In the photograph above is a Vortx^® wishing well. This is a miniature version of much larger devices that are popular as fundraising tools. These wishing wells are shaped like curved funnels, and they have one or more launching ramps (often two, diametrically opposed) from which one can release a coin. When the coin rolls down the ramp, it acquires a tangential velocity, and so travels in circles around the wishing well. As gravity pulls the coin down the curve, the circles get smaller. Since angular momentum must be conserved, the coin’s speed increases as it goes further down the funnel. Eventually, the coin gets to the narrowest point of the funnel, where it orbits fastest, and then falls into a collection box beneath the bottom of the funnel. The appeal of these devices is that the motion of the coin as it progresses from the top of the well to the bottom is quite fascinating, and it is a lot of fun to watch. Also, if there are at least two ramps, people have fun racing coins down the well. It turns out, however, that this type of device is also useful for illustrating several aspects of non-Euclidean space, which are important to discussions of certain exotic cosmic objects or of relativity. Below is a summary of four things you can illustrate with the Vortx^®, followed by a brief explanation of some of the ideas behind them.

When this demonstration is set up for use in class, the camera sits almost directly above the Vortx^®, to give a view that is nearly along the axis of the funnel. You can drop either a ball or a penny down either of the two ramps.

First, the Vortx^® resembles a diagram commonly known as an embedding diagram, which is a tool for illustrating the concept of curved spacetime. Whereas in Galilean relativity one can speak of spatial coordinates and time as separate entities, in Einsteinian relativity, we find that they are intertwined so that we cannot treat them separately, but speak of what our coordinate system represents as “spacetime.” (Some hyphenate this as “space-time.”) Einstein’s principle of equivalence (vide infra) leads to the idea that gravity is a manifestation of the curvature of spacetime, which means that spacetime, which is normally flat, is curved near objects that have mass. In the Vortx^®, if a coin starts out rolling around a circle near the top, by the time it has traveled down the side through a distance of, say, half the radius of its original circle, the new circle it makes has a radius that is greater than half the radius of the original circle. That is, it has traveled towards the center of the funnel by less than the distance it has traveled down the side of the funnel because of the curvature of the funnel. Similarly, for something traveling around a massive object such as a star, for a given change in the radius of its orbit, the new orbit is actually larger than the original orbit minus 2π times that change. The geometry around the star is non-Euclidean; spacetime is curved. In that this curvature is not really the result of 3-dimensional space curving through a fourth spatial dimension, the embedding diagram and the Vortx^® provide only a loose analogy to the physical reality, but a useful one nonetheless.

Another point you can illustrate is that whereas on a flat surface you could go in a straight line forever, on a curved surface it is quite possible to go in a straight line and end up at exactly the point from which you started. Whereas on a flat surface, you would have to turn left or right to do this, on the curved surface, without turning at all, you can end up back at your starting point.

Third, similarly to the way an object in motion on a flat surface, absent friction or any other external force, would continue forever in a straight line, an object in motion on our curved surface, absent any friction, would go around in a circle forever.

Fourth, you can use the Vortx^® to introduce Einstein’s geodesic law, and the underlying concepts of proper time and geodesics. Proper time is the time between two events as measured by a single clock. For a moving object, then, the proper time is the time as measured by a clock moving with the object as it travels, for example, between two points in spacetime. A geodesic is the shortest path between two points on a surface. On a flat plane, the geodesic between two points is a straight line. On a curved surface, it is that particular curve between the two points whose length is shortest. On a sphere, for example, the geodesic that connects two points is the path between them that lies along a circumference, or a great circle. Any other path is longer. Einstein’s geodesic law is a consequence of the form of the metric, the distance equation, for spacetime (it has minus signs where the metric for Euclidean space has plus signs). According to this law, for a body moving in spacetime, either free of any forces or subject only to gravity, the geodesic is the path for which the proper time is a maximum, and this is the path that the body follows.

A full development of all of these concepts is beyond the scope of this page, but a few of the ideas mentioned above require at least some explanation.

