'A Universe From Nothing' by Lawrence Krauss, AAI 2009

**Skolon** · Nov8-09, 08:47 AM

Thank you for yours answers.

Please tell me, what is the meaning of "flat" for cosmology?

**Chalnoth** · Nov8-09, 09:07 AM

Originally Posted by Skolon

Thank you for yours answers.

Please tell me, what is the meaning of "flat" for cosmology?

If you draw a triangle on a flat surface, its angles add up to 180 degrees.

If you draw a triangle on a "closed" surface, its angles add up to greater than 180 degrees. For example, if you draw a triangle on a sphere by starting from the north pole, drawing down to th equator, then across the equator a quarter of the way around, then back to the north pole, you've just made a triangle where each angle is a right angle: you've made a triangle whose angles sum to 270 degrees.

If you draw a triangle on an "open" surface, the reverse is true: the angles sum to less than 180 degrees. A saddle is an example of a surface that has this property.

When we're talking about the curvature of the universe, this is what we mean: we draw triangles by looking at the passage of light from place to place in the universe. The angles that they make tell us what the overall curvature is. And it's very very close to flat.

**Skolon** · Nov8-09, 09:18 AM

Thank you again, but I already knew that.

My question was about that: to measure the flatness of Universe the mean density of Universe if measured and it is compared with critical density. If I understood correct, this is how we can tell today that the Universe is "very close to flat".
But, every popular article about this subject stop here. I look for article that discuss about the effective shape of Universe for the flat case.

**Chalnoth** · Nov8-09, 12:21 PM

Originally Posted by Skolon

Thank you again, but I already knew that.

My question was about that: to measure the flatness of Universe the mean density of Universe if measured and it is compared with critical density. If I understood correct, this is how we can tell today that the Universe is "very close to flat".

Well, sort of. The way that this measurement is actually done, though, is by drawing triangles. Two measurements for this are the cosmic microwave background and baryon acoustic oscillation measurements.

The "triangle" drawn from the CMB is composed of the typical distance between the "acoustic peaks" which is set by the age of the universe at the time the CMB was emitted. Comparing this distance to the angle that we see and the distance to the CMB gives us our triangle.

The "triangle" drawn from BAO measurements comes from the fact that these measurements are measuring a typical distance between galaxies in the more nearby universe, a distance that is correlated to the same distance seen in the CMB. We can thus draw a different triangle as the typical separation between galaxies compared to the distance to those galaxies and their angular separation.

By contrast, other measurements of the contents of the universe that don't end up effectively drawing triangles, such as supernovae to measure the expansion rate as a function of time, don't measure the curvature at all.

**Schlofster** · Nov9-09, 10:42 AM

Originally Posted by Chalnoth

Er, well, no. A torus is not closed in that sense. It's flat.

Closed is a statement about curvature. Whether or not the universe wraps back on itself is a statement about topology. It is perfectly possible to have a flat universe with a topology that wraps back on itself. Just as it is, in principle, possible to have a closed universe that doesn't extend far enough to wrap back on itself.

Hi Chalnoth,
Please can I ask you to elaborate on this, as I am struggling to conceptualize it?
To me the property of being closed (angles of a triangle add up to more than 180) implies that the topology will wrap back on itself.
Can you give some examples of shapes where this is not the case - closed, but does not wrap back on itself?
Also, I seem to remember Krauss talking about flatness implying that it is infinite in spatial extent (or am I just misunderstanding him again?) - doesn't "infinite in spatial extent" rule out a topology that wraps back on itself?

Yes, I still don't quite have a handle on all this ;-)

Thanks :-)

**Chalnoth** · Nov9-09, 05:09 PM

Originally Posted by Schlofster

Hi Chalnoth,
Please can I ask you to elaborate on this, as I am struggling to conceptualize it?
To me the property of being closed (angles of a triangle add up to more than 180) implies that the topology will wrap back on itself.

Well, there are a few different possibilities here:

1. What if the curvature changes before it wraps back on itself?
2. What if it spirals in or some such instead of neatly meeting itself?
3. What if it just ends before meeting itself?

I mean, sure, in the simplest case with absolutely constant curvature in all dimensions, we'd be talking about a sphere, which obviously wraps back on itself. But what if it isn't quite so simple?

Originally Posted by Schlofster

Also, I seem to remember Krauss talking about flatness implying that it is infinite in spatial extent (or am I just misunderstanding him again?) - doesn't "infinite in spatial extent" rule out a topology that wraps back on itself?

