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:
e^{iΘ} = cos(Θ) + i*sin(Θ).
Taking the Pythagorean sum of the real and imaginary components:
cos^{2}(Θ) + (i*(sin(Θ))^{2} = cos^{2}(Θ) – sin^{2}(Θ)
which always has a magnitude of 1 on the unit circle in complex spacetime, and so
1 = cos^{2}(Θ) – sin^{2}(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 c^{2}dτ^{2},
c^{2}dτ^{2} = v^{2}dτ^{2} – c^{2}dt^{2}
which is typically denoted as
dS^{2} = dX^{2} – c^{2}dt^{2}
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 6D rather than the arbitrary and illogical 4D 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:
e^{iΘ}^{2} = (cos(Θ) + i*sin(Θ)) * (cos(Θ) – i*sin(Θ))
1 = cos^{2}(Θ) + sin^{2}(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})
and so
t = τ/γ
which is Lorentz time dilation. Lorentz length contraction is trivially
ct = cτ/γ
or
l’ = L_{0}/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 wavebased general relativity theory using the Eulerian approach, shouldn’t we? :)

(I’ll fix the original article accordingly.)
Joseph,
I’m following along but only tangentially.
Actually, your discussion related to sine waves does bring to mind a question I have for you (although it’s quite unrelated to your topic here). What would a meaningful, mathematical term look like that would describe the temperature of a planet? I’m entirely skeptical of the notion that we can derive a single number that represents the average temperature of the Earth, and then compare this number with analogous historical numbers. I figure that temperature readings from a single location on any planet throughout a relative day would present a sine wave of sorts. So shouldn’t there be some sort of standard deviation and perhaps a trend indicator? For example, to compare a single value average temperature for Mercury and Venus, we might have respective values: 176C and 467C. But Venus has very little variance in temperature throughout the entire surface area of the planet while Mercury has over 600C of variance. In terms of comparing Earth with itself, it seems like we could have two of the same average temperature values that are representative of wildly different environmental conditions.
Yes that’s a good question Thomas. So, firstly, temperature is a local intrinsic property of matter. The temperature of a substance is that where you stick the thermometer. When you have an object that has variable and nonuniform temperature, the equations of heat flow show that temperature is a function of time and position, which means that temperature is only when and where you measure it. This therefore relegates the concept of an average temperature as unrealistic – there are no equations that would give this meaning. There are only meaningful equations for temperature as a function of time and position. If you do have a large object all at uniform temperature, then fine.
In stars we have a similar problem as to planetary atmospheres, just not quite as bad. In a star, the temperature is generally radially isotropic, meaning at a fixed depth, and any longitude and latitude, the temperature should be the same. In a star the temperature just changes as a function of depth in the atmosphere, not as latitudinal or longitudinal position. But again the concept of an average is problematic here. At any particular depth, a thermometer will give a unique reading.
However, given that some stars emit radiation similar to a blackbody, the convention has become to integrate the star’s flux (energy output) and equate that to what would correspond with the equivalent flux if it were from a perfect blackbody hard surface; the corresponding temperature of the blackbody is called “effective temperature”. That terms is used all over astronomy to indicate the equivalent temperature of a source, given its flux, if it were emitting like a blackbody. This is the closest thing you might get to the concept of an average temperature from a nonuniform source, but it still doesn’t correspond to any particular physicallymeasurable temperature.
For planets, effective blackbody temperatures are calculated simply by equating the energy received from the Sun to what must be the energy output from the planet due to conservation of energy. Only the nearsurface air of Venus is 467C, just like how only the nearsurface air on Earth is +15C as a “time average”. If you stick the thermometer in the actual ground, or at a different height in the atmosphere, you will get a different result, notwithstanding that the “time average” of a temperature doesn’t actually have much meaning given that temperature is a local and temporal property. So for the Earth the effective blackbody temperature is 18C, and I think for Venus it is slightly cooler given its high albedo.
The fact that a locallymeasured timeaveraged temperature on a planet is different than the effective blackbody temperature is truly physically meaningless, because both temperatures are only conceptual, have no true measurable physical basis, and planets emit even less like blackbodies than stars do. Subsequently climate science has created an alarming greenhouse effect based on the comparison of numbers that have no true physical basis in empirical reality. Thus the alarming greenhouse effect is likewise – not based on reality!
Joseph,
I like your answer to Thomas.
Just to pick on you, ‘couse it is fun. You need not six dimensions, but four is barely enough!
Time (a sequency) is independent of the other three dimensions. but all use is consistent, collateral, or several other terms, all meaning that time is not different than itself in other orthogonal dimensions. Maxwell’s quaternions (actually Bill Hamilton) were sufficient for energy. Dual quaternions 6D can help for space a little. Go to Octonions that can handle dual coupled threespace 6D plus two orthogonal scalars i.e. “time and mass” or “time and temperature”.
Mathematics must be used to describe the physical. Earthling have discovered very little mathematics!
Joseph – Thank you, I appreciate your thoughtful responses. And again, you’re affirming my conjecture. What an odd phenomenon of modern times to have such marvels of technology but still have widely held beliefs that are: “not based on reality!”. That’s what got me started on this quest. I was contemplating how man practiced bloodletting for thousands of years, believing it was the right thing when in fact it was the complete opposite. I thought that must’ve been such a strange time to be alive, but they didn’t know they were wrong. So I considered that we wouldn’t necessarily be immune to such a possible belief now, and maybe there’s some commonly held idea that is the exact opposite of the truth. I wondered, what if increasing atmospheric levels of CO2 were the right thing to do, rather than trying to constrict it. With this assumption I’ve searched for contradictions and have yet to find any.
“So…we should probably develop a wavebased general relativity theory using the Eulerian approach, shouldn’t we? :)”
i may be a little off track,however..
ref. T.G.S. ‘Mathmos’‘galvanism’ …seems to ‘pop’out to me….
cant say i have grasped the ‘gravity of the point’ completly,..yet. :)