last updated: 12/14/2006
THE LORENTZ TRANSFORMATION
The Lorentz transformation is a mathematical system of equations utilized by Einstein in his special theory of relativity based on the premise that time and matter are altered as a consequence of relative motion. The Lorentz transformation was specifically designed to mathematically express the proposed effect of the contraction of (matter) measuring rods and the corresponding increase in time (clock speed) due to velocity. However, the equations also include a time alteration factor which requires time to act as a function of relative velocity. The sole purpose of this factor is to increase or decrease time by linking time to an observer's relative velocity and therewith to negate the apparent disparity between the principle of relative velocity and the constant propagation velocity of light waves.
Although this system of equations is often referred to as a transformation, from a mathematical perspective it is more than that. Not only does it modify the form of the original Galilean expressions, but more importantly it also changes the value of those expressions. In point of fact, the Lorentz transformation is also a transvaluation. An engineered system of equations used to alter the Galilean system of classical mechanics and hence the absolute value of time.
A valid mathematical transformation should change only the form of the equations it modifies while still maintaining the absolute value of the original equalities. The result of the Lorentz transvaluation is an alteration of the original equality and therewith an alteration in time for all reference systems in motion. A transvaluation factor [- ( v/cē) x] is used to alter the value of time from one frame of reference to another. Thus a mathematical inequality is introduced into the equations on the questionable assumption that simultaneity (time) is a relative concept. In fact there exists no mathematical justification for this unorthodox transvaluation of time other than its obvious design, that being to equivocate the presumed contradiction between the principle of relative velocity and the constant propagation velocity of light (see note 1).
There is second more troublesome aspect of the Lorentz transvaluation of time that becomes apparent by way of its practical application. When the Lorentz transvaluation of time [- ( v/cē) x] is applied to differing frames of reference, e.g., K and K', a curious inconsistency with respect to time occurs. When K' is traveling in the same direction as the light wave, time slows down for K' relative to K. However, when K' is traveling in the opposite direction as the light wave, time speeds up for K' relative to K. This inconsistency is difficult to reconcile intelligently. A time piece (clock) would not be aware of its direction of motion relative to a light wave and should therefore respond to relative velocity in a uniform fashion, i.e., by either slowing down in both instances or by speeding up in both instances (see note 2).
The hypothesis on which the special theory of relativity was based, the proposition that time and distance are dependant on the condition of motion, provides no explanation for this curious inconsistency. Needless to say, herein lies special relativity's fatal flaw, a flaw portending its ultimate demise. Indeed, as a consequence of the aforementioned inconsistency, the sad but true fact emerges that the special theory of relativity is unable to provide a viable solution to a scenario wherein K' encounters two independent light waves traveling in opposite directions. Employing the Lorentz transformation in such a scenario requires that the K' clock both slow down, and in the same time frame, speed up relative to the K clock, both a logical and practicable impossibility (see note 3).
Furthermore, because velocity is a function of distance traveled divided by time, the operative factor in determining the time alteration between K and K' is distance, not velocity, i.e., the distance K' travels relative to K and not the velocity at which K' travels relative to K. The magnitude of the time alteration is the product of the distance traveled by K' multiplied by the constant 3.33 x 10-6 sec / km, i.e., the time in seconds it takes light to travel one kilometer. For instance, if K' travels a distance of 30,000 km relative to K the magnitude of the time alteration [- ( v/cē) x] will be .10 seconds (30,000 km x 3.33 x 10-6 sec / km) regardless of the velocity of K' relative to K. That is to say K' could be traveling at a velocity of .2 km / sec or 200,000 km / sec and the magnitude of the time alteration would remain .10 seconds with respect to that 30,000 km distance. This is so because the velocity of K' is dependent upon the total time traveled by K'. As such, the time unit of the velocity factor cancels out [- ( v/cē) x] = [- (x - x') (t / x)] and therewith the time alteration between K and K' is reduced to a function of distance. This would also hold true for the contraction factor (sq rt, 1 - vē/cē), the effects of which are only measurable at extreme values of v.
In light of the aforementioned fact, an experiment to test the validity of special relativity could be structured solely on the basis of distance traveled without concern to achieving extra ordinary velocities, hence a less daunting task. As can be discerned from the above example, traveling a distance of 30,000 km at a speed of .2 km / sec should produce a time alteration of .10 sec in a period of approximately 42 hours. Unfortunately it cannot be determined from special relativity theory whether an increase or decrease in time (clock speed) should occur, since the theoretical light wave, from which the direction of motion is established, could indeed be emanating from any or all directions.
In addition there is another paradox pertaining to the practical application of the Lorentz transformation. This difficulty concerns the Lorentz contraction factor (sq rt, 1 - vē/cē). Note that the measuring rod of K' contracts in proportion to the relative velocity v of K' and therewith whatever is measured with this shortened measuring rod is longer in terms of the number of units measured. But what is it that K' is desirous of measuring? It is the distance that a light wave travels relative to the frame of reference of K' in time period t'. However, since the value of t' is also subject to the effect of the contraction factor (sq rt, 1 - vē/cē), time with respect to K' must also contract (speed up). Consider this then, as the measuring rod of K' contracts due to it's relative velocity, the K' clock will speed up. One effect will precisely offset the other and K' will compute an identical velocity for the speed light with or without the contraction factor. As is apparent, the speed of light relative to K' can not be a function of the Lorentz contraction factor since the synchronic increase in the speed of the K' clock exactly offsets the contraction of the K' measuring rod. The inclusion of the contraction factor (sq rt, 1 - vē/cē) in the special theory of relativity therefore appears pointless.
