There is a third option; which is that the equations will produce mathematically consistent results for some set of input values and mathematically inconsistent results for a different set of input values.
In order to see the problem we have to look at mathematical sets. Consider three sets, where each element of each set is a collection of four values, e.g., x, y, z, and t. Set A is defined as the collection of all combinations x, y, z, and t. Mathematically, this set is defined as A:{(x, y, z, t) for all Real x, y, z, t}.
Now consider that Set A is the union of two non-overlapping Sets B and C. Set B is defined as a subset of A such that B:{(x, y, z, t) for all Real x, y, z, t, where t=x/c}. Set C is defined as a subset of A such that C:{(x, y, z, t) for all Real x, y, z, t, where t <> x/c}. No member of Set B is a member of Set C, and visa versa. Elements such (0, 0, 0, 0) and (299 759 458, 0, 0, 1) are members of Sets A and B. Elements such as (234, 21, 2, 43) and (1, 1, 1, 1) are members of Sets A and C.
Consider one final set, Set D, which is defined as the collection of all combinations of ξ, η, ζ, and τ, and is written as D:{(ξ, η, ζ, τ) for all Real ξ, η, ζ, τ}.
Einstein’s Special Relativity equations are based on the premise that they result in a 1-to-1 mapping of all elements from Set A into Set D. This mathematically consistent 1-to-1 mapping is a requirement for his theoretical predictions. Now reconsider the ξ derivation in Einstein’s 1905 paper where he begins with ξ=cτ and concludes with ξ=(x-vt)/sqrt(1-v^2/c^2). Since he began his derivation with ξ=cτ, we know that this implies τ=ξ/c. And since the derivation is one of algebraic replacement, we can readily show that
- Equation 1: τ=(x-vt)/c*sqrt(1-v^2/c^2).
But, Einstein also gives us a separate stand-alone equation in Section 3 of his 1905 paper,
- Equation 2: τ=(t-(vx/c^2))/sqrt(1-v^2/c^2).
When elements from Set B are mapped into Set D, Eq1 and Eq2 produce the same result for τ. The mapping from Set B into Set D is mathematically consistent. For example, (x, y, z, t) is mapped into (ξ, η, ζ, τ where τ is the result of Eq1) and into (ξ, η, ζ, τ where τ is the result of Eq2). This is a 1-to-1 mapping since Eq1 and Eq2 both produce the same result.
However, when elements from Set C are mapped into Set D, Eq1 and Eq2 produce different results for τ. In other words, you can map one element from Set C into two elements in Set D. For example, (x, y, z, t) is mapped into (ξ, η, ζ, τ where τ is the result of Eq1) and into (ξ, η, ζ, τ where τ is the result of Eq2). This 1-to-2 mapping is mathematically inconsistent with Einstein’s premise of a 1-to-1 mapping. This is a mathematical inconsistency since both equations produce τ and were derived as part of the same system of equations – in one case explicitly and in the other implicitly. [In my papers, when I suggest that ξ/c does not equal τ, this means that Eq1 (which is τ= ξ/c) does not equal Eq2 (which is the stand-alone τ). This might be better stated as τ <> τ.]
Why hasn’t this been found before?
Consider that most SRT equation derivations set out to map elements from Set A into Set D. However, in each of Einstein’s derivations, he implicitly or explicitly states x=ct. As a specific example, in Einstein’s 1905 paper, this occur when he states t=x’/(c-v). This is simplified as x=ct and defines a definite mathematical relationship between x and t in the source set’s elements. This effectively limits the validity of the source elements to those elements that are part of Set B. In a case where the derivations actually produce equations that map elements from Set B into Set D and the experiments used to validate the equations only use elements from Set B, the problem would not be detected. I believe the experiments and practical applications to date have come from this subset, or Set B.
Finding the problem requires understanding that inherent in his derivation is an implied equation for τ that produces values different from his explicit equation for τ. Since I suggest that experiments or practical applications have not be performed using elements from Set C, this explains why his equations appear to “work” and why the problem has not been previously discovered.
