From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Densitized Pseudo Twisted Forms Date: Wed, 7 Nov 2001 03:15:50 +0000 (UTC) Newsgroups: sci.physics.research Summary: Variant frameworks for relativity on manifolds In article , eric alan forgy wrote: >On Tue, 30 Oct 2001, John Baez wrote: >> Well, like I already said, if the E field is zero, this volume form >> will be ZERO. This means you can't use it to define a Hodge star >> operator. >Ok. That is fine. So avoid the Hodge :) I am not 100% sure of that though. >If E is 0, vol is 0, and that means Hodge is 0 (I think). This just means >you can't define *^{-1}, that's all. Okay, I guess you're right. More precisely: there are two common ways to define the Hodge star operator. One is to insist that vol = a ^ *b and the other is to insist that vol = a ^ b for all differential forms a,b. These two definitions give the same thing when ** = 1, since then * is its own inverse. In situations where ** = -1 these two definitions differ by a sign - one of those nasty signs that infests the subject of differential forms! Anyway, now we're talking about situations where the metric is degenerate. In these situations vol = 0. So, the first definition works when we take * = 0, but then we can't define the inverse of the Hodge star operator. The second definition doesn't work at all - that's the one I had in mind. So, in these situations the first definition is a bit better than the second, and then everything you say above is true. But alas, even the first definition doesn't allow us to invert the Hodge star operator. And this causes lots of problems. >What I am trying to figure out is >whether A and E are the variables to use versus, say for example A and >vol. There are dozens of different formulations of general relativity. It sounds like you want to invent one where a connection A and a volume form vol are the basic variables. You are welcome to try. However, a pretty reasonable requirement is that you be able to write down the Lagrangian for general relativity using these variables. I don't see how to do it using just A and vol! Let me review a few of the most popular formulations: 1) The Einstein-Hilbert formulation uses the metric g as the basic field. From this you can write down formulas for the volume form vol and the Ricci scalar R, and the Lagrangian for general relativity is R vol. 2) The Palatini formulation uses a tetrad field e and an so(3,1) connection A as the basic fields. If F is the curvature of A, in 4 dimensions the Lagrangian for general relativity in this formulation is tr(e ^ e ^ F). Here the "tetrad field" is (locally speaking) an R^4-valued 1-form, and I'm using "tr" as shorthand for some process that turns the stuff in parentheses into an ordinary 4-form. Here I'm talking about 4 dimensions; in different dimensions we'd change the gauge group and Lagrangian in an obvious way. E.g. in 3 dimensions A would be an so(2,1) connection and the Lagrangian would be tr(e ^ F). 3) The EF formulation: like the Palatini, but we work with E = e ^ e, an so(3,1)-valued 2-form. We need to introduce a Lagrange multiplier field such that varying the action with respect to this field gives an equation of motion guaranteeing that E is actually of the form e ^ e for some tetrad field e. So the basic fields are A, E, and this Lagrange multiplier field. I'm too lazy to write down the actual Lagrangian. 4) The Ashtekar formulation. This is like the Palatini, but instead of working with the so(3,1) connection we work with just its "left-handed part", aka "self-dual part". 5) The CDJ formulation. Like the Ashtekar, but we work with E instead of e. Anyway, these are a few, and there are lots more. But I don't know any where the only fields are a connection and a volume form. If you want one like that, you'll have to invent it yourself! Now, I'm too tired to answer your other questions, so at this point I'll turn you over to the Wizard. Hey, Wiz - come here and deal with this Eric Forgy dude! [John Baez exits stage right. Growling under his breath, the Wizard enters.] [To be continued in another post.] ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Densitized Pseudo Twisted Forms Date: Wed, 7 Nov 2001 03:23:21 +0000 (UTC) Newsgroups: sci.physics.research When John Baez gets too busy, the wizard enters and continues explaining math and physics to Eric Forgy. The Wiz is much less patient than Baez, so right away Eric knows he's in for trouble. John Baez wrote: >> Someday you'll learn all the finite-dimensional irreps of GL(n) and >> you will see that not just tensors but densitized tensors and >> densitized pseudotensors play a natural role in physics. Eric Alan Forgy wrote: >Wow. That certainly sounds interesting to me. I cannot imagine how >finite-dimensional irreps of GL(n) relate to this stuff. The Wizard