Over on measure space, Toby brought up an old Usenet discussion.

It is a fun blast from the past. I started leaving comments about it there, but decided to move them here.

Eric: Oh oh! This is interesting. In this post, I was (not very successfully) arguing that “maybe” $vol$ should be used as a fundamental variable. At the time, I didn’t know it, but later, when Urs and I started working on this stuff together, we realized that to define a Lorentzian manifold, the “causal structure” gives you everything you need “up to volume”. What that means is that by actually specifying

a prioria volume form (as I was suggesting) and specifying the causal structure, i.e. poset, then you’ve got your spacetime. I should put thought into that and find a home for it on some page here on the n-Lab. For now, this will have to suffice as a reminder to myself.

It is funny to resurrect discussions from nearly 8 years ago, but in this post, Toby responded to something I said with:

Eric: Yeah, this is more like what I am thinking about. It seems like E can be deduced from vol in a fairly simple manner (based on a perhaps flawed intuition). Integrating E over a surface gives the area of the surface. Sounds a lot like some kind of restriction of vol from n-d to 2-d.

Toby: Volume is not sufficient to define area, unfortunately. Suppose that the world were stretched to double size along the N/S axis and squashed to half size on the E/W axis. Then no volumes would change, but many areas would!

Eric: I don’t see that I made any reply to this, but now I would say that I was not suggesting you could get area from volume. I was suggesting that *maybe* you could get $E$ from $vol$ somehow. What if we thought of $vol$ as a pseudo 2-form-valued 2-form? Could somehow integrating over a 2-surface result in a pseudo 2-form?

Toby: I'll take your word for it on what you meant or didn't mean, rather than try to justify my assumption these many years later (^_^).

But you can't expect to get a *form* (a local concept) as the result of an *integral* (a global concept), can you?

Eric: Probably not, but who knows what kind of possible tricks we could cook up? :) I’ve got cobwebs in the corner of my mind that thinks about this stuff, but I asked myself, what if instead of “integrating” you do some kind of “partial evaluation” of a bivector on the volume form, but some dust flew off and I remembered something along the lines that

$i_B vol = \star b,$

where $B$ is a bivector corresponding to the 2-form $b$.

Eric: So continuing the thought from above, if $E$ can be written as the Hodge star of an $(n-2)$-form, then (I think) there is a bivector $X$ such that

$E = i_X vol.$

Oh oh! Wait. The existence of a Hodge star is less important than the existence of this bivector. So I guess the more important question is that given a volume form, is there a bivector such that

$E = i_X vol?$

*Toby*: As long as $vol$ is nondegenerate (everywhere nonzero), yes.

If $vol$ is just something that measures volumes, then I would say that it ought to be a nondegenerate pseudo $n$-form. That is enough to define a sort of Hodge star just as you have above; given a $p$-form $\omega$, you get a pseudo $(n-p)$-vector field $v$ by demanding

$\omega = i_v vol .$

Now, if $vol$ actually came to you from a nondegenerate metric, then you can go further and turn your pseudo $(n-p)$-vector field into a pseudo $(n-p)$-form. If your manifold is oriented, then you can go further yet and turn your pseudo $(n-p)$-form into a $(n-p)$-form. And now we've recreated the standard Hodge star on an oriented (semi)Riemannian manifold. But in a way, the most fundamental Hodge star, the one that can be defined using only a nondegenerate pseudo $n$-form $vol$, turns $p$-forms into pseudo $(n-p)$-vector fields.

In one of the posts in that famous thread, you and John discuss how to do the Hodge star if the metric (and thus $vol$) might be degenerate. You were able to make it work, in effect, by defining the *inverse* of the usual Hodge star instead of that usual Hodge star itself. Since really, the choice of one over the other is only a matter of convention even when $vol$ is nondegenerate, we can take that attitude here. In that case, the *most* fundamental Hodge star, one makes sense given *any* pseudo $n$-form $vol$, is one that turns a $p$-vector field $v$ into the pseudo $(n-p)$-form

(1)$\star v = i_v vol$

(or similarly turns a pseudo $p$-vector field into an $(n-p)$-form, etc).

And now that I think about it, I believe that your post with John was wrong (even though you convinced him). That's because you allowed a possibly degenerate metric to define a possibly degenerate pseudo $n$-form $vol$, which means that your metric must have been (as it standardly is) a metric that takes inner products of vector fields (rather than of $1$-forms). But then you used it to take the inner product of $p$-forms, which you can't do if that metric is degenerate. So I don't think that, even in the presence of a metric and an orientation, you can turn a $p$-form into an $(n-p)$-form, unless the metric is nondegenerate. Another way to see the problem is this: while even a degenerate metric will turn $p$-vector fields into $p$-forms, only a degenerate metric will turn $p$-forms back into $p$-vector fields, but that is what you'd have to do to extend that most fundamental Hodge star (1) any further.

Eric: I think it was actually John who had the definition of Hodge a little bit backwards in a way that required the inverse. (Edit: Here is the relevant post.)

Toby: My interpretation of that post (and the ones that it followed up) was that you came up with the idea, and John came up with the precise formulation in the course of conceding that your idea worked.

