Figure 1: Orbits due to Curvature due to Darts
You can demonstrate a form of orbital motion by letting a small marble roll in a large bowl.
For an appropriately-shaped bowl, this is a moderately accurate model of a planet orbiting the sun. The height of the bowl represents the gravitational potential. At each point, the slope of the bowl represents the gravitational field. The fact that the marble rolls (rather than sliding freely) introduces some nonidealities.
If you use a glass bowl, you can demonstrate this to the whole room using an overhead projector.
Although this models the classical physics to a fair approximation, it does not correctly model general relativity. The curvature of the bowl is not a good model of the spacetime curvature that produces gravitation in general relativity. Not at all.
There are several ways of seeing that it would be wrong to consider the bowl a model of general relativity.
To repeat: any alleged connection between this model and general relativity is essentially 100% wrong.
Here is a convenient way to correctly demonstrate the motion of a free particle in curved space.
We use a two-dimensional surface as our model universe. To model the path of a particle, apply masking tape to this surface.
For reasons explained in section 4, the tape will follow a geodesic (straight line) in the two-dimensional universe. Meanwhile, masking tape is so thin that it can bend as necessary in the extra dimension (the embedding dimension). Be sure you use masking tape, which is designed to be non-stretchy (as opposed to something stretchy like electrical tape).
1) Let’s start with an uncurved universe. Use a flat sheet of construction paper or (if you’re using an overhead projector) a blank acetate foil. Lay out some tape and see what happens. Put down a couple of inches of tape to get things started, and then lay down the rest bit by bit. Allow the most-recently attached bit to guide the placement of the next bit. Hold the supply of tape slightly slack; do not try to "force" the result to go in any particular direction. Observe that the result is automatically straight, to an excellent approximation.
As long as you don’t allow "crumpling" or "air pockets" under the tape, it should guide itself quite well.
2) Lay out another line of tape that is initially parallel to the first. Extend it, letting the tape itself do the guiding, and observe that the two lines remain parallel for a long ways.
3) Make a hump under the sheet, in such a way that it doesn’t stretch the sheet. This creates extrinsic curvature without creating any intrisic curvature. As a rather dramatic example of this, you could give the sheet a cylindrical shape. (Wrap it around an oatmeal carton if you need help maintaining a cylindrical shape.)
Observe that this has no consequences for the geodesics; things that start out parallel remain parallel, et cetera. This is important because it demonstrates extrinsic curvature. In particular, it shows that extrinsic curvature has no effect on the path of particles in our two-dimensional model world. That contrasts with intrinsic curvature, which will be demonstrated next.
4) On top of a flat sheet, put a large bowl, upside down. I have a large salad bowl that works beautifully.
Lay out a geodesic that starts on the flat sheet and heads toward the bowl. When it reaches the bowl, it will refract.
5) Lay out a geodesic that starts out parallel to the previous one. It will hit the bowl with a different impact parameter, and refract differently.
*) Et cetera. You get the idea.
Note: If you are not using an overhead projector, choose the color of the paper and the color of the bowls to contrast with the color of the tape. If you can get multiple colors of non-stretchy tape, so much the better.
Another suggestion: You can pile smaller bowls onto the back of the larger bowl, to change the shape of the potential.
Remark: The embedding world’s gravity has no effect on the tape. This model would work perfectly in the weightless environment in a spaceship.
Related remark: The tape does not care whether the curvature is a bump or a pit. You could put the bowl under the paper, right-side-up. Then cut a hole so the tape can drop down and follow the curvature of the inside of the bowl. In all cases the geodesic will be bent toward the region of high intrinsic curvature.
The model in the previous section demonstrated the qualitative effects of curvature, but we have to refine it a bit if we want a really accurate model of, say, planetary orbits.
The world-line of a particle in orbit is best described as a helix. It goes around and around in two spacelike dimensions, while moving steadily forward in the timelike direction.
As pointed out by David Bowman, if you try to project that helix onto a two-dimensional model, there will be problems. Spacetime is curved (by gravity) in the timelike direction as well as the spacelike directions. So if you build a model that represents the two spacelike directions, suppressing the timelike direction, you can’t properly represent the sort of curvature that leads to planetary orbits.
We can do much better if we use our model to represent one spacelike dimension and one timelike dimension. We will illustrate the gravitational field of a planet that exists for a long time (a long ways along the time axis). The gravitational field decreases as we move away from the planet in either direction along the X axis.
This can be modeled using darts, as shown in figure 1. The time axis runs vertically up the page, and the spacelike X axis runs horizontally across the page. The five ribs are made of two darts each, for a total of 10 darts. See section 6 for hints on how to fabricate darts.
I emphasize that the tape follows geodesics determined by the local curvature of spacetime at each point. Given a arrangement of darts, each trajectory is completely determined by its starting point and initial direction; there is no other choice involved.
Wheneve the tape crosses a dart, the curvature of spacetime will deflect the trajectory toward the center. As it continues along, it will "orbit" the center, as you can see in the figure. In D=1+1, each trajectory looks like a sine-wave when viewed from above the page. One trajectory starts with an X value slightly left of center and completes a half-cycle before exiting at the top of the diagram. The next trajectory starts with an X value slightly right of center, and also completes a half-cycle. The third trajectory starts farther right of center, and completes only a small fraction of a cycle.
