Spacetime Table

2015 is the Centennial of General Relativity (GR). GR is the modern way astronomers and physicists think about gravity. Today, 100 years after it was first deduced by Einstein, general relativity plays a major role in astrophysics (think: black holes), is used in your everyday life (think: GPS navigation), and is used in our everyday language to tell stories (think: "gravity well"). The spacetime table provides an intuitive, heuristic way to understand how we think about modern gravity.

This spacetime table is a great hands on demonstration for talking about gravity with people. It's fun to play with, portable, and easy to maintain and repair (an important design consideration -- I like to let people experiment with it, and kids LOVE it; I don't want them to worry about hurting it).

This particular design for a table is small enough to carry around in my backpack or luggage, and is quick to set up and take down. It is built around a PVC frame with slip-fit joints.

(1) 1/2" Schedule 40 PVC

• 8 pieces, each 12" long
• 4 pieces, each 24" long (cut these in half if you want 8 legs for stability)
• 4 pieces, each 27" long (these are the legs, so choose a height you like; cut 4 extra if you have 8 legs)

(2) 8 PVC elbows, 45 degrees

(3) 4 PVC T-joints (8 if you want 8 legs for stability)

(4) 16 small clamps (this is about the right number; you can experiment with what works best for you)

(5) Large mass (rock, heavy sphere or ball bearing). It works best if it is roughly no bigger than your hand.

(6) Ping pong balls

(7) 2 yards of Spandex -- look with the "performance fabrics" at your local fabric store; the spandex in this project is 82% Nylon/18% spandex. I usually use black, but a pattern (grid) helps people see the distortions. Amount you get should be enough to stretch slightly across the longest dimension of your table.

I cut all the PVC to the correct lengths using a rotary pipe cutter, to make clean, square ends.

The cutter does leave a bit of a warped lip after cutting, so I beveled the edges all off with a file. This is an essential step because the portability of the table depends on the parts slip-fitting together. The bevel insures the pipes go in and out of the joints easily.

I put a wrap of colorful duct tape around the pieces that are my legs, so they are easy to identify from the long side pieces when all the parts are in a pile on the floor.

(1) The basic design is an octagon, with 4 legs.

(2) 4 Sides are made from the 24" long pieces, and 4 sides are made with 2 of the 12" long pieces on either side of a T-joint.

(3) Connect adjacent sides of the octagon with the 45 degree elbows. Alternate the long pieces and the assembled sides from the T-joints, until you have completed the octagon shape.

(4) Put the leg pieces in the bottom pointing port of the T-joints.

• Drape your spandex over one side, and secure it with 3 clamps.
• Stretch and secure the spandex to the opposite side also with 3 clamps.
• Next fasten the left side and the right side, also each with 3 clamps.

At this point, the remaining 4 sides can be fastened with clamps. Work your way around the table, adjusting the clamps and pull of the spandex to make it flat.

Overall, the spandex should be taut, but should bounce lightly when struck with your hand. Placing your heavy weight in the center should make it depress roughly 6 inches or so.

Your heavy mass can be anything that will stretch the surface of the table. Different masses will stretch it different amounts, giving different behaviours on the table. I made a lightweight mass by cutting a slot in an old racquetball and filling it with pennies; a smooth, palm-sized river rock works well too.

There are two modes of demonstration with the table.

(1) FLAT SPACE. This demonstrates motion in the absence of masses ("in the absence of gravitational forces"). A particle that is acted on by no forces travels in a straight line. If you roll a ping pong ball straight across the table it will go across without changing direction (until it falls off the edge of the Universe on the far side).

(2) CURVED SPACE. This demonstrates motion in the presence of masses ("under the influence of gravitational forces"). Place your heavy mass or rock in the middle of the table. The mass curves spacetime. Now if you roll a ping pong ball across the table, it does NOT travel in a straight line! Instead, its path is curved, bending toward the mass in the center of the table!

(3) CURVED SPACE ORBITS: With practice, you can make your ping pong balls "orbit" the mass in the center of the table. Depending on how you roll the ping pong ball, you can get circular or elliptical orbits (like planets, moons and asteroids have), or highly elongated parabolic or hyperbolic orbits that do not close (like many comets have). Note that if you roll two ping pong balls at the same time, the inner one goes around faster than the outer one, just as Kepler's Laws of Planetary Motion predict.

Enjoy your spacetime table!

General relativity is the modern understanding of gravity, where astronomers think of gravity not as a force that pulls you, but rather as a curvature of space and time. The way you move through space is a consequence of the shape of spacetime. Particles travel on straight lines when spacetime is not warped, when it is "flat." If the spacetime has a shape, if it is warped, then a particle moves along the shape of spacetime, which is "curved." So what warps spacetime? Mass tells spacetime how to warp.

The spacetime table is a two dimensional model showing how the shape of space (the table surface) changes when masses are put in the space, and how the shape of space affects the motion of other masses. The two-line mantra of the modern gravitational astrophysicist is this: Matter tells spacetime how to curve, spacetime tells matter how to move.

If your large rock is in the center of the table, the fabric of the table (the fabric of the Universe is spacetime) stretches and curves in response.

If you roll a pingpong ball across the table, its path curves toward the rock in the center of the table!

There is no (measurable) force between the rock and your ping pong ball (a fact you can prove to yourself by putting them both on your dining room table and seeing if they roll together). So why does the ping pong ball curve toward the rock? Because the shape of the table (the shape of spacetime) is curved, and it moves in response to the shape of spacetime.

You can read a lot more about how we think about general relativity in my Centennial of General Relativity series of blog posts (this is the first post) or in my series of short 3-minute Centennial of General Relativity vides (this YouTube playlist).

Happy Centennial Year!