...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Thursday, August 29, 2002

Ah, the Hodge star operator. What did I ever do without it? Not much! I promised to write about it a long, long time ago, when I was beating my head against a wall trying to grasp it, but haven't gotten around to it until now. Mea culpa.

So what is it? Well, it took me quite a few days to figure it out, and I'm going to try to give a brief overview of my picture of it at this point. First, we'll need the concept of differential forms. If you don't know what those are, none of this will probably make sense. Briefly (I talk about this more in previous posts), a '1-form' is just something that eats a vector field and craps out a number. A 1-form looks like this: df. If you have a function f, you can take its exterior derivative, and get an object called df, naturally enough. On R^n, this is df = [Partial derivative with respect to u] * f * dx^u, where Einstein summation convention is used, and the dx^u's are basis 1-forms on R^n.

Anyway, we've got 1-forms. If you take the exterior derivative of a 1-form, you get a 2-form, and so on. Also, there's something called a wedge product that you can do to forms (and vectors, too!). It's meant to be a generalization of the idea of a cross product of vectors. This just has to satisfy u /\ v = -v /\ u, and u /\ v has to form 'an algebra'. But never mind the technical details! The point is, if you wedge two 1-forms together, you can get a two-form, and so on.

(By the way, 2-forms look like so: dx/\dy (i.e.,two one-forms wedged together, as I said above), and say 3-forms look like dx /\ dy /\ dz.)

Now, if you take a 1-form, and take its exterior derivative, you get a 2-form, as I said above. But, if you actually try this on R^3 (meaning, good ol' space), you get something that looks a hell of a lot like a curl! Similarly, if you wedge two one forms together on R^3, you get a 2-form, you get something that closely resembles a cross product!

Except it doesn't look exactly the same. The difference is, first of all, is that when people say 'curl' and 'cross product', they are talking about vectors or vector fields, not 1-forms. But this is actually not a complete disaster, because if the manifold we're playing on is equipped with a metric, we can use that to make 1-forms into vector fields, and so everything's cool. Except it's not.

The problem is that, as I noted above, we're not dealing with 1-forms, we're dealing with 2-forms. And those aren't the same as one-forms, and they most definetly aren't the same as vectors! (Or we could be dealing with '2-vectors' - meaning two vectors wedged together, but the point is the same: the result of the wedge product of two vectors ain't a vector, it's a 2-vector.)

So, we need a way to convert two-forms to one-forms, and vice versa. This is where the Hodge star operator comes in. Visually (on R^3), a 2-form can (with caveats) be pictured as a little 'area'. That is, say you picture a 1-form as a little arrow (huge caveats here), and you put two of them end-to-end, they form part of a parallelogram. That parallelogram 'is' a 2-form. The Hodge star operator will convert that area into an arrow, which will be perpendicular to the two other arrows.

The full-blown definition goes something like this. A Hodge star operator, denoted by *, on an n-dimensional manifold M equipped with with a metric and an orientation, is a linear operator from p-forms to (n-p)-forms on M. (Example: Make M = R^3. Then n=3, and if m is say 2, then *: 2-forms -> (3-2)-forms = 1-forms.)

The Hodge * operator has to satisfy v /\ *u = vol, where v and u are p-forms on M, *u is the 'Hodge dual' of u, that is, a (n-p) form, is the inner product of v and u (you need a metric to define it), and vol is a volume form on M. Whew.

A 'metric' is just a way of taking 'inner products' (aka 'dot products'). Given two vectors (or other things), a metric will spit out a number. Metrics have many, many uses! You need a metric to define distances, and to define volumes, and areas, and angles, and other things. For instance, if you want to be able to tell if two things are 'perpendicular', you need to define 'perpendicular' first, and that calls for a metric! (If the inner-product of two unit vectors is 1, they're perpendicular - remember, 'inner-product' is just a fancy way of saying 'dot product'.)

