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Minkowski's Space-time Equations
by Miles Mathis
October 15, 2004
In Appendix 1 of the book Relativity, Einstein derives the spacetime equations of Minkowski from the light equations, like this:
r = [x 2 + y2 + z2]1/2
Where r is straight line distance, found by a three-dimensional Pythagorean theorem.
r = ct
(This is equivalent to x = ct in the light equations of Lorentz and Einstein. Light does not have to travel along orthogonal axes, obviously. It can travel in any direction the the 3-dimensional continuum.)
so,
x2 + y2 + z2 - c2t2 = 0
Then, he assumes r' = ct'
But I have shown that the two equations
r = ct and
r' = ct'
cannot both be true, because the assumptions of Relativity demand that the x and t variables be in inverse proportion. However, it turns out that the one that is wrong is the first equation, although that seems very odd initially. The reason is simple: the variables in that equation do not stand for how the unprimed system sees itself. In Relativity, the variables in that equation stand for how the unprimed system sees the primed system. The unprimed equation states the situation as seen from the "stationary observer." Therefore, these equations are measured from a distance.
r ≠ ct
r' = ct' and
r = ct'2/t
r = [x2 + y2 + z2]1/2 = ct'2/t
r' = [x'2 + y'2 + z'2]1/2 = ct'
x'2 + y'2 + z'2 - c2t'2 = γ ( x2 + y2 + z2 - c2t'4/t2)
[x'2 + y'2 + z'2 ] - 1 = γ{[x2 + y2 + z2] - t'2/t2}
c2t'2
c2t'2
That last equation is the corrected space-time transformation. r' does not equal r, so the transform γ must be used to complete the equality. This transform cannot be generated simply from the two light equations or from Minkowski's quadratic equation; more information is necessary. Einstein used the equation x' = x - vt to develop a transformation equation, but I have shown that this method was faulty. So if you use γ = gamma = 1/√1 - v²/c² to complete the transformation, you will compromise the math once again: gamma is incorrect. The true value for the transform for one degree of relativity is
α = alpha = t/t' = 1/[1 - (v/c)].
The value for two degrees is
ω = omega = t/τ' = 1 - wv/c2
(1 - w/c)(1 - v/c)
To see how I derive these new tranforms, you must read my papers on Special Relativity.
In his lecture of 1908, called Space and Time, Minkowski claimed that by setting his "four dimensional space equation" equal to one instead of zero, as follows,
instead of x2 + y2 + z2 - c2t2 = 0
let c2t2 - x2 - y2 - z2 = 1
he was able to thereby generate gamma without Einstein's lengthy theoretical proof or any of his equations. In fact, he said in that lecture that Einstein's Relativity Postulate "seemed to him very feeble" when compared to his own method. His method, he claimed, simply expressed an invariance within a predefined mathematical group, which he called Gc (the group of objects defined by the limiting speed of light). Minkowski claimed that an "easy calculation" allowed a mathematician to derive gamma simply from the above equation. [This is before he introduced √-1 to complete his math.] He did not supply this easy calculation in the lecture; however, the mathematician A. Sommerfeld completed the math in the publication of the lecture (English translation 1923, Methuen &Co.; reprinted 1952, Dover Edition).
Since I have proven in a previous paper that both Lorentz and Einstein made the same basic error in conceptualizing the Michelson Morley Interferometer, and since I know that gamma is a phantom, I went into Sommerfeld’s proof expecting the worst. I soon found it. It is no surprise to find that Minkowski’s absurd hyperbolas have needlessly complicated the problem, and Sommerfeld is quickly lost. Sommerfeld is obviously consulting the first part of figure 1 in the lecture more than the text, for near the beginning of his proof he assumes that OC’ (the “world line” of x’) is equal to OA’ (the “world line” of t’). But Minkowski states that “OC’ = 1 and OA’ = 1/c” (p.77, bottom, Dover). Sommerfeld also assumes, at the very beginning, that the angles of these lines to OBB’ are equal, which they cannot be if what Minkowski says is true [Sommerfeld assumes that the two angles I have pencilled in here as β are equal; but Minkowski himself contradicts this in the text]. With that, Sommerfeld’s proof has already collapsed.
