Saturday, March 26, 2005

General Relativity Lite

When I taught secondary school physics, I always tried to take the students a little farther than required by curriculum guidelines. The goal was to give them just enough so that future learning might be easier. In complex topics such as relativity, cosmology, string theory, fractals, or chaos, the aim was turn on in their head a little light bulb so they might say “OK, I sort of get it.” I’ll try that here with general relativity. It's a little tricky without a diagram, but do you want to try it?

General Relativity: Curved space and a Theory for Gravity

Background (and you don’t really need this)

You may remember that an object’s weight is mg, where m is the mass and g is the Earth’s gravitational field strength. Remember also that Newton’s second law is F = ma. A subtle point about these formulas is that the two m’s needn’t be the same thing. The m in mg is the gravitational mass, that quality of a body that causes it to be attracted to another object. The m in F=ma is the inertial mass, that quality of a body that causes it to resist being accelerated. Writing g in N/kg and giving it a value for Earth’s surface of 9.8 N/kg gives both the m’s the same unit (kg) and the same value (because a N/kg equals a m/s2, and the gravitational acceleration is 9.8 m/s2 down at Earth’s surface.)

Does it actually matter that these m’s might be different? Not for general relativity, because Einstein started general relativity with the assumption that they ARE the same thing. That led to …

Curvature of Space

Being in an accelerated frame of reference, like a box accelerating "upward" in empty space at 9.8 m/s2 is equivalent to being in a gravitational field where g=9.8 N/kg. That is, you can’t tell the difference. All experiments must come out the same. If you release a ball in that accelerating box, it seems to fall to the floor. (It travels up with the speed it had in your hand, but the floor’s speed is increasing, so the floor comes up and hits the ball. To you inside, it looks as if the ball is falling.)

But what if you release two balls, one from each outstretched hand? In the box, the balls fall on parallel paths to the floor. They hit the ground the same distance apart as they were when you released them. But on Earth’s surface the balls fall toward Earth’s centre, so should converge. But Einstein supposed that both experiments should come out the same. To rescue the assumption, he makes a great intellectual leap (but one that is simple and logical.) He says that in the gravitational field case, the balls DO fall on parallel lines, EVEN THOUGH they appear to be converging, because space is curved! How does this work? Imagine you and your friend walking north on Earth’s surface. You are walking parallel to your friend but converging. So parallel lines converge on (positively) curved surfaces. If VOLUME is curved, that could happen in space. So Einstein proposes that space is curved in the vicinity of masses.

So what’s gravity? Remember dropping the ball in the box that was accelerating upward in empty space? Inside, you observe (and feel) a force down even though there was really no force down on objects in the box. (Same thing in a rotating frame of reference…you feel a force to the outside (which you call centrifugal force) even though there is no such force. Objects in rotating frames of reference are tending to travel STRAIGHT, the paths they travel in the absence of force.) In accelerated frames of reference fictitious forces are felt.

Are you ready for General Gelativity Lite - An Explanation for Gravity?


If space is curved, then when you think you are travelling a straight line, you are actually travelling a curved path (through space-time, actually). So when you think you are in a non-accelerated frame of reference, you are really in an accelerated frame of reference. Therefore you feel a fictitious force. That “force” is gravity.
In Twenty Five Words or Less

If this phrase means something to you, welcome to the over-50 club! These days (i.e. for the last 35 years!) most contests that people enter by writing their names on a coupon or an entry blank are sweepstakes. When I was young, contests on the back of cereal boxes required some additional element, such as drawing a picture, writing a joke, or composing a poem. I must have been about ten when I entered a contest printed on the back of a box of Nestle's Quik, a chocolate milk powder. I had to write, in twenty five words or less, why I like Nestle's Quik. I don't remember what I wrote, but I won a watch!

It was disappointing that, a few years later, the contests changed to becoming straight draws. (Of course, my earlier win could have been drawn from the hat, too...I'll never know.) Evidently, the Canadian government very strongly disagreed with the change, because they brought in the infamous skill-testing question rule. Apparently, voters (or some prominent legislators) objected strongly to people winning money without actually demonstrating that they deserved the prize. From then on, to claim the prize you had to earn your prize by computing 9 times 8 divided by 4 minus 3.

I still see the skill-testing question on some sweepstakes entry blanks, so I guess the law is still on the books. I wonder if it has ever been enforced.