I just went and looked at the equations from this article again and realized I mixed them all up. The equations for the Lorentz transformations are mixed up with the derivation of the Minkowski Spacetime Metric. Did no one notice that?! Why the heck didn’t I?
Let’s do it again. But if you haven’t read the original article, you still really need to because all the groundwork is laid out there for why this is an important derivation.
The complex plane and the unit Euler circle:
The Eulerian spacetime equation is:
eiΘ = cos(Θ) + i*sin(Θ).
Taking the Pythagorean sum of the real and imaginary components:
cos2(Θ) + (i*(sin(Θ))2 = cos2(Θ) – sin2(Θ)
which always has a magnitude of 1 on the unit circle in complex spacetime, and so
1 = cos2(Θ) – sin2(Θ).
Substituting for cos and sin,
1 = (v/c)2 – (t/τ)2,
moving to an infinitesimal in time,
1 = (v/c)2 – (dt/dτ)2,
and then multiplying through by c2dτ2,
c2dτ2 = v2dτ2 – c2dt2
which is typically denoted as
dS2 = dX2 – c2dt2
which is the Minkowski Spacetime Metric. The dimensionality however isn’t the same as it is in the standard interpretation of relativity, but I didn’t discuss this in the other article. Logically, each real space dimension must have a mathematically corresponding imaginary time dimension, and so spacetime is 6-D rather than the arbitrary and illogical 4-D of the traditional method. Motion occurs in all 3 spatial dimensions and so therefore each space dimension must have a corresponding time dimension.
Now, taking another approach, we can consider the complex conjugate of Euler’s equation:
|eiΘ|2 = (cos(Θ) + i*sin(Θ)) * (cos(Θ) – i*sin(Θ))
1 = cos2(Θ) + sin2(Θ)
1 = (v/c)2 + (t/τ)2.
Then rearranging for t/τ,
t/τ = √(1 – (v/c)2)
and the usual thing is to denote
γ = 1/√(1 – (v/c)2)
t = τ/γ
which is Lorentz time dilation. Lorentz length contraction is trivially
ct = cτ/γ
l’ = L0/γ.
Note that the fundamental form of the Eulerian spacetime equation is complex and is based on waves, i.e. sines and cosines. That’s just like electromagnetism, heat flow, and quantum mechanics. So…we should probably develop a wave-based general relativity theory using the Eulerian approach, shouldn’t we? 🙂
(I’ll fix the original article accordingly.)