A 'volume form' is a funny thing. On an n-dimensional manifold M, a volume form is a 'nowhere vanishing n-form'. But what it really is is a way to measure volumes! Or at least, that's one use. On R^3, an example of a volume form would be dx/\dy/\dz. You'd integrate it 'over M' to get the volume of M, hence the name. There's another issue hiding here, and that's 'orientations'. Turns out, to properly talk about a volume form, you need an 'orientation'. That's just a way of telling between left and right, really. The 'standard volume form' on R^3 is dx/\dy/\dz. dy/\dx/\dz is also a volume form, and it's just as good, but it's in the opposite orientation from the conventional standard, as it turns out. The way to tell if something's in the standard orientation is to see if it's an even or odd permutation away from the standard. (This is actually a theorem, but a very handy one.) For example, on R^3, we have dx/\dy/\dz. Let's relabel that as dx^1/\dx^2/\dx^3. The 'standard' is then (1, 2, 3). An odd permutation of that, for instance (1,2,3) -> (2,1,3) -> (2,3,1) -> (3,2,1), where we had to flip two numbers three times, an odd number, gives a 'negative' orientation, dx^3/\dx^2/\dx^1. If we were to do an even number of 'flips', we'd get a positive permutation - meaning, it gives a 'right and left' the same as the standard, instead of the opposite way.

Anyway, I won't go into actually calculating Hodge duals, because that's generally something of a pain in the ass, but I'll give a few examples: *dx = dy/\dz, *dy = dz/\dx, *(dx/\dy) = dz, *(dy/\dx) = -dz, etc.

So the Hodge star operator can turn 2-forms into 1-forms. (And 3-forms into 0-forms, which are just functions, and vice versa.) This is what we needed! Now we can take the wedge product of two 1-forms, apply the Hodge star, and get a 1-form. Or take the exterior derivative of a 1-form, and apply the Hodge star. And if we do that, then what only looked sort of like curls and cross products will be curls and cross products!

What's more, the exterior derivative of two-forms on R^3 looks like the divergence of a vector field, except it's a 3-form. Applying the Hodge star, we actually get something that is the divergence. And it turns out that the exterior derivative of a function is just a way of talking about its gradient without using a dot product. So, first of all, it turns out that on R^3, the exterior derivative is a grand generalization of gradients, curls, and divergences, and second, hodge stars let us pretend two different kinds of forms are 'the same'. (Well, ok, we need a metric in there to get from forms to vectors, but whatever - we need a metric to define the Hodge star operator anyway.)

Now, as I've briefly mentioned a few times, a wedge product is a generalization of a cross product, but to actually get a 'cross product', we need a Hodge star, and for that, we need a metric and an orientation on our manifold, in the case of the cross product, R^3. There's actually a nice visual reason why we need a metric and an orientation to define a cross product.

People often say 'a cross product of two vectors is a vector perpendicular to both of the input vectors, and its direction is given by the right hand rule'. There's a lot hiding under the bonnet here! First, you can only claim the result of a cross product of two vectors is a vector, and not some kind of '2-vector' in R^3. See, it turns out that the 'algebra' formed by the wedge products on an n-dimensional vector space V, called "/\^pV", and it has n!/[p!(n-p)!] dimensions.

The thing of it is, if V is 3-dimensional, that is, n = 3, then /\^2V (the 'algebra of two-forms', if you will) has dimension 3!/2!1! = 6/2 = 3, which is again 3-dimensional. So we can, if we want, pretend the wedge product of two vectors is a vector, instead of thinking of it as it really is, which is a 2-vector - they have the same number of dimensions, and using a star operator, we can turn one into the other and forget we ever used it. But this only works for n=3! 3D is special that way!

Anyways, here's where the visual stuff comes in. The wedge product of two vectors in 3-space 'looks' like a parallelogram formed from the two vectors. Now, if we want to change that to look like a cross product, we'll need to somehow change that parallelogram into a vector, and what's more, it'll need to be perpendicular to both of our original vectors. Not only that, but of course there's two such perpendicular vectors in 3-space (pointing in opposite directions to each other), and we need to pick one.

Well, we need a hodge star to do all that. The metric comes in getting things 'perpendicular', and a choice of orientation comes in to pick which of the two possible vectors we'll call 'the' cross product.

Whew. My fingers are tired, I'm confused, and so on. Watch out: having just proof-(ha!)-read over this, I think I made a muddle of distinguishing 1-forms and vectors. Untangling that mess is left as an exercise for the sadomasochistic reader. If you want, you can make this resemble what I 'promised' I'd deliver yesterday, by adding a supply of your own sex, violence, and bridges. Also Shakespeare.