Minkowski himself does a partial “easy calculation” on the second illustration of figure one, which consists of the “parallel bands.” These bands create triangles that are strictly equivalent to those of Michelson and Lorentz, although this is difficult to see at first. You can ignore the first illustration of the figure, the one with the hyperbolas—which is ironic. Just import the terms from the first illustration into the second one. Here I will simplify Minkowski's math even further, to make it completely transparent.
PP = OC = x
QQ’ = OC’ = vt
QQ = OD’ = x’
x = ct
x’ = ct’
x = √x'² + (vt)²]
c = √x'² + v²t²]/t
c² = (x'² + v²t²)/t²
c² = x'²/t² + v²
c² - v² = x'²/t² = x'²c²/x²
(c² - v²)/c² = x'²/x²
x/x' = 1/√(1 - v²/c²)
That is the easy calculation that Sommerfeld should have inserted, a calculation simpler than, but equivalent to, Lorentz and Einstein. It is simple, but mistaken. It is mistaken not in the way that Sommerfeld is mistaken. Sommerfeld is mistaken in that he does not follow the explicit instructions of Minkowski. The calculation above is mistaken because x ≠ ct and because both the v and the t in vt are undefined, as I have shown in my other papers. Minkowski imports the conceptualizations of Maxwell, Michelson, Lorentz and Einstein, but he does not recognize that their conceptualization are all faulty. He therefore makes no effort to correct them.
The question then becomes, is it possible to substitute my corrected calculations into Minkowski’s conceptualization? It is possible, and I have shown how to do so above, simply by substituting the correct transform for gamma and by dumping x = ct. However, I do not find that Minkowski’s setting of the four dimensional equation equal to 1 is at all helpful. Quite the reverse. Not only does it make the math more difficult, for no reason, it throws a blanket over the algebraic mistakes, making them much more difficult to unravel. [My simplification above is not difficult, but that is only because I knew how to quickly gloss the mess Minkowski made of the problem.] Like Einstein and Lorentz before him, Minkowski throws much more math at the job than it requires. He ends up offering a variant math that is just as dense and unnecessary as the math that came before. His solution is therefore not an improvement. Sommerfeld was not able to penetrate Minkowski’s arguments, nor has anyone else clarified this problem in the intervening century since the lecture. Minkowski is treated as one of the fathers of the new spatial math, and his conceptualizations stand to this day, faults and all.
Mathematicians like Minkowski’s four dimensional spacetime continuum for several reasons. The first reason is that Minkowski convinced fellow mathematicians that his equations more elegantly expressed the invariance of Relativity. And in a sense they do. It is not difficult to be more elegant than Einstein, whose exposition of the problem and its math was far from masterly. By a seemingly innocuous trick, Minkowski appears to create an invariant group that expresses c as a limiting aspect. But it does so at an enormous cost, a cost that no one has yet tabulated. Historically, Minkowski’s seemingly innocuous trick has both bolstered the incorrect math of Einstein and Lorentz, and hidden it from view. Because Minkowski failed to correct the mathematical errors of Relativity, his claim to confirm its transforms by a variant method has been disastrous. And because his method is mathematically even more subtle than that of Einstein, he made the problem that much more difficult to penetrate and argue. You have seen that some of the finest mathematicians of the 20th century were not able to understand Minkowski, including the man given the job of supplying extensive footnotes to the international publication of the lecture.
And this is even before Minkowski introduced s = t√-1 . The reason for doing so is again purely mathematical: the quadratic differential equation becomes symmetrical. But this is completely beside the point. The symmetry is a manufactured symmetry. Minkowski says, "Thus the essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 3 x 10 km = √-1 seconds." As if we are in need of mysticism in physics! Besides, this equality is not mystical, it is false. Clothing a theory in false math is not pregnant in manner, it is impudent in manner. Making the quadratic equation symmetrical implies that time (or at any rate, s) travels orthogonally to x, y, z. But there is no reason to assume this. A mathematician will say that there is no reason not to assume this, if we want. It makes the math more stylish, if nothing else. But there is a very good reason not to do it. Importing more and more abstract math into a problem that is already in need of repair is very dangerous. The reason not use Minkowski’s math is simple: it hid the t variable in Relativity, making it impossible to correct the underlying algebra for a century.
Beyond that, I can show simple empirical proof that time does not travel orthogonally to x, y , z. Go here for the proof.
For much more on this problem, you may now visit my multi-part analysis of the Einstein field equations, which revisit this problem in finer detail, including a line-by-line critique of Einstein's 1916 paper.