This contention is further corroborated by the reality that the Lorentz contraction factor (sq rt, 1 - vē/cē) is completely unnecessary in the Lorentz equations achievement of their desired result i.e., equivocating the principle of relative velocity with the constant velocity of light. Aided by the following illustration, we see that the law of the transmission of light is satisfied both for reference body K and for reference body K' without inclusion of the Lorentz contraction factor, to wit:
x' = x - v t & t' = t - (v/cē) x;
substituting for x where x = c t, yields
x' = c t - v t & t' = t - (v/c) t
or
x' = t (c - v) & ct' = t (c - v)
from which by substitution we conclude that x' = c t'.
This illustration likewise confirms our earlier assertion that it is the algebraic inequality of the transvaluation factor [- (v/cē) x], used to modify the time expression t' = t, that gives the Lorentz equations their capacity to equivocate the presumed contradiction between the principle of relative velocity and the constant propagation velocity of light. The contraction factor (sq rt, 1 - vē/cē) is totally superfluous to this outcome (see note 4).
In view of the above-mentioned deficiencies in the Lorentz equations, their veracity in resolving the apparent contradiction between the principle of relative velocity and the constant propagation velocity of light is conspicuously diminished and as logic would have it, should be evaluated with appropriate skepticism.
Note 1. The Lorentz equations for the co-ordinate system K' undergoing a state of uniform translatory motion along the x-axis relative to K are stated as:
x' = x - v t /(sq rt, 1 - vē/cē),
y' = y,
z' = z,
t' = t - (v/cē) x /(sq rt, 1 - vē/cē), where
x' = c t' & x = c t.
The Lorentz Equations employ two separate modifying factors:
I. The Lorentz transformation (contraction) factor (sq rt, 1 - vē/cē) modifies the Galilean equations in the form of an equality. It mathematically expresses the proposed effect of contracting matter (measuring rods) and a concurrent offsetting increase in time (clock speed).
II. The Lorentz transvaluation (time alteration) factor [- (v/cē) x] modifies the Galilean equations in the form of an inequality therewith requiring time (clocks) to speed up or slow down depending upon their direction of motion relative to the photon wave. The transvaluation (time alteration) factor [- (v/cē)x] becomes [- (v/c) t] when c t is substituted for x.
Note 2. In the Lorentz equations the transvaluation factor [- (v/c) t] is used to modify the Galilean time equality equation t' = t so that t' = t - (v/c) t. When v has a positive value, that is when K' is traveling in the same direction as photon wave c, the transvaluation produces a time unit inequality in the form of t' (second) > t (second), whereupon the t' second is longer in duration than is the t second. This causes K' to experience time at a slower rate than K. Each second K measures is measured as less than one second by K' based on the value of t', that is [t - (v/c) t].
Conversely, when v has a negative value, that is when K' is traveling in the opposite direction as photon wave c, the value of t must be modified by the factor [- (-v/c) t]. The transvaluation therewith produces a time unit inequality in the form t' (second) < t (second), whereupon the t' second is shorter in duration than is the t second. This causes K' to experience time at a faster rate than K. Each second K measures is measured as more than one second by K' based on the value of t', that is [t - (-v/c) t] or [t + (v/c) t].
Note 3. In the scenario wherein K' encounters two independent light waves traveling in opposite directions, one photon wave is traveling in a positive direction i.e., ( c ) and one photon wave is traveling in a negative direction i.e., ( -c ) that is with respect to co-ordinate systems K and K'. Of course K' would of necessity be traveling in the same direction as one of the photon waves and in the opposite direction as the other photon wave. Assuming v to be positive, that is K' is traveling in the same direction as is photon wave ( c ), the Lorentz transvaluation produces a time unit inequality in the form of t' (second) > t (second) i.e., [t' = t - (v/c) t] so that the K' clock must run slower than the K clock. However, concurrently with respect to photon wave ( -c ) the Lorentz transvaluation produces a time unit inequality in the form of t' (second) < t (second) i.e., [t' = t - (v/-c) t] or [t' = t + (v/c) t] so that the K' clock must also, in the same time frame, run faster than the K clock.
Note 4. The Lorentz transvaluation (time alteration) factor [- (v/cē) x] was derived from the classical Galilei transformation equations expressly for the purpose of mathematically reconciling what appeared to be a contradiction between the principle of relative velocity and the law of the transmission of light. The transvaluation factor [- (v/cē) x] was deduced from the Galilei transformation equations by solving for t' where t' is not equal to t. That is to say: where no absolute standard of time exists with respect co-ordinate systems K and K', and where K' is in motion at velocity v relative to K then,
x' = x - v t, where x' = c t', x = c t and t' # t
so that
c t' = x - v t or t' = x/c - (v/c) t or t' = t - (v/c) t or t' = t - (v/cē) x.