Disclaimer: This is from memory, muddled, confused, and unlikely to be particularly educational (or entertaining, but hey, one never knows.) It's likely riddled with errors, too, both intentional (where I fudged stuff to avoid explaning tricky stuff) or unintentional (where I either just don't get it, or simply screwed up.) If you want the real thing, check out Baez and Munian's book, which is excellent, has far more details, and is very clear.

I shall update this tomorrow with a storied tale of courage under fire, blood, bridges, tears, sweat, sex (neither goats nor sheep will be involved, to the relief of some of my readers and perhaps to the consternation of others), bombs, hair-pulling, Shakespearean flights of poetry, and yet more sex to end the final act on a high note. There will also be exterior derivatives, integrals, and 'gauges' discussed.

I might have overreached a bit in the first sentence above. Don't sue me if there aren't any bridges!

Thursday, August 22, 2002

I spent about six hours straight on one problem, involving a line integral, of all things. Line integrals tend to be relatively easy to do, right? I thought so too. Still do. Nonetheless, I just couldn't get this thing to come out right. And I didn't even need it to get 'the answer', because I saw what it should be using vigorous squinting! But I just couldn't get it come out right.

My mistake: cocking up a dot product, repeatedly, for six hours.

I'm going to go beat my head on the wall now.

Wednesday, August 21, 2002

Holy donkey poo. I've just finished working out what an integral means (a particular integral, not 'integral' in general). Damn, this thing was hairy. Hmm. I guess a brief bit of catch-up again is advisable. To my surprise, I've actually Accomplished A Goal, and I now know how to write down Maxwell's equations on a (sort of) arbitrary semi-Riemannian oriented manifold. It's really very simple: dF = 0, and *d*F = J. Explaining what that means is a different question altogether, though.

Anyway. I'm now reading the next chapter, with the rather terrifying title "DeRham Cohomology Theory in Electromagnetism". So far, it's all about getting down and naughty with 'potentials', both scalar and vector. A scalar potential is a function Phi such that E = dPhi, where E is the electric one-form (you can get away with calling it a field if you squint at it). It's called 'potential' because it has a lot to do with such things as 'potential energy' and whatnot. Moving on. If all you've got is Phi, you'd need to integrate to get E. And it turns out you have to integrate E along a path, say from point p to point q, to get Phi. That's all fine and dandy. But can we actually do that? What can stop us? (This is all the stuff the book asks.) Well, we'll be in deep shit if there isn't a path between p and q, for starters! That is, our manifold (or space, or whatever) damn well better not be scattered around the room in chunks - we'll need to be 'connected'. Next, it's fairly obvious that 'in general', we'll get different Phis if we go and integrate along different paths. This is Bad Mojo - we want phi to be defined reasonably uniquely (it'd be perverse to have an infinite number of Phis for every E). So the question is, what conditions need to be in place for our integral to be 'path independent', that is, giving the same answer for any reasonable path? The way to attack this turns out to find a crafty way of writing down an integral that integrates along all paths, and then differentiate it with respect to the 'change in paths', kind of, and then see what will make it zero. That way, we'll get the conditions on E that will make it path-independent.

Actually doing that is a total bitch, requiring strong calculus-fu. There's all the differentiations, and chain rules, and integration by parts, and products, and watching after notation, and hair-pulling, and other indignities, all to do one damn integral. By the time I got done with it (which was ten minutes ago, and I started last night), I felt like poor Polly Nomial. A quote from the linked Saga:

...She was being watched, however: that smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. Was she convergent? He wondered. He decided to integrate improperly at once. Hearing an improper fraction behind her, Polly rotated and saw Curly approaching with his power series extrapolated. She could tell at once from his degenerate conic and his dissipative terms that he was bent to no good.

It was like that, but scarier, different in the details (for one, I'm not a female nor a polynomial) and it lasted a long time. Ahem. Anyway, in the end it turns out if dE = 0, it's going to be path-independent. This is great, because dE has to be 0 in the first place, because that's a consequence of Maxwell's equations.

I'm a lazy bum, so I still don't have anything ready to post on the stuff I listed earlier.

