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In the beginning, God said that the four dimensional divergence of an antisymmetric second rank tensor equals zero and there was light.

Can someone explain what is the meaning of this quote by Michio Kaku?

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migrated from math.stackexchange.com Jan 21 '13 at 2:16

This question came from our site for people studying math at any level and professionals in related fields.

I didn't downvote, but I suspect the fact that you're getting about one downvote per minute has to do with the fact that you're asking about terms that you could read up on at Wikipedia. Perhaps you won't be able to understand the Wikipedia articles, but that's because you need some other concepts to understand these terms and an answer on this site can't teach you all those concepts if you don't already have them. –  joriki Jan 20 '13 at 23:56
There's a close vote with reason "off topic". It's not off topic, it's a question about mathematical terms. If you want to close it, "not a real question" seems more appropriate, which has among its subreasons "overly broad [...] and cannot be reasonably answered in its current form". Also, I'd ask everyone who's downvoting or voting to close to consider whether they'd be doing the same if the question were written in correct English. –  joriki Jan 21 '13 at 0:02
@joriki I'm the one who voted to migrate to physics.stackexchange. It's true that the OP is asking about mathematical terms, but I suspect that someone familiar with the underlying context (ie the physics of light) might be better equipped to answer the question. –  user2617 Jan 21 '13 at 0:08
In short, the quote is a reference to the special relativistic formulation of Maxwell's equations (without source terms), which in turn govern light. –  Qmechanic Jan 21 '13 at 8:20
@Dilaton: That won't ping me.. Also, I didn't immediately close it for exactly that reason -- I thought that it probably was OK if open. Though next time, if you saw a discussion about closing in chat, reply right there. DavidZ posted it in chat to garner second opinions anyway. –  Manishearth Jan 21 '13 at 12:35

3 Answers 3

The antisymmetric second-rank tensor being referenced is the electromagnetic field tensor. It is defined as follows. Let $\varphi$ be the electrostatic potential (a scalar field), and let $\underline{A}$ be the magnetic potential (a 3-vector) from classical E&M. Concatenate them into a 4-vector $\vec{A}$. Now define the tensor of interest as the exterior derivative of $\vec{A}$: $$ \mathbf{F} = \mathrm{d}\vec{A}. $$ We can write this component-wise with partial derivatives: $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$ You can see that if thought of as a matrix, the components $F_{\mu\nu}$ of $\mathbf{F}$ are antisymmetic.

Now the use of this is that the four equations that govern classical electromagnetism (and hence light) are equivalent to: $$ \partial_\nu F^{\mu\nu} = J^\mu $$ ($\vec{J}$ is the 4-current composed of electric charge concatenated with 3-current) and $$ \partial_{[\alpha} F_{\mu\nu]} = 0 $$ (the brackets denote the summing all permutations of indices with a sign given by the parity of the permutation).

Note that depending on your unit system there may be constants like $c$ or $\mu_0$ floating around in these equations.

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What's the advantage of starting with the requirement that $F$ can be written as $\mathbb{d}A$? And if you start that way, why write down $ \partial_{[\alpha} F_{\mu\nu]} = 0 $ as well? –  NikolajK Jan 22 '13 at 8:26

From this transcript, the full quote is

And so, if you go to Berkley, where I got my PhD, you can buy a t-shirt which says, “In the beginning God said, the four-dimensional divergence of an antisymmetric, second rank tensor equals zero, and there was light, and it was good. And on the seventh day he rested.” Ladies and gentlemen, this is the equation for light.

A rank two tensor is commonly known as a matrix, and a rank one tensor is commonly known as a vector.

An anti-symmetric tensor is a tensor in which exchanging two indices negates the tensor; for example $a_{ji}=-a_{ij}$.

According to Wikipedia, the divergence of a second order (rank) tensor is a first order (rank) tensor (I've extrapolated the result there to four dimensions) $$ \nabla\cdot\epsilon=\begin{bmatrix} \frac{\partial\epsilon_{xx}}{\partial x}+\frac{\partial\epsilon_{yx}}{\partial y}+\frac{\partial\epsilon_{zx}}{\partial z}+\frac{\partial\epsilon_{wx}}{\partial w}\\ \frac{\partial\epsilon_{xy}}{\partial x}+\frac{\partial\epsilon_{yy}}{\partial y}+\frac{\partial\epsilon_{zy}}{\partial z}+\frac{\partial\epsilon_{wy}}{\partial w}\\ \frac{\partial\epsilon_{xz}}{\partial x}+\frac{\partial\epsilon_{yz}}{\partial y}+\frac{\partial\epsilon_{zz}}{\partial z}+\frac{\partial\epsilon_{wz}}{\partial w}\\ \frac{\partial\epsilon_{xw}}{\partial x}+\frac{\partial\epsilon_{yw}}{\partial y}+\frac{\partial\epsilon_{zw}}{\partial z}+\frac{\partial\epsilon_{ww}}{\partial w} \end{bmatrix} $$ As for the physical meaning, that is more a topic for physics.

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It is similar to the t-shirt designs of a couple decades ago with Maxwell' Equations in place of "Let there be light" in the famous biblical quote, except this time some designer decided to use the slick spacetime tensor way of describing electromagnetic fields. In relativity, the electromagnetic field is described by a 4D antisymmetric tensor. Its divergence, meaning the antisymmetrized derivative, is zero in vacuum. (It's nonzero at charges and currents, but I guess God didn't invent charge until the next day.)

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