The embedding diagram gets its name from the idea that an n-dimensional space is embedded in the (n+1)-dimensional space that contains it. For example, a 2-dimensional space contained within a 3-dimensional space is said to be embedded in the 3-dimensional space. A surface can be flat, in which case it is 2-dimensional, or it can be curved. Observers on the surface can detect its curvature by noting such things as that parallel lines eventually meet, or that the sum of the angles of a triangle add up to more (or less) than 180 degrees – in other words, that the geometry of their surface is non-Euclidean. Such curvature is obvious to an outside observer in the 3-dimensional space within which the surface is embedded.

In the course of developing his theories of relativity (special and general), Einstein formulated what is now known as his principle of equivalence. This is that an observer in a particular reference frame cannot tell the difference between his reference frame being accelerated in a particular direction, and gravity acting upon it from the opposite direction. For example, if an astronaut inside a spacecraft parked on earth drops a ball, he sees it fall to the floor with the acceleration of gravity. If the spacecraft is in a region of space where there are no nearby objects to exert gravity on it, and the astronaut sets the engines to accelerate the spacecraft upwards at the appropriate rate, when he drops the ball, it falls in exactly the same way as it did on earth. Thus, there is no difference between an object’s inertial mass and its gravitational mass; the two are equivalent. This example involves a slight oversimplification, in that all gravitational fields radiate from a point. If an astronaut dropped two balls, for the spacecraft on earth, their paths would converge slightly, whereas for the spacecraft accelerating in the middle of space, they would be perfectly parallel. Also, on earth, gravity would be slightly stronger at the bottom of the spacecraft than at the top, but in the spaceship accelerating through space, the ball would have the same apparent acceleration whether it was dropped from near the top or the bottom of the spaceship. The principle of equivalence holds only for systems that are small enough that, for all intents and purposes, the gravitational field they experience is uniform.

Apparently, by thinking about how measurements in a rotating frame would compare to those made by an outside observer, Einstein realized that his priciple of equivalence implied that matter curves the space around it. Imagine a rotating turntable whose rim is marked in short, straight segments. Observers on the turntable measure the length of each of the segments and add them up to obtain the circumference of the turntable. Observers on the ground, however, see the segments moving in the direction of their long axis, and measure a shorter length than what the observers on the turntable measure (by the Lorentz contraction: L = L₀√(1-v²/c²)), and their value of the circumference is smaller than what the observers on the turntable measure (by the factor √(1-v²/c²)). Both sets of observers measure the same radius for the turntable, since it is perpendicular to the relative motion. Since the ground observers are in an inertial frame, when they divide the circumference by the radius, they obtain a value of 2π. When the observers on the turntable do this, they get a value that is greater than 2π. Thus, the geometry in the reference frame of the turntable is non-Euclidean. The turntable is an accelerating reference frame, and by the principle of equivalence, this is equivalent to its being in a gravitational field (which would point outwards from the center in all directions; the acceleration points inwards, towards the center; this is analogous to the linear acceleration vs. gravity in the example of the rocket ship above). Therefore, where there is a gravitational field, geometry must be non-Euclidean; spacetime is curved.

References:

1) Sartori, Leo. Understanding Relativity: A Simplified Approach to Einstein’s Theories (Berkeley and Los Angeles, California: University of California Press, 1996) chs. 3, 8.
2) Feynman, Richard P.; Leighton, Robert B. and Sands, Matthew. The Feynman Lectures on Physics, Volume II (Menlo Park, California: Addison-Wesley Publishing Company, 1963) ch. 42.
3) Thorne, Kip S. Black Holes and Time Warps: Einstein’s Outrageous Legacy (New York: W. W. Norton and Company, Inc., 1994) chs. 1-3.
4) Peebles, P.J.E. Principles of Physical Cosmology. (Princeton, New Jersey: Princeton University Press, 1993) p. 249.
5) Norton, John D. Gravity near a massive body, at https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html.