May 31 2012

Reading Greene’s The Elegant Universe

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Notes from The Elegant Universe by Brian Greene:

We are accustomed to three dimensions

Any path of movement we follow, regardless how complicated results from some combination of motion through what we might call the “left-right dimension”, the “back-forth dimension”, and the “up-down dimension”. Every step we take, we implicitly make three separate choices that determine how we move through these three dimensions.

Einstein asked us to ponder a fourth dimension – time.

Events are specifed by telling where and when they occur.

Pretty basic stuff, yes?

Or is it?

In 1919, Theodor Kaluza posited that there may be more than three spatial dimensions

The Garden Hose Example:

Imagine a few hundred feet of garden hose is stretched across a canyon, and you view it from 1/4 mile away.

At this distance, the thickness of the hose is difficult to discern.

From your distant vantage point, you would think that if an ant were constrained to live on the hose, it would have only one dimension within which to walk: the left-right dimension along the hose’s length.

If someone asked you to specify where the ant was at any given moment, you would need to give only ONE piece of data: the distance of the ant from the left (or right) end of the hose.

From a quarter mile away, a long piece of garden hose appears to be a one-dimensional object.

Zooming in, however, we find that the hose has thickness. The ant now has two dimensions in which to operate. “left-right” and “clockwise-counterclockwise”.

Now we realize that to specify the ant’s location, we now require TWO bits of information.

There is a distinct difference between the dimensions of the hose.
-direction along its length is long, extended, and easily visible
-the direction around the thickness is short, and “curled up”. to examine and be aware of this dimension, you need to examine the hose with greater precision.

This underscores a subtle and important feature of spatial dimensions. They can be large, extended, and easy to see, therefore directly manifest, or they can be small, curled up, and much more difficult to detect.

Kaluza sent this idea off to Einstein, as he realized it provided a framewok for weaving together general relativity and electromagnetic theory.

In 1926, Oskar Klein (a mathematician) showed that the spatial fabric of our universe may have both extended and curled up dimensions.

This next bit is a little difficult to visualize, but bear with me.

Kaluza and Klein suggestd that the extra circular dimension exists at EVERY POINT in the extended dimensions, just as the circular girth of the garden hose exists at every point along its unfurled horizontal extent.

The similarity with the garden hose is manifest, though there are important differnces.
– universe has thre large extended dimensions, compared with the garden hose’s one
-more importantly, we are discussing the spatial fabric of the universe itself, not just an object within it.

If the additional curled up, circular dimension of the universe is extremely small, it is much harder to detect than the manifest, large extended dimensions.

Even though the circular dimension is very small, it is NOT MERELY A CIRCULAR BUMP. Rather, it is a new dimension that exists at every point in the familiar extended dimensions just as the others do.

To specify the location of a microscopic ant, we’d now need 5 pieces of information (if we factor in time as a dimension).

To give you an idea of how small these dimensions would be, we’re talking Planck length. So, undetectable by current scientific equipment.

Unfortunately, the Kaluza-Klein model ran into some difficulties, and wasn’t revisited again until the 1970’s, where the difficulties were ironed out.

As quantum mechanics was formulated, and the strong and weak forces (not known about during the time of Kaluza and Klein) were discovered, some physicists suggested that Kaluza’s original proposal had run into difficulties because he had not know about these forces., and had therefore been too conservative in his revamping of space.

The mid 1970’s brought about intense research into higher dimensional theories.

one idea brought in two extra dimensions curled up into the surface of a sphere. Again, these are tacked onto every oint on the familiar extended dimensions.

another idea brought about a donut, or torus shape.

It turns out though, that in order to resolve the fundamental disconnect between quantum mechanics and general relativity, string theory requires that the universe have extra dimensions.

without these extra dimensions, some calculations attempting to resolve the two were coming up with negative probabilities.

However, if the string has more dimensions within which to vibrate, then this sorts itself out. Basically, a string needs approximately 9 space dimensions (possibly 10). Add a time dimension, and there you have it.

These extra six (or so) dimensions can’t just randomly crumple in on themselves. They would tend to fall under the distinction of Calabi-Yau structures.

So, all of this brings about some questions, of course.

1. Why? nobody really knows at this point

2. Why aren’t all the dimensions extended? see above

3. Could there be other time dimensions too? its entirely possible. :)

And, I have an as yet unanswered question to bother my professor with this week, unless I get really impatient and just e-mail him:

4. Is there a possibility of massively huge extended dimensions that we’d equally have no way of directly experiencing, and has anyone done any research on this?

The last one, think in terms of this. If you place an ant on a sphere with a 1 mile diameter, it will think that it is walking a straight line / on a flat surface, and will not realize that the surface upon which it is walking is curved.

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