Ah Interesting-- Sorry, I seemed to have missed bullet point one in your original post. If we lived in a closed 3D space, and were able to observe the whole thing, then we would see some sort of repeating pattern in the CMBR. That's really interesting (and cool that somebody has looked into this!)
I guess I don't know enough about what cosmologists mean by ``flat.'' There are different kinds of flatness, and they have different interpretations.
For instance, if you wanted to look at a surface, and just assign a single number to each point to describe the curvature, then things like cylinders are flat.
If you take the General-relativity definition of flat, then I would believe (although, I don't really know!) there are a lot of manifolds which are Ricci-flat.
If you take the curvature to be zero (in the curvature tensor sense), then there are only a few closed manifolds which are flat (see http://en.wikipedia.org/wiki/F... for all 6 examples in dimension 3)
A mathematical note: Tori can be flat.
If the universe (and here we are talking about the large spacetime structure, not any of the weird tiny compactified extra string dimension stuff) is globally flat, it can still have the structure of a torus.
The torus when viewed as a 2 dimensional space in 3 dimensions, is not flat-- it has some positively curved parts (think the outer edge of the donut) and some negatively curved portions (think the saddle like regions on the inner ring of the torus.) However, the total curvature (when I sum up all contributing curvatures) on the torus is zero. This is related to a mathematical fact that the total curvature of any surface is given by a topological quantity called the genus. In simpler terms, no matter how I deform the torus, the sum of the curvature will be zero. This is very different from the sphere, whose total curvature is always 2\pi.
So, a flat universe would imply that we cannot live on a 4 sphere, because such objects must always have at least some positive curvature. However, there are examples of tori that have no curvature.
In the 2 dimensional case, it is best to see this from the ``Pac-Man'' perspective. The pacman game is played on a flat surface, and whenever you head off the top of the screen, you arrive at the bottom, and whenever you go off the left side of the screen you wind up on the right hand side. This describes a possible shape for the universe, and this shape is the torus! To see this, imagine that you took the playing field, and glue the top and bottom sides together. That would give you a cylinder. Taking the left and right sides and gluing them together would give you a torus. Now that we believe the pacman game is played on a torus, notice that the original interpretation was a flat surface. So , there is a flat representation of the torus.
To avoid some confusion and people trying to draw flat tori in 3 dimensional space, it can't be done. Every surface viewed in 3 dimensional space will necessarily have some positive curvature around its maximal value. Sorry folks!
In fact, of all the 2 dimensional (compact) surfaces, the only one that has a flat representation is the torus. So, if the universe is compact (and 2 dimensional, which seems unlikely,) there is hope for a Pacman world.
Sure. I don't work in physics, but here is my understanding of the holographic principle.
Imagine that you are in a bathtub. There is a certain kind of physics that dictates the motions of waves in the bathtub. Now, you might believe that you need to understand the entirety of water to predict its future motion. You could develop a theory of water in bathtubs, and run experiments to verify if they are true.
After a lot of thought, you might come across the realization that in order to understand the mechanics of the water in the bathtub, it is only necessary to understand the way the surface of the water moves, or maybe even how the water interacts with the edge of the bathtub. This means that you've reduced the dimension of your theory in some way. While this analogy isn't true, there are examples of where it is-- for instance, the physics of harmonic oscillators, like strings, drumheads, etc, can be understood by looking at the boundaries of those oscillations.
Now, in physics, there are several ways that holography shows up. The most famous of these holography theories is called the AdS/CFT correspondence. It conjectures that a certain 5 dimensional string theory can be understood as a 4 dimensional field theory on the boundary. Now, I think that this perspective is interesting to physicists not because of the dimension change (dimensions in theoretical physics usually have little correlation with the observable dimensions of spacetime) but because it was one of the first known correspondences where a string theory reproduced the results of a field theory. Quantum Field theory is the most validated theory of physics we have, but it is thought to have foundational errors. String theory is suppose to offer a way out, but is... hard.
Hope that helps!
"Update on shooter incident. Responding agencies continue to investigate the situation. The scene is outside of Building 32 (Stata) and 76 (Koch) near Vassar and Main Streets. Injuries have been reported. The situation is still very active and we ask everyone to stay inside. "
Link to Original Source
The school has posted a message indicating that the Genesis system is down, but has yet to publicly acknowledge that a breach occurred.
Link to Original Source
Pretty harsh. Completely awesome.
Sweets and Flowers
(QUESTIONER): Vice President Cheney yesterday said that he expects that American forces will be greeted as liberators and I wonder if you could tell us if you agree with that and how you think they'll be greeted and also what you meant you said before that some Iraqi opposition groups might be in Baghdad even before American forces?