If $vol$ is a non-degenerate $n$-form, then I wonder if the interior product becomes invertible, i.e. given a pseudo $(n-2)$-form $E$, we get a bivector

$X = i_E vol.$

Can we recover $E$ via

$E\stackrel{?}{=} i_X vol.$

Put another way, if we define

$\tilde\star E = i_E vol,$

is $\tilde\star^2 = Id$?

Eric: Ok. That was a little embarrassing, but in my defense, I can only concentrate for a few minutes at a time before life intervenes :)

At least with standard definitions, $i_E vol$ doesn’t make sense. In the next few minutes before life intervenes again, I wonder if I can say something that makes a little more sense.

Start with something safe:

$(a,b) vol = a\wedge \star b$

means that $\star 1 = vol$. **BZZT!** Time’s up! What I am thinking about is somehow inverting these products, but right as I’m headed to bed, I’m reminded of the “exterior geometric algebra” (which is essentially “geometric algebra” using forms), which means I should probably review the stuff Urs was talking about way back here.

Eric: Here is a statement by Pertti Lounesto:

The constitutive relations depend on the medium. They do not determine the geometry (of Lorentz signature).

I actually disagree with this statement and that disagreement is the basis for a comment on electromagnetic field. In fact, there is a very pretty duality between curved manifolds and inhomogeneous media. Somewhere (and it wouldn’t be hard to reproduce, plus, I’m sure I wasn’t the first), I showed that the Schwarzschild metric could be mapped to inhomogeneous electric permittivity and magnetic permeability. I even used this fact to develop a boundary condition for numerical simulations. I called it a “Black Wall” :) It was neat. The electromagnetic wave approached the wall and appeared to slow down. With some tweaking it might have been amenable to practical numerical models, but the way I did it was very rough to demonstrate the idea and I did not pursue it further.

Eric: Due to the magic of Google, I find where I talked about the “black wall” here

*Toby*: How about kinetic energy $K = \frac{1}{2} m \|v\|^2$? Here $v$ is a tangent vector, but $K$ is a scalar, so we need to introduce geometry in the form of the metric that takes the norm of $v$. More explicitly and in components, $K = \frac{1}{2} m g_{ij} v^i v^j$. Normally one thinks of $g$ as the geometry, but perhaps you would like to think of the $m$ (maybe even the $\frac{1}{2}$???) as part of the geometry too? That is, in (what one would normally regard as) local coordinates, the metric that appears in $K$ is

$\left( \array {
m & 0 & 0 \\
0 & m & 0 \\
0 & 0 & m }
\right )$

but could conceivably be something more complicated (for an ‘inhomogeneous’ particle?). I think that the situation is analogous.

Eric: Ah ha! Good question. I don’t know the answer :)

Here is a thought. To model a particle in an electromagnetic field, you replace the momentum

$p\to p-e A,$

where $e$ is a constant and $A$ is the vector potential (see kinetic momentum).

The pure electromagnetic field case is what you get when you set $m = 0$ and the pure free particle case is what you get when you set $A=0$. I think you are on the right track looking at the “mass tensor”.

*Toby*: Eric, I've been rereading the Usenet thread on pseudoforms, and I'm looking particularly now at this post. There's a lot of stuff in there, but I'm looking at the idea (towards the bottom) that, if forms are defined on a cellular complex, then pseudoforms are defined on the Poincaré dual complex. And I'm doubting that. I'm thinking that you thought so (back then) because you were using the metric on space(time?) to identify a pseudoform with its Hodge dual (a form) and that the dual complex was needed to perform the Hodge dual. But I'm thinking that a form or pseudoform should be definable on any given underlying combinatorial structure, only with all of the little arrows for one pointing perpendicularly to the little arrows on the other.

But I really don't know for sure, this I don't have a copy of your thesis. Can you send me one? And can you say anything about this issue of the dual complex?

Eric: That thread really was a lot of fun and I’ve received several emails over the years, even as late as last year, from people saying how much they enjoyed it :)

A copy of my thesis is here on my wiki web! :) It is embarrassingly incomplete though. I do not talk about dual lattices, but what I do talk about and what Urs and I developed subsequently supports your idea about the (lack of) necessity of the dual lattice.

In Section 3.4: Orientation, I talk about inner and outer orientation on a *given* lattice. I developed a notation for inner and outer orientation on simplices that I was pretty happy with. Regular forms have *inner* orientation whereas pseudoforms have *outer* orientation. By discussing dual lattices, I was giving a nod to others in computational electromagnetics that build models on a lattice and its dual.

Later, Urs and I developed everything we needed for discrete differential geometry (including pseudoforms) on a single diamond complex without ever introducing a dual complex. But the dual complex of a diamond complex is also a diamond complex, so I thought that we *could* discuss Poincare duality if we ever got around to it.

So yeah, I would tend to agree with you that the dual lattice is not necessary to talk about discrete pseudoforms. What is important is to discuss inner and outer orientation.

- John Baez describes numerous choices of variables for gravity (post)

Revised on July 26, 2009 at 16:11:51
by
Toby Bartels