Can we analytically continue the Standard Model?

[jackmell]

Why can't we analytically continue our model of the Universe past the Big Bang in a way analogous to how the Euler sum is analytically continued into the zeta function? If this were possible, this extension would encompass a larger phenomenon but would reduce down to our Universe when certain parameters are reached just like the larger phenomenon which is the zeta function reduces down to the Euler sum when Re(s)>1. In this analogy, the reason GR fails at the Big Bang singularity is in some as yet unknown way, similar to why the Euler sum fails when we attempt to use it past it's region of convergence.

This just seems to make sense to me.

Any chance they're trying to do something like this with Quantum Cosmology or Loop Quantum Gravity and I just do not understand it?

 

[zhermes]

The success of all new theories lies in them reducing to existing (observationally correct) solutions. E.g. special & general relativity reduce to Newtonian mechanics at low speeds and energies. So yes, that's what people are looking for in a new unifying theory.

But, extending theories beyond the big bang doesn't make sense, because there isn't universe beyond the big bang. I also don't understand the relevance of the zeta function.

 

[Chalnoth]

Why can't we analytically continue our model of the Universe past the Big Bang in a way analogous to how the Euler sum is analytically continued into the zeta function?

Analytical continuation is only possible when the singularity isn't "real", but is only there as an artifact of the particular way in which we are modeling reality. That is to say, in mathematics, there are singularities, and then there are singularities. If I expand the function:

$$f(x) = \frac{1}{x + 1}$$

in a power series about $x=0$, then the solution will work well between $-1 < x < 1$. There is a "real" singularity here at $x = -1$, and it will be there in any expansion we use, but the singularity at $x = 1$ in the expansion only pops up because of a peculiarity in how we put the series together.

This is very analogous to singularities in General Relativity. Let's examine the Schwarzschild metric, for instance:

$$c^2 ds^2 = \left(1 - \frac{r_s}{r}\right)c^2 dt^2 - \frac{dr^2}{1-\frac{r_s}{r}} - r^2\left(d\theta^2 + \mathrm{sin}^2\theta d\phi^2\right)$$

In this metric, we see two apparent singularities. One is when $r \to 0$. This is the singularity at the center of the black hole, and is a real singularity (it appears no matter what coordinate system we use). But there's another singularity in this particular metric, one at $r = r_s$. This is at the horizon to the black hole. This singularity, however, is just an artifact of these particular coordinates, and we can get rid of it by just transforming to a different coordinate system (such as that of an infalling observer).

We know that the singularity at the center is a "real" singularity in General Relativity because it turns out that you can actually work out the geometry of a black hole without reference to any coordinate system at all. This sort of coordinate-independent work is analogous to just considering the function $f(x) = 1/(x+1)$ to find singularities instead of looking at any particular expansion. And when we do this, we find that the curvature, which is a coordinate-independent quantity, has a singularity at the center of a black hole.

So no, we're quite certain, unfortunately, that little mathematical tricks won't get rid of the singularity at the center of a black hole. Getting rid of the singularity requires an actual change in the theory of gravity.

 

[jackmell]

Ok. I just don't know enough to mount a decent defense.

My understanding is that the singularity at the Big Bang may not be a "real" singularity but rather only a limitation in our model of Cosmology and that a more complete model may not have a singularity there. If so, then that to me is suspiciously similar to what we encounter when we attempt to use the Euler sum to "model" the zeta function: we encounter a singularity at Re(z)=1 but a more complete "model" which is the zeta function does not have a singularity at Re(s)=1 (although it does of course at s=1).

The two (singularities in the Euler sum and Cosmology) just look similar to me and I wonder if there is a connection in principle?

 

[Chalnoth]

Ok. I just don't know enough to mount a decent defense.

My understanding is that the singularity at the Big Bang may not be a "real" singularity but rather only a limitation in our model of Cosmology and that a more complete model may not have a singularity there.

It's "real" in the sense that General Relativity demands it (as far as I know, this is an actual proof, and is relatively independent of other model assumptions). But we know that GR is wrong, so a proper theory of gravity is expected to not have any such singularity.

The problem, then, is that you need a new theory of gravity. Simple analytic continuation under GR isn't going to work.