"Questionable" article in Scientific American?

SW VandeCarr

You're too fast Sylas. I was editing my post when you responded. Earlier, in a recent response to a post of mine on the topic "Flat universe?" you stated that an infinite universe was always and will always be infinite. I don't see how this is consistent with a singularity at the beginning of time and a finite history.

sylas

Imagine a function of time. It should be smooth, monotonically increasing, and with f(0) approaching 0 as t reduces back towards 0. Here's the simplest example of such a function:

[tex]f(t) = t[/tex]

For a more realistic function, I'll have to give the differential equation

[tex]\frac{df}{dt}= \sqrt{0.27 f^{-1} + 0.73 f^2}[/tex]

The differential equation has a problem when t=0. In fact, it has a singularity. But that's okay. We can just let the differential equation define f for positive values of t, with boundary conditions so that f(t) approaches 0 in the limit as t approaches 0. When f is very small, df/dt is large, so it aligns very closely with

[tex]f(t) = (1.5 t \sqrt{0.27})^{2/3}[/tex]

in the neighbourhood of 0.

We call this function "scale factor", and usually add a couple of constants to scale f and t as we choose, but the above functions will do.

Now. Imagine an infinite 3D cartesian space with specks laid out in an infinite 3D grid pattern, one at every point with integer co-ordinates. This represents a static infinite flat space, filled with evenly spaced galaxies.

To apply expansion, suppose that the specks are moving apart from each other, so that this is their position at a time t when f(t) = 1. Give each speck a name (x,y,z) corresponding to its co-ordinates at this time, and at all OTHER times, suppose that the speck is located at (x.f(t), y.f(t), x.f(t)).

Voila. This is expansion. The speck at location (0,0,0) never moves, and all the others move away from it, with a velocity that is the product of their distance when the scale factor was equal to 1, sqrt(x²+y²+z²), and to the function f(t).

Of course, you can also convert the co-ordinates to see where all the specks are relative to any other speck; and curiously, no matter which speck you pick as your origin, the expansion looks exactly the same. Each speck would see itself as the center of the expansion.

Also, as you run time backwards towards 0, you always have a perfectly spaced rectangular grid, with distances of f(t) between the points.

At t = 0, of course, there's a singularity. Every point is at zero; which introduces a strange kind of discontinuity. At every other time, the grid is infinite and expanding.

That's the flat Big Bang model: flat meaning we can use nice simple cartesian co-ordinates like this.

Cheers -- sylas

PS. Sorry for being so fast. I've been monitoring the forum and happened to see your post. Yes, the Big Bang involves "superluminary" expansion, in the sense that the distance between widely separated specks (or galaxies) can be greater than the speed of light; in fact there's no limit on the speed of separation. Ignore this detail; it is not actually in conflict with relativity and why it isn't is a question for another time.

Finding the path of a photon moving between specks in this simple model can be fun; pick your velocity c and just make sure the photon is always moving at this speed relative to any speck it is moving past. Don't worry about anything special in the way of relativistic conversions; they don't matter. The technique outlined in this postscript works.

SW VandeCarr

Thanks for your detailed responses Sylas. The main issue I have is with what I quoted above. You have this "strange kind of discontinuity" where your scaling factor seems to break down. What's the analogue of this in the standard Big Bang theory - inflation?

In any case, you seem to be going from a finite perspective to an infinite one; something that you said couldn't happen in the "Flat universe?" thread.

While the math seems to work after the strange discontinuity, does the physics work? Looking backwards in time we are going from galaxies to quarks. Are quantum level phenomenon describable in the simple expanding grid you described given the uncertainty of position/momentum?

sylas

No... the analogue is the Big Bang "singularity", the origin of time. Inflation would involve a some fun and games with the function f over a small finite interval of time close to zero in which the function looks exponential, but that doesn't actually change much in this simple description. An indefinite exponential would have no singularity, but that isn't what inflation usually proposes.

No, it's always infinite. The time t=0 is not defined; it involves a divide by zero, which is a singularity, it is simply an undefined instant, where size has no meaning. Every point in the infinite universe converges to every other point... at the limit... singularity. The "size" is something like 0/0.

No, the physics breaks down before you get to t=0. In particular, everything falls apart very badly at the "Planck time", which is about 5*10^-44 seconds. And in fact it all gets hairy well before that, because descriptions of the universe have such enormous densities on such small scales that you simply must use a quantum theory of gravity. Which we don't have as yet.

