## Thursday, August 13, 2009

### An Expanding 4D Sphere

A direct link to the above video is at http://www.youtube.com/watch?v=ZKh2y93hwa4

How old is the universe? Most scientists currently peg it at around 13.7 billion years. A light year, of course, is the distance light travels in one year. If I look through a telescope, then, what's the furthest I should be able to see? Intuitively, we would presume it to be no more than a distance of 13.7 billion light years. Here's a video that explains how cosmic expansion complicates this: because everything is moving away from everything else as the universe expands, currently observable particles can theoretically be as far away as 42 billion light years in any direction, and early stars can be as much as 36 billion light years away in any direction.

A direct link to the above video is at http://www.youtube.com/watch?v=zO2vfYNaIbk

In The Holographic Universe, I showed a way of visualizing how our spacetime is not completely flat, but instead has a very slight curve to it. It's easy to confuse this statement to think we're saying that space has a slight curve to it, and this can be the start of some confusion. In the above video we see that we're at the center of a 3D sphere with a radius of as much as 42 billion light years. If we're thinking about 4D spacetime, though, we're thinking about how that 3D sphere is on the surface of a 4D hypersphere (this relates to the recently proved Poincare Conjecture, which we talked about in "Why Do We Need More Than 3 Dimensions?"). In the video for The Holographic Universe, I showed how this slight curvature could create the observable universe horizon that we're talking about above - if time has a slight curve to it, then it's like we're in the middle of the ocean, and the horizon we see around us is the furthest distance back in time we're able to see. In the wikipedia article on The Cosmological Horizon, it says this:

it has been said that the observable universe is many orders of magnitude smaller than the greater universe that lies beyond the limits of our perception.

Imagine that the entire cosmological horizon is modeled by a sphere that is the diameter of a quarter (24.26 mm in diameter). If Alan Guth's inflationary model of early era cosmology is correct, the universe that lies beyond this “quarter-sized” horizon would conservatively be a sphere as large as the Earth globe itself.
If this is really the scale of curvature we're talking about here, then spacetime for our purposes is flat: if our universe were the size of a quarter and its curvature was the equivalent of the curvature of the earth's surface, imagine how sensitive a measurement you would have to make to be able to register that curvature! But spacetime does indeed have a slight curve to it, and that's an important piece of the puzzle we're putting together.

A direct link to the above video is at http://www.youtube.com/watch?v=hMLVjFrtq6Q. You can watch the video from about 5:50 if you want to jump to the section where I show a way of visualizing how our spacetime is curved.

In What's South of the South Pole? and The Map and the Territory, we looked at how tricky it can be to create useful visualizations of concepts like these. Visualizing a 3D sphere on the surface of a 4D hypersphere boggles the mind. The beauty of the approach I'm using with this project is that these are all really spatial dimensions that we're talking about: this means that as per the point-line-plane postulate, which can be used to visualize any number of spatial dimensions, we can simplify this concept to imagine that our 3D universe is a point, moving on the surface of an expanding 4D plane, and that plane has a slight curvature to it which takes it into the fifth dimension. That slight curvature gives us the impression that our universe has a certain size, but that size is an illusion - like the boat in the middle of the ocean, looking at a horizon all around them, we have to understand that there is still much more beyond that horizon which exists -- even though we can't see it from our current point of observation.

In Where Are You? I made the point that each of us is right at the center of our own version of the universe, and as metaphysical as that may sound, the above discussions show a scientific reason for why this is so.

Enjoy the journey,

Rob Bryanton

Next: When's a Knot Not a Knot?