I don't think he actually said that. If he did, he's wrong. But yes, if it so happened that our universe was infinite in spatial extent, then it clearly couldn't wrap back on itself.

**GoldPheonix** · Nov9-09, 09:58 PM

I should state that I've met Krauss on several occasions and listened to several of his lectures and colloquiums; I do research under one of his previous colleagues (This isn't to say at all, however, that he remembers me). Krauss is an interesting fellow, to say the least.

Originally Posted by Schlofster

I think that I might be starting to see a pattern emerge - it seems to me that the intellectual abilities that scientists like Krauss are so far beyond what ordinary humans are capable of, he cant even articulate them at all in a way is intelligible by the majority of humans on this planet.

Ordinary humans would understandably respond with incredulity, and since they pay his salary, I suggest that people like him should only make statements that are logically intelligible ordinary people.
I think that public lectures like this just serve to polarize the voting public, against science.

I am rather interested though in what the evolutionary consequences of this cavernous gap between the intelligentsia and the rest of the human population would be.
Is it possible that we eventually see a divergence in the species - this may also be accelerated if people are selected for establishing a human colony on another planet like Mars.
Presumably the criteria for selecting the people for this mission would be heavily biased in favour of intellect.

You should send him an e-mail... He'd probably actually respond to you if you phrased it that way.

Originally Posted by Schlofster

what I said is that most of the mass of protons and neutrons can be attributed to virtual particles.. it is that mass that we can calculate using the theory of the strong interaction and powerful computers..

_______________________________
Lawrence M. Krauss
Foundation Professor
Director, Origins Initiative
Co-Director, Cosmology Initiative
ARIZONA STATE UNIVERSITY
College of Liberal Arts and Sciences
School of Earth and Space Exploration
http://krauss.faculty.asu.edu

On Nov 3, 2009, at 5:04 PM, Jessica Lee wrote:

------ Forwarded Message
Date: Wed, 28 Oct 2009 06:48:23 -0700
Subject: A Question For Prof Krauss.

Hi Jessica,
Please could I ask you to send this question on to Prof. Krauss?

Hi Prof Krauss,
I loved your recent public lecture <http://www.youtube.com/watch?v=7ImvlS8PLIo> at AAI 2009, but I don't understand something,

and I was hoping that you might be able to spare a few moments to clear it up for me.

First of all let me state that I am no physicist, so I am rather certain that my
misunderstanding is due to an incomplete grasp of the mathematical model that you
attempt to describe in natural language. I do understand that it is actually impossible to
do this since the mathematical model cannot be expressed in natural language.

In the lecture you say that you have calculated the mass of the universe, and have calculated that
most of the mass in the universe is as a result of virtual particles "popping in and out of existence" in empty space,
while matter particles constitute a very small fraction of the mass of the universe.
Considering this, and since you are able to calculate the mass of the universe to be a finite quantity,
it seems to me that a conclusion that the universe is infinite in spatial extent would be logically incompatible with a universe of finite mass.
I say this because if empty space itself constitutes most of the mass of the universe, then if the amount of empty space is infinite,
then the mass of the universe must also be infinite.

Thanks very much & kind regards,

------ End of Forwarded Message

Tell me how this goes. I'm interested to see what he replies with.

By-the-by, he's neither wrong nor right. We don't know if the universe's curvature is zero or not. We know that, right now, however, it is very small and close to zero (and possibly negative, if memory serves).

**George Jones** · Nov10-09, 10:53 AM

Originally Posted by Skolon

Yes, maybe a torus is mathematically flat shape but is a strange kind: a torus shape Universe is closed (going on any direction you will finally arrive at the same point) but for each direction other "diameter" of Universe exist. So, it seems like each direction have its own properties.
In our Universe all directions seem to have the same properties, at least until now. Or I don't know well?

Yes, this model is closed, and is different in different directions (anisotropic).

Originally Posted by Chalnoth

Er, well, no. A torus is not closed in that sense. It's flat.

Closed is a statement about curvature. Whether or not the universe wraps back on itself is a statement about topology.

I don't think so. "Closed" is a statement about topology, not about curvature. A universe is closed iff its spatial sections are topologically compact. I think that this usage is standard. I cant think of how to use (the) curvature (tensor) to distinguish between "closed" and "open."