Big Bang cosmology is basically a "classical" theory in physics, involving general relativity for space and time and gravity. You can then talk about particles in the universe with quantum mechanics (nucleosynthesis, for example). The theory involves the universe expanding (in precisely the sense given above; everything moving apart from everything else) and then over time the rate of expansion changes and the energy densities drop all according to classical physics up to the present day.

Big Bang cosmology doesn't actually explain the "initial moment" or the singularity. It just says that the universe starts out from a state of extreme density that we can't describe very well, and it cools and expands from then to now.... and the rate of expansion if extrapolated backwards has this singularity. The great issue now is to have a physics which doesn't fall to pieces in this way. But there's not any need to go from finite to infinite. We really don't know if the universe is finite or infinite; but whichever it is, it has always been thus. Personally I have a profound philosophical inclination to finite... but that's not science. I know others have a philosophical inclination in the other way. We really won't have a good handle on that until we get some kind of unification of relativity and quantum physics.

Cheers -- sylas

SW VandeCarr

Well, there are mathematical models for going from zero to infinity in zero time such as the Dirac delta. At least I can say that the probability of finding my twin in an infinite universe is zero because the probability of finding any finite collection of atoms in an infinite universe is zero unless you're included in that collection yourself.

Chalnoth

Uh what? No. You're assuming here that there exists only one twin in the entire infinite universe. That's not the case. Instead for each finite (but large) volume of space, you'll have a twin.

sylas

Shrug. I am not talking about any old mathematical function, but a specific physical theory.

The Dirac delta is an unusual beast; not really a function in the usual sense; but more particularly it isn't a model for the universe.

There is no model of the universe which goes from finite to infinite. I've explained multiple times now the case for conventional cosmological Big Bang models, which may be either finite or infinite; they never change from one to the other.

What you mean by the last sentence is really unclear. Under the assumptions I have given... an infinite flat homogenous topologically simple universe ... the existence of an infinite number of "twins" is pretty much inevitable. The good old ΛCDM model is of this form, if you take curvature as identically zero and extend homogeneity arbitrarily beyond the visible horizon.

If by probability of "finding" a given collection of atoms you mean the probability of you meeting your own twin; sure... but that isn't what we've been talking about.

Cheers -- sylas

SW VandeCarr

The probability of any finite subset in an infinite set is zero. If you isolate (interact with) a finite subset of n elements the probability of given element is 1/n.

EDIT: OK, for an infinite number of twins you can define a non zero probability of a twin in an infinite universe.

SW VandeCarr

I wasn't talking about going from finite to infinite, but from zero to infinite which is what the Dirac delta distribution does. I'm not saying it's a model of the universe, just an example of going from 0 to infinity in zero time. My last sentence is explained in my response to Chalnoth. However, I agree that if the number of twins is infinite, than there is a non-zero probability of a twin. As far as the physics is concerned, you said there is essentially no physical theory for an expanding infinite universe.

sylas

I have said precisely the opposite. Emphatically and repeatedly. Your continued distortions of what I have said is getting very old.

There is a perfectly adequate physical theory for an expanding universe, which covers both the case of an infinite universe and a finite universe.

I have said this repeatedly and in some detail. What science does not have is a unified theory of physics applicable to the conditions of extreme density in the very very early universe. That is, we can handle the expansion, but not the point of origin. The theory for the expanding universe works just fine, and it not limited to either finite or infinite cases.

Sylas

S.Vasojevic

Is there any sense of talking about twin (both physical and mental) without 'twinning' entire Hubble volume he is in?

Chalnoth

In this sense, it's just saying that there is going to be, on average, an identical configuration of matter of a given size some distance away.

physicsexplor

from philosophy to science, there is a long path to go. we are curious so we speculate about things. a scientific jouranl has tired to inspire people to go through that path. i think that this curiousity is leading scientists and philosophors to move forward. i am also living for those unsolved questions but I think that we should act collectively for enormous mutual benefit and economies of scale.
if you donot agree we can correct each other.

Galteeth

Because of an article published in may 2003? The loli sciam article you like was published in 2008.

While the article seemed pretty far out, perhaps the name Max Tegmark was the reason for publishing?