(28.02.2013)starfox schrieb:  So, ich hab mal etwas gesucht und schließlich so etwas wie eine Beschreibung der Formel gefunden.
Offenbar macht sie in unserer Welt so wie sie aufgeschrieben ist nicht allzuviel Sinn, aber vllt sind die Ponys ja klüger als wir Menschen^^

Spoiler (Öffnen)

It is present since Twilight's childhood (you can see it scribbled on the blackboard in the auditorium), and at present times, it appears often in Miss Cheerilee's blackboard in her classroom, many times. I wonder what is she teaching to those little fillies? ;-J

At first glance, it doesn't directly resemble anything what I know. I haven't seen exactly this equation anywhere in my study of Physics. But at the second glance, it contains many familiar symbols, all of which are related to Maxwell's Equations for electromagnetism :-> The symbols used in it are real mathematical symbols, correctly used.

Differential operators

For example, the inversed triangle symbol seen in the numerator is called "nabla", and it symbolzes the "del" differential operator. For three dimensions of space in Cartesian coordinates, it can be expanded like this:



Just a vector sum of partial derivatives for each independent direction of space. When applied to a scalar field, it produces a vector field called gradient, which shows the direction of quickest change of value. When applied to a vector field, it produces a scalar field called divergence, which shows where the vector field is radiating from (or where the sources and sinks are). So let's see now what Miss Cheerilee applies it to.

Electric displacement field

In Miss Cheerilee's Equation, it's applied to a symbol of capital D. This symbol is also known in Electrodynamics: It designates the so-called electric displacement field. At present, this quantity is used a little bit differently than originally by Maxwell in 19th century, because he used it to describe the displacement of electric fluid. At present, we don't treat electricity as a fluid, but a stream of particles (electrons). But no matter the interpretation, the law is the same.

D is a vector quantity (because it has a direction of displacement), so in Miss Cheerilee's Equation, the "del" operator means the divergence of that field: how much it spreads out and from which places. So it's something about radiation of electric fluid from matter. In this way, it's very similar to Gauss's Law, which is the first of Maxwell's Equations for electromagnetism shown here in a modern form:

I need to emphasize that this is not how Maxwell had them written originally! These are really modified versions by Heaviside. This is a whole story of how these equations evolved through history, and how they were modified several times, and there are evidences that some parts of these equations has been purposedly hidden (or censored) from public. But this is a whole another story. If you're interested, we can talk about it someday in some other place, because it's not directly related to ponies. I'm mentioning this just to signal you that there's more in our Science too that meets the eye, and what you can find officially in Physics books is not necessarily everything what's there; some things are still hidden in shadows ;-) So it's worth to be like Twilight Sparkle and dig through these dusty old books sometimes ;-)

Epsilon and mu

The other two symbols from Miss Cherilee's Equation, that is, Greek letters epsilon and mu, are also important in Electrodynamics, as you can see in the following version of Maxwell's Equations:


(See? I told you that there are many different forms of Maxwell's Equations :-P )

The former (epsilon) is called electric permittivity, and the latter (mu) is magnetic permeability. In official books you'd be told that both are constants, and that they're some special features of Nature itself which have to be taken from measurements. This is a lie. They're no more special than the units we've chosen arbitrarily for measuring electric charges, forces, distances and time. If we've chosen different units, we'd have to tweak these constants to match the reality again. They're just conversion factors resulting from our choice of units, and we can even chose units in a way that these constants will be all 1 and drop out of these equations, leaving just the bare law of physics laying before our curious eyes. But, back to the subject...

In Miss Cheerilee's equation, these constants are multiplied together in the denominator. What happens when you multiply them together? You'll get the speed of light squared, which appears also in wave equation, or in the famous Einstein's formula for mass and energy equivalence (E = m c2). So Miss Cheerilee is apparently comparing the spreading out of electric fluid from matter with the speed of light. Why? I wish I could know 9_9, but it all seems plausible for me.

Curvature of space

Also the small little triangle added to the whole fraction has a meaning in Mathematics and is used in Field Theory. It is just the "del" operator described above, taken twice in a row (or squared), that is, the second derivative. It is called Laplace's differential operator (or laplacian in short), and it calculates the curvature of space (or how much does it deviate from the average level of equilibrium). It is used widely in Electrodynamics for calculating electric potentials for a given distribution of electric charges in space. It works this way: When you have an electric potential field (which is scalar), and apply the "del" operator once, you'll get the gradient of this potential (the rate of change), which is a vector field (because it can change differently in different directions). This vector field is just electric field. When you apply the "del" operator once again (to get the laplacian), you'll get the divergence of that electric field, or "how much it spreads out". And it spreads out from electric charges, so it should be somehow related to where these charges are located in space.

So you can clearly see that Miss Cherilee's Equation has plenty of connections with Maxwell's Equations for electromagnetism. Unfortunately, it doesn't match any of the official forms of the four Maxwell's Equations known at present. Also, I couldn't transform it to any of them myself.

Well, there's always an option that this equation is one of those missing Maxwell's Equations from his original Field Theory, concerning gravity :-> The picture of a planet with rings or orbit below the equation seems to suggest it a little bit ;-J Well, I'll have to massage these equations a little bit further in some near future...

Ein paar Pics dürfen natürlich auch nicht fehlen

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cute^^ (Öffnen)

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