Saturday, August 17, 2002

Humor time!* First, you owe it to youself to carefully read this opinion piece from America's Finest News Source, The Onion**. Next, you should take a very careful look at this scan of an interestingly tasteful advertisment for a personal lubricant (it's from France, which explains everything). Also, you may be amused to know that when I took a gander at my referrer logs the other day, I discovered that one of the most common search engine queries used to find this page was ... wait for it ... "hairy balls".

There is a reason for this, and it isn't that this site is secretely a gay pr0n repository that Google has discovered and outed thanks to their PigeonRank technology. No. That's not it at all. Really. You have to believe me! Ahem. See, some time ago, I wrote a blog entry which made reference to hairy balls. It did so at considerable length, and contained many repetitions of the phrase "hairy balls". It also contained a few instances of the word "cheerleaders". Well, apparently this was enough to propel my humble blog onto the third page of Google's links to web sites where one can find hairy balls, in the illustrious and colourful company of alternatively-oriented pr0n sites.

This, I think, is hilarious in and of itself. It's even more funny (but also a little sad) that the people who have come to my site in a desperate search for hairy balls (they went all the way to the third page of search results!) must be terribly dissapointed. To see why, you'll have to go and read that old post.

In other news, I think I have sorted out the archives, so old posts should now be findable. Also, I've written a little more about 'point set surfaces', so that's getting close to being posted. Neck-n-neck, however, is a tale of my struggles with 'Hodge star operators', whatever the hell they are. I dunno which will get posted first, but it's probably one of those two -- the other stuff I listed out earlier is a bit further off.

Oh, and I forgot to complain about this last week, but the geese and ducks that live in the park where I sometimes drag myself to jog (well, ok, waddle is perhaps a better description) have absolutely no respect for me. They diss me all the time. For one thing, the geese take their sweet time getting off the foot paths so I can get past, and they hiss and beat their wings threateningly as they do. Bastards. And both they and the geese routinely moon me as I run past while they are swimming about in the lake. No respect whatsoever. There was a wonderfully fitting snippet of rap lyrics that I wanted to put in here (something about those who diss homies getting caps in their asses***) that I can't quite remember. Darn.

* -- This is much like Hammer Time, but not quite as rocking, and without MC Hammer in da house. Which is, I think you'll have to agree, a good thing. You don't want MC Hammer in your house.

** -- And no, for the record, I've never even thought about the anatomical details of certain famous Star Wars characters until I read that Onion article. Also, as for any long-time reader of the Onion, the combination of "Onion" and "penis" are for me indelibly associated with this this classicOnion article.

*** -- Oh lord almighty. If Google thought this site contains interesting material concerning "hairy balls" before, it's going to be even more certain of it now. Good thing I haven't yet thought of a way to use goats, sheep, or other exciting barnyard animals for metaphors yet! I mean, just look through the potential keywords scatered throughout this entry... I swear, it's not intentional!

Monday, August 12, 2002

Well, I've been silent a little while, for a lot of assorted reasons. Which I'm not going to go into. Anyways, I just figured I'd write down for posterity a few of the things I've got kicking around on my hard drive at the moment, in various stages of (dis)composition.

  • There's a paper I've been trying to read which goes into more detail about what "1+2=3" means than I thought existed. It's hilarious, entertaining, and thought provoking. I've referenced it obliquely before, but hadn't read much of it at the time. I've read a bit more now, and have started trying to summarize some of what it seems to be saying into a more-easily-readable form (where more readable just means 'less funny greek letters').

  • I've recently read a set of papers about something called "point set graphics" (following a trail of links from a SIGGRAPH2002 summary). Say you have a physical thing you want to represent in your computer. The conventional approach is to scan it, getting a set of points corresponding to the surface of the thing, and then 'triangulate' it - to fit a polygon mesh to it, connecting the dots, so to speak. Then that's your model. This is what's used in games and such, by the way - the models are made up of triangles. In 'point set' graphics, you don't 'triangulate', you just work with the more-or-less raw points the whole time. Why anyone would want to do this is going to be the subject of a future posting, if I ever get it written.

