Reading:
Chapter 1-1, 3-1 (ZG4)
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Key Question: | How do we ask and answer questions about our Universe? |
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Key Principle: | Physical and Mathematical Models |
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Key Problem: | How big is the Universe? |
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Key Quote: | Its big, really big. |
The speed of light in vaccuum, c, 186,282 miles per second, or 299,792.5 kilometers per second, the ultimate speed limit, the light barrier. For us "twiddle mathematicians", 3 x 10^8 m/s. Call it what you will, what is it? When we look out in the Universe, the farther away something is, the farther back in time we see it. It takes light time to reach us and report on its origin. Is there a deeper meaning to this?>
Actually, just try and graph space versus time without a way of relating changes in distance to changes in time - the conversion factor must have units of velocity. Lets call it "c", for "conversion factor". Thus, in calculus terms
or, since changing the sign of the "delta x" dx doesn't change the sign of "delta t" dt, we can just as well deal with the squares of these quantities
This little differential equation ('cause the little dx and dt are differential quantities, just as dx/dt is a derivative) is know as the metric, and is the basis of Einstein's Theory of Relativity! In three spatial dimensions x, y, z, time t, we can define a "proper distance" ds such that
which for the path light takes, it is easy to see necessarily that ds^2 = 0!
When you see a first-order derivative such as
you should picture a straight line graph with slope v. (Remember elementary calculus?) The important thing is to actually visualize this, and think of a ball rolling down that line with constant velocity, for example. What about second derivatives? Think of a graph with a curve, like a parabola at its vertex. The rate of curvature is given by the second derivative
where you can also think of R as a radius of curvature. Hey, a circle of radius R has constant radius of curvature, with second derivative R. In three dimensions a sphere, etc.
Hey wait a sec, didn't we learn in first semester Physics that the acceleration is the second derivative
Does that mean that if we have a spacetime curvature with radius R, that this gives an acceleration a? Einstein's Theory of Relativity says "yes", and so the mass of the Earth curves spacetime at the radius of 9.8 meters per second per second at its surface! Neat, huh?
Note, we are not going to be doing anything with this in the class (or on homeworks or tests), but I thought it would be illustrative to show how a simple line of questioning, and careful consideration of the meanings of mathemetical representations, can lead to deep and important discoveries about the Universe. This is only a cartoon picture of relativity, for example, but it has a good deal of merit to it.
Note that you should be comfortable with simple derivatives, powers of ten and scientific notation, and the notions of velocity and acceleration.
No worries. I mostly talked about the stuff in the course guide, and some of the topics listed above. Oh, and I showed some cool laserdisk pictures too. You can find photocopies of my lecture notes on reserve in the Physics, Math & Astronomy Library, 3rd Floor, North Wing, DRL.
We start our main work, first looking over the history of Astronomy from the Greeks to Galileo, then looking at how the Sun, Moon, Planets and Stars move on the sky, and how we build models to explain those motions. And we see how those clever Greeks figured out the size of the Earth, and other pretty advanced things!
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smyers@nrao.edu Steven T. Myers