Originally Posted by Chalnoth

Just as it is, in principle, possible to have a closed universe that doesn't extend far enough to wrap back on itself.

I'm not sure what this means.

Originally Posted by Chalnoth

But in any case, it's still perfectly possible that the circumference in each direction is identical.

But this model is still anisotropic. Along the axes of symmetry, there are geodesics that close, but there also are geodesics "between" these axes that wind around the torus, never closing.

A flat three-dimensional torus is a model for a closed, homogeneous, anisotropic universe. The only flat universe that obeys the Cosmological Principle is one for which space is R³.

**Chalnoth** · Nov10-09, 12:38 PM

Originally Posted by George Jones

Yes, this model is closed, and is different in different directions (anisotropic).

Well, that doesn't have any meaning in observational cosmology, as when we refer to "closed" we exclusively refer to the curvature, not to the topology (which is unobservable). Well, I mean, a topologically closed universe could, in principle, be observed. But the fact that it hasn't so far seems to indicate that if our universe is topologically closed, then we won't ever be able to tell (because the distance is just too great, by current observational limits).

Originally Posted by George Jones

But this model is still anisotropic. Along the axes of symmetry, there are geodesics that close, but there also are geodesics "between" these axes that wind around the torus, never closing.

In what sense? Given that a torus is finite, any geodesic must necessarily come back to itself eventually. Some will take longer than others, of course, but it must eventually occur.

Originally Posted by George Jones

A flat three-dimensional torus is a model for a closed, homogeneous, anisotropic universe. The only flat universe that obeys the Cosmological Principle is one for which space is R³.

In the sense that the word "flat" is used in observational cosmology, a torus is quite flat. But I thought you said it wasn't anisotropic? Of course, it is anisotropic in the sense that not all geodesics that wrap around a torus come back after the same distance is crossed (like a sphere). But it is very much flat in terms of curvature.

**George Jones** · Nov10-09, 02:01 PM

Originally Posted by Chalnoth

Well, that doesn't have any meaning in observational cosmology, as when we refer to "closed" we exclusively refer to the curvature, not to the topology (which is unobservable).

If "closed" is defined using curvature, and is not equivalent to topological compactness, I guess I need a precise, mathematical definition of "closed" in terms of curvature, and at least one of:

a space that is closed in terms of the precise, mathematical curvature definition, but that is not compact;

a space that is compact, but that is not closed in terms of the precise, mathematical curvature definition.

Originally Posted by Chalnoth

In what sense? Given that a torus is finite, any geodesic must necessarily come back to itself eventually. Some will take longer than others, of course, but it must eventually occur.

Here's what I had in mind. For simplicity, consider only two spatial dimensions. The flat torus is constructed by putting an equivalence relation on the 2-dimensional plane, with (x, y) ~ (x + n , x + m) for all integers n and m. Now consider the geodesic y = pi*x in the plane, which goes the point (0, 0). The only way that this geodesic can close in the torus is if the line goes through a point in the plane that is equivalent to the point (0, 0), i.e., if the line goes through a point (n, m) for some integers n and m. But a line that goes through (0, 0) and (n, m) has a slope that is a rational number, which can't be equal to the slope pi of the original line.

Consequently, this geodesic in the flat torus induced by the geodesic y = pi*x in the plane doesn't close in the torus.

Originally Posted by Chalnoth

In the sense that the word "flat" is used in observational cosmology, a torus is quite flat.

I agree, and I didn't write anything that contradicted this.

Originally Posted by Chalnoth

But I thought you said it wasn't anisotropic?

I didn't writes this, and I didn't mean to give the impression that I believed this. Quite the opposite; in the first sentence of my previous post, I explicitly stated that the flat torus is anisotropic.

Originally Posted by Chalnoth

Of course, it is anisotropic in the sense that not all geodesics that wrap around a torus come back after the same distance is crossed (like a sphere).

Originally Posted by Chalnoth

But it is very much flat in terms of curvature.

Yes.

**Chalnoth** · Nov10-09, 04:13 PM

Originally Posted by George Jones

If "closed" is defined using curvature, and is not equivalent to topological compactness, I guess I need a precise, mathematical definition of "closed" in terms of curvature, and at least one of:

From the parameter k in the FLRW metric:

k < 0: open
k = 0: flat
k > 0: closed

This is the terminology usually used in observational cosmology. Clearly these terms were inspired by topology, but bear no direct relationship.