  • I'm continuing my reading of Baez and Munian, and am now on the part where they rewrite Maxwell's equations. It's getting more difficult, mainly because I haven't quite digested a lot of the concepts yet - metrics, orientations of manifolds, Hodge stars, and other funny-looking gobbledygook that is probably going to make obvious sense eventually but for now just looks funny and/or intimidating. Again, more on this later.

  • Inspired by being annoyed with the phrase 'closed timelike curve', I've decided to try to use the definitions of the terms in the phrase to get a handle on it. And by handle I mean something better than "'Closed timelike curve' is fancy-talk for time travel, 'mmm-kay!" I've actually written a post on this tonight, but I'm unhappy with it, so I'm not posting it for now. Later, perhaps, when I've had a chance to spank it into a more tender shape.

And that's about it for now.

Sunday, August 04, 2002

The archives are driving me nuts. Blogger's system for them is insane. You can see the resulting clusterfuck in my manual attempt to get archives listed over on the right. Why May 2002 can't be listed as one damn month on one damn page, and can only apparently be listed in three chunks, I don't know.

If I had money, I'd sign up for a web host that allows perl, and go to, say, MovingType. But alas, I don't.

In my browsing, I came across this thread. Unfortunately, anyone who buys the claims made on the pages linked to in the first post is being taken for a ride. This is a bogus patent (so far as I can tell). For what it's worth, you may be distressed to know that the USPTO often issues idiotic patents, including for perpetual motion machines (if you don't literally call it a perpetual energy machine, and throw in lots of Star-Trekesque jargon to cover the fact that you're full of shit).

In the second of the linked pages provided by sIntax in the title post, the following is said:

The complicated physics of how the MEG works is explained in the paper "The Motionless Electromagnetic Generator: Extracting Energy from a Permanent Magnet with Energy Replenishment from the Active Vacuum," which can be found at Tom Bearden's website

Following the link, we start reading this paper with the impressively ponderous title.

The first sign of trouble comes right on the first page. Notice that all of the author names have claimed degrees listed next to them. This should set off loud warning bells for you! Basically, if you see people making big claims, and loudly trumpeting degrees, be very, very careful. You don't see degrees next to names in actual scientific publications, such as Physical Review Letters or what have you. You don't see "Richard Feynman, PhD" on his Lectures on Physics. You do see them on popular diet books and on creationist tracts. The presence of degree claims on an argument is not a foolproof BS detector, but it's enough to raise my hackles.

I'm not going to do a paragraph by paragraph critique, because I don't have the time, nor, to be honest, the expertise. But there are a few howlingly stupid things that even a beginner like me can see in there. So those I'm going to highlight. Oh, and notice that there are like, five equations in the whole forty page paper! Basically, all the arguments are not mathematical, they're verbal. Which is a very, very bad sign in a paper attempting to overthrow electromagnetism!

Anyway, moving on to the body of the paper. Second paragraph, first page:

Since the present "standard" U(1) electrodynamics model forbids electrical power systems
with COP>1.0, we also studied the derivation of that model, which is recognized to contain flaws
due to its 136-year old basis. We particularly examined how it developed, how it was changed,
and how we came to have the Lorentz-regauged Maxwell-Heaviside equations model
ubiquitously used today, particularly with respect to the design, manufacture, and use of
electrical power systems.

Ok, this is BS. Putting 'standard' in quotes does not mean that E&M "is recognized to contain flaws due to its 136-year old basis." The picture of the electromagnetism as being "Yang-Mills theory with the gauge group U(1)" is a rather modern one, given that Yang and Mills published their paper on what are now known as the Yang Mills equation in the 1950s. Classical electromagnetism is not 'recognized to contain flaws', except in the sense that it's not a quantum electrodynamics, that is, it's not a quantum theory. Moving, on, though...

Now, look at pages 6-9, where they apparently describe how stuff works. On page 6, we get this howler:

The conservation of energy law states that energy cannot be created or destroyed. What
is commonly not realized is that energy can be and is reused (changed in form) to do work,
over and over, while being replenished (regauged) each time. If one has one joule of energy
collected in one form, then in a replenishing potential environment one can change all that
joule into a different form of energy, thereby performing one joule of work.