replies: NO??? [He throws a thunderbolt at the ceiling in frustrated rage, wondering how the educational system has sunk to such a sad level.] This is the WHOLE POINT of tensors and these fancy "densitized" and "pseudo" tensors! All these gadgets are supposed to transform in some reasonable way under coordinate transformations. We have to start by seeing how they transform under *linear* coordinate transformations, before we get into fancier stuff. Right??? But what's a linear coordinate transformation? It's an invertible n x n matrix with real entries. These matrices form the group GL(n)! So: to completely classify various flavors of tensors and their funky "densitized" and "pseudo" versions, we need to list all the ways something can transform under GL(n). More precisely, we need to understand all the REPRESENTATIONS of GL(n)! Or more precisely still, the FINITE-DIMENSIONAL representations - assuming you aren't yet interested in tensor-like gadgets with *infinitely* many components. Of course, it's enough to understand the IRREDUCIBLE finite-dimensional representations, since all the fancier representations are direct sums of these. So, you need to consult your local math expert and have them explain the finite-dimensional irreducible representations of GL(n). Each one of these gives a flavor of "tensor-like gadget". You will find the classification involves Young diagrams and little bit of extra fluff. A Young diagram is just a bunch of boxes arranged in a pattern like this: ******** ***** ** * * * The number of boxes in your Young diagram specifies the RANK of your tensor - that is, the number of indices, if you like indices. The pattern in which they're arranged tells you the "SYMMETRY" of the tensor - e.g., completely symmetric with respect to permuting indices, or completely antisymmetric, or some fancy mixture. Finally, the little bit of extra fluff tells you the DENSITY WEIGHT of your tensor, plus whether or not it is PSEUDO. The density weight tells you how your gadget transforms under dilations, while the pseudoness tells you whether or not it picks up an extra minus sign under reflections, besides what it would usually get. Your obsession with differential forms means that you only love representations of GL(n) where the Young diagram consists of n boxes arranged in a vertical column: * * * * In other words, you're extremely fond of tensors that are rank n and completely antisymmetric with respect to permuting indices. Also, you scorn them unless the extra fluff is trivial, meaning that the density weight is zero and there's no pseudo-ness. It's all very well and good that you love this particular sort of representation of GL(n) so much - they're great! But, it's terribly limiting to say that you'll never ever talk about any other sort of representation of GL(n). Heck, even the metric tensor is not of this form - it's a completely *symmetric* tensor of rank 2, so it goes along with a Young diagram like this: ** >Would you mind >recommending a reference that specifically relates irreps of GL(n) to >densitized tensors and densitized pseudotensors? [The Wizard scratches his head and thinks a minute, then becomes impatient.] Hmm, err... doesn't everyone ALREADY KNOW THIS STUFF? I can't remember where I learned it! It was so long ago. Maybe I was born knowing it. Or maybe as a child, I once wanted to understand what all this stuff about "pseudovectors" and "densities" was about, so I asked my local wizard, and he gave me a lecture just like this, and then I studied Young diagrams and it all fell into place. [He scratches his head some more and stares out the window.] Hmm... a "reference", eh? You want a "reference". Ah, if only life were so simple: you just ask for a "reference", and some wise old wizard tells you where to look... wouldn't that be easy! [Grumbling, the Wiz opens a cabinet and pulls out a tome. Blowing the dust off the cover, he shows it to Eric.] How about this: Morton Hamermesh, Group Theory and Its Application to Physical Problems, Addison-Wesley, 1962. Let's see... Chapter 10: "Linear groups in n-dimensional space; irreducible tensors". Sounds promising. Let's see how it starts... yes, this has the stuff you want. First he works out the finite- dimensional irreducible representations of GL(n), and then he moves on to fancier groups like O(n) and Sp(n) - don't worry about those just yet. The only trouble is, he doesn't quite come out and talk about concepts like "density weight" and "pseudo-ness" - that's buried in the stuff where he goes from reps of SL(n) to reps of GL(n). Surely there must be SOMEONE who has explained this more clearly, but I don't know where... like I said, this is just one of those things everyone knows... now GET OUT and let me get back to work! [He tosses the book to Forgy, pushes him out the door, and slams it.]