is the motion axis for objet. The dashed lines are referene lines for the purpose of determining the transformation equations for. To avoid onfusion the referene lines for are not shown. The lines for would simply be orientated relative to P. We know from the previous paper and /or the above diagram that: X = ( P V T) 1 V / 1.40 '

2 X = ( P V T) 1 V / 1.50 ' Now if we analysis the motion of relative to we get the diagram of Figure below: In Figure, the details were left out for larity. The veloity of relative to is u and ' the axis of motion is labeled P sine the motion is the result of the projetion of T. " The transformation equation for the point X is: X = ( P ut') 1 u / 1.60 '' ' A vetor veloity Addition formula All the axes of motion in spae/time have diretion and therefore any veloity addition formula must be developed from vetor onsiderations. We shall proeed with the simplest whih is shown below in Figure 3.

3 In Figure 3, the motion of both and relative to A is shown. A vetor labeled UT onnets VT and VT. When V equals V, UT is zero and when V equals ( V lies on the T axis) UT equals T '. From s point of view the veloity of in units of, is UT. This follows from the definition of veloity in the previous paper. It learly T ' ranges from zero to one as viewed by. It is easy to develop an expression for UT. We start by onsidering the diagram in T ' Figure 4 below. In Figure 4 a perpendiular is dropped from the apex of the triangle to the VT line. From basi geometry we quikly arrive at the following equation.

4 UT = T ( V os( )) ( sin( ) ' V θ θ + V θ θ ) 1.70 T ' T From the previous paper we know that T = 1. Letting equal 1 and defining T ' os( θ ) U = UT (the veloity of relative to ) we finally arrive at: T ' 1 U ( os( )) ( sin( )) = V V θ θ + V θ θ 1.80 os( θ ) At this point it is instrutive to alulate some atual veloities and see how they ompare to the Einstein formula. The hart below show several ombinations of veloity. Vb is the veloity of relative to A. U is the veloity of relative to Vb U V (eq 1.80) V Einstein A program for alulating values is available at From the hart a number of things are lear. For very small veloities both eq 1.80 and Einstein s formula give idential results. The largest variations between 1.80 and Einstein our when one or both veloities are in the mid range. One dimensional representation If the veloities vetors above are onverted to a momentum/energy representation, eah would have a omponent along the line perpendiular to the momentum axis of their vetor sum (the motion axis of relative to A). These two omponents anel when the

5 vetors are added and therefore a veloity representation using only the omponents of the vetors along the axis of motion is instrutive. This is illustrated in Figure 5 below. In Figure 5, The omponents of the veloity vetors are labeled V and U. It is assumed that point P, observer A and observer are all at the same point in spae/time at the intersetion of t and t, and are at the points shown in Figure 5 after a time T on A s lok. The veloity A would use for the point P relative to is given by the distane between P and : P V T 1.90 The time P is at that distane aording to A is T. From the previous paper and/or Figure 5 we know that: VP T'' = T T = T.00 The omponent of veloity of relative to from A s view is then: U P VT = VP T.10 Now dividing the numerator and denominator by T and noting that P/T is the atual spae/time veloity of the point P relative to A ( V ), we arrive at: U V V = VV 1.30

6 It is interesting to note that X ' = ( P V T) γ and T' = T'' γ where γ = And hene taking X over T yields equation.30. Of ourse X /T is the onventional method of determining relativisti veloity omposition. onlusion 1 V 1 The spae/time geometry developed in my previous paper lends itself to a simple veloity addition formula. The similarity between the results of the formula and the Einstein formula would explain why there is no substantial experimental evidene ontrary to the Einstein formula results. The Einstein formula is the result of onsidering only the omponents of the spae/time veloities along the final axis of motion. Experiments with high veloity partiles do not diretly measure veloity but rather the results of momentum/energy. Using the atual spae/time veloities to alulate energy/momentum would require spae/time vetor onsiderations. One should be able to test my theory with a properly devised experiment.