Ok, this is bullshit. Energy 'replenished'? And 'regauged' is a synonym for 'replenished'? Um, no. They are apparently hoping that no one reading this paper has ever seen the word 'gauge' before, and will get intimidated into buying the jargon. Here's a brief and incomplete (hey, I'm just starting to learn about this myself) sketch of picking 'gauges'. Looking at the magnetic field B as a two form, we can define a vector potential to be a one-form such that B = dA, where d is the exterior derivative. The thing is that A is not unique, in the sense that we can add df (f being a function, d the same as above) to A without changing B. (This is an identity from how exterior derivatives work: d(A + df) = dA + d(df) = dA, because d(df) = 0.) This sort of thing is called 'gauge freedom', and works for the electromagnetic field, too (i.e., I used the magnetic field above as an easier example). Placing conditions on A (adding on df's, stuff like that) is known as 'picking a gauge'. It does NOT do anything about field energy!

That's not even mentioning that what the quoted paragraph is suggesting is something that violates some rather well-respected laws of thermodynamics...

Mind you, these characters are not using normal E&M theory, or so they say. They claim that the normal theory is wrong, and that their never-actually-articulated theory is right - but given that they never actually lay out precisely what their theory IS, it's rather hard to discuss...

Page 10 contains some real howlers.

So any amount of energy can be collected from any nonzero scalar potential, no matter
how small the potential's reaction cross section, if sufficient intercepting charge q and collecting
points x, y, z are utilized. In short, one can intercept and collect energy from a potential
indefinitely and in any amount, and in any form taken by the interaction, because the potential is
actually a set of EM energy flows in longitudinal EM wave form, as shown by Whittaker {1} in
1903 and further expounded by Bearden {24, 26}.

This is simply wrong. They're trying to get infinite energy out of electromagnetic fields! This just doesn't work: the energy in an electromagnetic field looks like E^2 + B^2 (E and B being electric and magnetic field strengths respectively). And 'reaction cross section' for scalar potential? Wtf? And what, precisely, is 'suffient intercepting charge'? If they mean taking some electric field, and sticking an arbitrary amount of charges in there, then they are indeed going to strengthen the field, because electric charge gives rise to electric field. But you'd need an infite amount of charge to get an infinite amount of field, of course.

Later on the same page:

For the magnetic vector potential, some preliminary comments are necessary. First, for
over a hundred years it has been erroneously advanced that the magnetic vector potential A is
"defined" by the equation

B = (del) � A [2]

This is easily seen not to be a definition at all, since an equation says nothing about the nature of
anything on its right or on its left, but merely states that the entire right side has the same
magnitude as does the entire left side. For an expression to be a definition, it must contain an
identity (:=) sign rather than an equal (=) sign. Hence in seeing what is attempted to be defined,
we rewrite equation [2] as

B := (del)�A

This is idiotic. First, to reconcile notation, B = (del) x A is just the same as B = dA above, since the curl (the (del) x part) is the special case of the exterior derivative on R^3 for one-forms (well, ok, you've got to pick a right-hand rule, too, and pretend that one-forms are vectors, but whatever). But moving on. First, a magnetic vector potential is indeed defined by B = dA. Note the article 'a' instead of 'the' -- as observed above, A is not uniquely defined due to 'gauge freedom'. But that's not what they are objecting to! They say that this is a bad definition since "an equation says nothing about the nature of anything on its right or on its left, but merely states that the entire right side has the same magnitude as does the entire left side." First, that's wrong, because equations aren't just about magnitudes - in this case, you've got direction to worry about, too. But ultimately, they're complaining about the use of an equals sign instead of a 'defined by' sign. This is insane - they THEMSELVES choose to use an equals sign, and then they complain about it!

Further, people generally use this relation (B = dA) to define A, not B. That is, A is the one-form such that the equation holds.

I don't think I need to go on. These people don't hold PhDs in physics, that much is obvious. PhDs in theology from Bob Jones University, maybe. Physics, no.

This 'paper' is full of bullshit arguments, vigorous handwaving, and no math that wouldn't induce peals of laughter in anyone even marginally familiar with some of the terms they use. Don't be taken in.

There ain't no such thing as a free lunch.

References used:

J. Baez and J. Munian, Gauge Fields, Knots, and Gravity, World Scientific Publishing, 1994.

Gauge transformations

Lorentz gauge

Friday, August 02, 2002

Yay. Blogspot works again.