People warned me that writing a blog would take a lot of time. Hopelessly naive as I am, I thought, well, I would just not post anything if I am too busy. It seems, I underestimated the persistent interest of my fellow readers.
So, I wrote last week that I volunteered to bring an Honest Question to our gravity lunch. This meeting takes place here at the Department pf Physics at UCSB every Friday at noon. Lately, the discussion has been mostly about black thingies. I tried to come up with a question that would roughly fit into the string dominated atmosphere and the time constraint, and eventually settled on "Why do we live in 3+1 dimensions?"
The last time I wrote this question down, someone was so nice to tell me that 3+1=4. Therefore, let me point out that the question actually consists of two parts: a) why 3 spacelike dimensions and b) why Lorentzian signature -- I will only discuss a) in the following.
But of course I had to make the question more complicated to be appropriate for the gravity lunch. To do so, I picked the paper
Relaxing to Three Dimensions
Authors: Andreas Karch, Lisa Randall
Phys.Rev.Lett. 95 (2005) 161601 (hep-th/0506053)
Last time I was at PI, I happened to hear a seminar by Lisa Randall about the paper, you can find it online at the streaming seminars (click on "seminar series", then "filter by presenter" Lisa Randall - comes under L not R, and hit "search" -- they promised they are working on an improvement...).
While writing this post, I also found an article about the paper from economist.com
A braney theory: An explanation for the anthropic principle comes a little closer
Here is a quick summary: The idea is to take a 9+1 dimensional non-compactified spacetime. Fill it with gases of d-branes, each with an energy density and pressure. And let it expand with a Friedmann-Robertson-Walker (FRW) ansatz, i.e. homogeneous and isotropic. The paper is quite impressing, as it only contains 4 equations, which are the FRW equations in higher dimensions.
Now the question is what happens to the gases of the d-Branes.
- When the branes don't interact, the energy density will dilute slower the larger the dimension of the brane - because it can dilute only into the dimensions it does not occupy. In terms of the FRW scale parameter a, the density goes with a power -9+d.
- But they can interact, i.e. branes can meet anti-branes and decay. This decay goes slower the larger the dimensionality of the brane - because there is less space to decay into. In terms of the time t, the density goes with -9+d.
- Then the question remains whether d-branes do find each other to interact. It turns out from dimensional arguments that they will generically find each other and attempt to decay when 2*(d+1) is larger or equal 9.
From 3. it follows that 3-branes are those with the largest dimensionality that will not interact. From those that will not interact, they are also those whose energy density will dilute the least. For d larger than 3, the 9-branes do always overlap and therefore are gone. The 8-branes are apparently more complicated, but can be argued away. The only argument for the latter that I understood was that in some scenarios there just are no 8-branes. Let's assume that works.
Then, in terms of energy densities, 3-branes and 7-branes will dominate. Such the conclusion of the paper. I understand why one would like to have 3-branes. As to the 7-branes, the paper states
A configuration that is a natural candidate for four-dimensional gravity is the intersection of three 7-branes where the intersection has spacetime dimension four.
Among others, a question I raised on Friday was why this is natural. I learned that the physics on such an intersection allows chiral fermions. That explains why they write it is natural to live at this intersection. But not why it is natural.
More importantly, I fail to see why the densities of the gases are relevant for the question why we experience three dimensions. Even if the higher dimensional thingies decay, the lower dimensional ones are still around, no matter what their density is. Why is the energy density the selection criterion?
And another point that I still don't understand is how it is possible that the ongoing time-dependence in the bulk does not influence the physics (locally localized gravity) on the brane or brane/intersection. I mean, one has to make sure that things we call constant actually are constant (restrictions apply).
Bottom-line: I like the paper, I like the idea and the minimalistic setup. Unfortunately, it seems to me some of the arguments are more wishful thinking than strict conclusions.
I have certainly heard weirder things. I once sat through a seminar while the speaker explained that our universe has 10 dimensions because Pi^2 is approximately 10, and we live on a 3-dimensional submanifold because Pi is close by 3. No, I can't recall the name of the speaker, and I never heard of him again.
On the other hand, it is indeed puzzling that dimensional regularization works only in 4 dimensions, isn't it?
For those of you who are interested, here are some further references
Why do we live in 3+1 dimensions?
Ruth Durrer, Martin Kunz, Mairi Sakellariadou
Brane Gases in the Early Universe
S. Alexander, R.Brandenberger, D.Easson
On the dimensionality of spacetime
Oh, and due to a certain lack of volunteers, I agreed to bring another question for the next gravity lunch. Any suggestions?