It is interesting to think about "what is the shortest path from point A1 to point A2". The path in the presence of the darts is very different from what it would be in the absence of the darts. Actually in the presence of the darts (spacetime curvature) there are a couple of different geodesics that connect A1 to A2.
Hint: The stores around here sell masking tape that is 3/4" or 1" wide. Narrower tape is better for this demo. So you may want to divide the tape in half lengthwise. Do this while it was still on the roll, using a very sharp knife to cut through many layers.
If you attach the darts to a piece of acetate foil, you can set the whole thing on an overhead projector so everybody can see it. But letting people play with it hands-on is the best.
We need to define what we mean by a straight line in a curved universe, and to explain why the tape automatically follows such a path.
Masking tape has the wonderful property that it very non-stretchy, as you can verify by trying to stretch a piece. (If your masking tape is stretchy, replace it with some that isn’t.)
Next, note that it has definitely nonzero width. It is non-stretchy across its width, and also across innumerable diagonals, so it can hold itself straight, the way a truss holds itself rigid with cross-bracing.
This notion of straightness, defined by cross-bracing, has numerous good properties. For one thing, if you stick an initial piece of tape to a surface, it defines a unique way of laying down the next piece, and then the next. And it is reversible: You can retrace the such a tape-path in the reverse direction and get the same result.
It is intriguing that in mathematics, lines are defined to be straight and have zero width, but in physics, if you want to make sure it is straight, it needs to have nonzero width (so the cross-braces have some leverage). You can pass to the limit of infinitesimal width, but not zero width.
For thousands of years, mathematicians have asked us to draw straight lines, but have depended on physics to actually draw them. (Borrowing a sentiment from reference 1.)
see reference 1 page 249 for a picture of cross-bracing.
Straightness is sometimes defined by the proposition that a straight "line" is the shortest path between two points. More precisely, it should be defined an extremal path (either shortest or longest). You can show that the tape satisfies this definition, by the following argument: The two edges of the tape have equal length. The tape was made to have that property, and it retains that property because it is non-stretchy. If you choose a hypothetical path that is not a geodesic, then nearby paths to one side will be slightly longer, and nearby paths to the other side will be slightly shorter. You cannot make the tape follow such a path without buckling.
Similar notions of curvature play a role in the expansion of the universe, as discussed in reference 2.
Suppose you are helping a buddy with a plumbing project that involves cutting PVC pipes and gluing them into fittings. Your buddy sees no reason to buy a pipe cutter, since he has a perfectly good hacksaw. The problem is, if you want the joint to be strong, the pipe needs to be cut square, and this isn’t always easy to do. It’s not too hard under favorable conditions in the shop, but not so easy out in the field when the pipe is at a funny angle and there are other constraints.
The desired result can be obtained more easily and more reliably if the pipe is first properly marked ... then all you need to do is cut along the mark.
So now we come to the heart of the matter: What is the quick and easy way to create a "cut here" guide on the pipe?
Answer: Masking tape.
As discussed in previous sections, masking tape follows a geodesic. We assume the pipe is accurately cylindrical (otherwise the glue joint would fail anyway, so the whole exercise would be pointless). The geodesics of a cylinder are helices, including the zero-pitch helix which is a circle.
If you start the tape perpendicular to the axis of the pipe, the tape will come around and meet itself, making a zero-pitch helix, and you know you have succeeded. Cut parallel to the edge of the tape.
If you start the tape at an angle, it will come around and make a non-zero-pitch helix, and you know you need to re-do the taping.
This trick with the tape is super-accurate, super-quick, and super-easy.
Many of the following fabrication ideas are due to Paul Fuoss.
Maple is an excellent material for making darts. (Any hard, fine-grained wood will do.) The grain should run down the long axis of the dart; this makes fabrication harder but makes the final result nicer. We found that Paul’s compound miter saw was the smart way to make them. We tried making them with a table saw but the compound miter saw was a much better choice.
Attach the hardwood piece (from which you are cutting the darts) to a much larger carrier piece, using wood screws, so you can hold it very securely without endangering your fingers. (Holding the piece securely is vastly more important for miter cuts than for regular cuts, where slight vertical motions are harmless.)
Each dart is a thin pyramid. The base of the pyramid is an equilateral triangle, 1/2" on a side. The pyramid is about 4" long in the other direction.
Set the saw so that the blade is inclined 30 degrees to the vertical. Then set it so that the arm is angled 3.6 degrees away from perpendicular to the fence. That gives a slope of 1/4" (half the base of the pyramid) per 4".
Make the first cut. Cut just enough off one end of the stock so that the shape of the edge is determined by the cut. Then flip the stock over. If the stock was to the left of the blade to begin with, it will be to the right of the blade now. This may require unbolting the stock from the carrier and rebolting it (although you might avoid this step by using an extra-fancy carrier). Mark the starting point for the second cut. The base of the pyramid should be against the fence. Make the second cut. The first dart should fall free.
Jessica MacNaughton pointed out that you can make darts that are just as functional (but perhaps not as beautiful) out of clay. If you don’t have access to a compound miter saw, this may be your best option.