Friday, October 20, 2017

Misunderstanding can be constructive

During a discussion, the interlocutor may say something, and if it is not clear, you may come up with the wrong idea. When something is not clear or complete, we try to fill the gaps. Sometimes when I do this in discussions about physics I come up with a mathematical model completely different from the idea the interlocutor had in mind. Here are two examples, one about causality and general relativity, and the other about Maxwell equations independent on the space metric. Both happened to me when discussing with Tim Maudlin.

The first time it was back in 2005, at the FQXi essay contest, where we shared the third prize. I was attracted by his essay, where he describes his idea to reconstruct general relativity in a way emphasizing time as fundamental. I didn't understand what he was doing, so I asked some clarifications on the comment section of his essay. He said at some point
The idea is a bit like Causal sets, but the actual implementation is quite different. The most fundamental structure is light-like rather than time-like, and the Causal sets it is time-like.
I completely misunderstood that in a way that suggested me an idea to recover the causal structure of general relativistic spacetime, in a way that works for both continuous and discrete, and for any dimension and even variable dimension. By our exchange of emails his idea was completely different. He already published a first volume about this, but the part about relativity I understand it will be in the second one. So because my idea was completely different from what he said, I made it into a paper
and published it in Journal of Gravity, vol. 2016, Article ID 6151726, 6 pages, 2016.
It shows how you can start with a simple relation and what you get is a generalization of the causal structure of spacetime, which works for both discrete and continuum, and for any number of dimensions, even variable from point to point. I also know how to add gauge theory on top of this without complicating with a single bit the definition of the fundamental relation, maybe I will do this someday.

The second time was now. Tim wrote
And then it further turns out that there is an even simpler, less mathematically committed formalism in which you can see this same fact (by "less committed" I mean you don't need any metric). As far as I know, I just invented that formalism in the last month, but maybe someone else has already figured it out. So I do a lot of playing around with basic questions and basic formalisms, trying to be very clear about the fundamental physical postulates (the "ontology") being made. But since the very distinction between the physical postulates of a theory and the mathematical representations of those postulated items has been systematically ignored and confused and erased in contemporary practice, these questions tend not to even be asked, much less answered.
In my reply I suggested a way to get rid of the space metric:
I tried for long time to find a way to be completely metricless in 4D, but I only found partial commitments.

In 3D, E and B look like vector fields (well, B is not a vector field, but a pseudovector, or equivalenty a 3D 2-form, hence the right-hand rule), but in special relativity they are parts of the same 2-differential form F, which I like. But I prefer the U(1) fiber bundle formulation with a connection that corresponds to the potential, which I agree with you is more fundamental than the field.

If you have something written about this metric independence in 3D, I would be interested to read it. Here is how I see it, maybe it's the same as you.

In 3D this works indeed, if you write the equations in terms of the 3D Hodge duals of E and B, *E and *B and forget about the metric and Hodge. And it is not committed to the 3D metric. But if you want to use only the potential A, then the equations contain the 3D Hodge dual, which commits you to the 3D metric.

Now if you look at the *E and *B formulation in 4D, there is some partial commitment to the 4D metric in the Minkowski case. If you are interested in Galilean invariance, you are aware that Maxwell's equations are not invariant. However, let's see what we get. The Galilei group, considering the Galilean spacetime as a 4D vector space V, is the group of transformations preserving a degenerate metric of rank 3 on the dual of V, which gives the foliation and the space metric in each slice, and a 1-differential form whose annihilator or kernel is the space slice. Since you can eliminate the space metric, you will only need the 4D 1-form that gives you the time and space slices. This would extend the Galilei group. In the Minkowski case decomposed in space+time, you will need a vector field that gives you the time, and also a slicing, which can be given by a 1-form which doesn't annihilate the vector field defining the time. These encode the partial dependence of the 4D metric. In both cases you can get independence of the metric if you refer only to the 3D metric, having given the slicing and the time.

Here is what I know about the commitment to the metric in 4D, which interests me more. One thing you precisely know is that conformal symmetry of Maxwell's equations reduces the commitment to just the causal structure of the metric. This is broken back to the full metric if you plug the stress-energy tensor of the electromagnetic field in Einstein's equation. But this is regained in conformal gravity (Gerard got great results in this), where in the Einstein equation you replace the Einstein tensor with the Bach tensor. And this actually works for the full standard model without Higgs and masses, but there are results indicating that you can get a mechanism identical to Higgs by breaking conformal symmetry going back to the metric.

You can look at the differential formulation in 4D, dF=0 and d*F=J, in more ways. You see that * which is the Hodge dual operator in 4D this time, which is built out of the metric. But you can formulate it so that it is not committed to lengths. You can also avoid using the codiferential *d*, which depend on the metric, and consider instead that you have two fields F and F', where F' is actually *F. For more on this I recommend to look up "premetric electromagnetism". But I am not very attracted by premetric electromagnetism, because if you consider F' as independent also for other gauge fields F, they will lead to several independent metrics.
 But again, this was not what Tim had in mind:
There are spaces with the same dimensionality, in terms of degrees of freedom, but no one-to-one mapping from element to element. And where the expression in terms of the differential forms uses star-d-star, which requires the Hodge, I use my two different fundamental operations, Lift and Drop, each of which can be defined without any metric. The thing is defined for discrete spaces: what a continuum limit version would look like, if it is possible to even define one, I don't know.
He didn't show me how he does this in discrete spaces, but I was happy to find how it works for continuum spaces. Clearly the Maxwell equations in term of *E and *B is known, what I don't know if it is known is the full construction, which includes the 4D Galilean or Minkowski spacetime. If it is new, maybe somebody will find this useful.

1 comment:

lifeisthermal said...

"Unitary flow", the name suggests you would appreciate this:

Which I think relates to this post. My calculations are instantaneous, so time is excluded, only heat flow and geometry is used to find equivalence between heat flow(temperature potentials in points of T^4)and the force of gravity. It is an approach where the idealized model of earth as a heat engine, with only dU=Q+W, produce exact solutions. I was surprised at first that a non-interacting cavity according to the shell theorem is an exact calculation of earth temperature and the force of gravity, but then shaming myself for doubting the first law.

We have measures of temperature/heat flow, we know the power of the force of gravity and combining it with the power density of the heat source(sun), we have everything we need to balance energy in heat and work.