November 23, 2020 Filed in: Articles

Philip Freeman (

The authors were inspired to write this article while they worked on Perimeter Institute’s new black hole resource together. To learn more about general relativity and black holes and find ready-to-use, hands-on activities you can do with your class, download the free resource here.

One of the most startling and remarkable discoveries of the 20

One common analogy used to introduce general relativity is the idea of a “mass on a rubber sheet” or “bowling ball on a trampoline”. The bending of the surface caused by the mass pulling the sheet down is used to illustrate the curvature of spacetime in general relativity. A rolling marble on the surface follows a curved path, or “orbits” the central mass, giving convincing evidence of the parallel between the sheet and the action of gravity. But there is a problem with this demonstration – it isn’t showing what it claims to show. In this article we argue that this analogy, as used, is fundamentally flawed and creates significant misunderstanding for both students and teachers. We explain where the problems arise, and how to avoid these problems and still provide a strong visual model and deeper understanding of how general relativity works.

We all use analogies ― they give our brains a break and make the unknown seem more familiar. If you’ve never seen a giraffe, being told that it is like a long-necked horse is pretty helpful. But analogies are also dangerous because they are inherently wrong. A giraffe is like a horse in some ways, but it is different from a horse in many important ways. Without explicitly stating the limitations and boundaries of the analogy, the risk is that it is overextended, and misconceptions are introduced. Many misunderstandings about physics in general, and modern physics in particular, come from overextended or poorly chosen analogies.

The bowling-ball-trampoline analogy has several problems, some of which are shown in the xkcd cartoon in

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Second, why does the marble roll down into the well? Because it is being accelerated by the Earth’s gravity and the force from the fabric (see

General relativity is a geometric theory and not a theory of forces. General relativity holds that the path of an object moving near another massive object appears deflected but is in fact following the shortest path (called a geodesic) with no force acting and no acceleration. The geometry of spacetime causes this straightest possible line to appear curved compared to expectations of our usual flat geometry.

Ultimately this difference results from the distance between points on the surface having a different relationship than the one we usually take for granted, so that it no longer follows the same geometric rules. We see such changes when we try to make three-dimensional surfaces match with our familiar two-dimensional plane. Anyone who has done any sewing of clothing knows that making flat fabric fit a human body is not trivial. The various notches and gussets of a pattern are necessary because of curvature. Similarly, airplane routes along ‘great circle’ paths look like massive detours on a flat map but are indeed the shortest route possible.

In a demonstration of general relativity, trajectories should curve because they are following the shortest paths in a space with curvature and this can be shown with the bowling-ball-trampoline analogy. We can illustrate a “straightest line” path on the trampoline by laying down a strip of tape so it is as flat as possible. Any deviation from the “straight” (or shortest) path will wrinkle the tape.

Even the brains of physicists who have been studying general relativity for years can’t handle visualizing three dimensions of space mixed with time all acting in a non-Euclidean way (you are not alone!). To simplify it they sometimes remove time completely and also remove one dimension of space.

An embedding diagram shows how a slice of 2-D space (a plane) is curved in general relativity by embedding the slice in a make-believe third dimension.

Looking at

Can we ‘fix’ this analogy to help understand general relativity?

A good analogy for general relativity should rely on curvature, not on a Newtonian description of gravity through forces. When illustrating curvature in general relativity, you can bend the fabric up or down (as shown in

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Either way you bend the fabric the same curvature is created, and all the effects on geometry are the same. Rather than bending the fabric down using a massive object placed at its centre, if you bend the fabric up you remove the confusing role of Earth’s gravity causing the bending and the misleading effects of the forces from gravity and the surface. To bend the fabric up, you can place a support like a flask or funnel with a marble on top beneath the fabric as shown in

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Placing strips of tape illustrates how straight paths become bent by curved space. The upward bent fabric and the downward bent fabric have

Probably the most obvious, of course, is that space isn’t a two-dimensional piece of fabric! A surprising number of people take this analogy literally, so be aware of this risk.

Another limitation is the fabric model leaves out more than just one spatial dimension. Bending fabric up is a great way to illustrate the curvature of a 2-D slice of space. In

The problem is that this embedding does not show us how spacetime is warped. Time has been completely removed from the analogy. General relativity predicts warping effects on time that are similar to those on space, and even more pronounced in their effect. Objects at rest or moving differently are following different paths in this combined spacetime.

One of Einstein’s key insights is that we can’t speak of space and time separately. The two are forever intertwined as “spacetime.” General relativity requires considering the path of an object in both space and time and to visualize the warping of “space” and “time,” spacetime diagrams are essential.

Let’s explore how spacetime diagrams can help to show differences in spacetime paths by considering three balls at the top of a cliff. One ball is dropped from rest and the other two are launched horizontally from the same location at the same time, but with different speeds (see

While this is familiar to us, it is not easily explained using tape on stretchy fabric the way we showed. Since the two moving balls are launched from the same location and in the same direction, we would represent them with pieces of tape starting in the same direction, and thus we would expect the balls to travel along the same path, which we know is not the case! And a stationary ball doesn’t follow any path, never mind a curve, but it is clearly affected by gravity.

Even an object at rest is moving through time. This is why gravity is still apparent even when you are not moving through space. You are always moving through time, and thus the curvature of spacetime can explain gravity.

This is also shown in

In order to continue to stay in one place and not follow the ‘curved’ straightest path (i.e. in order not to fall) we would need to exert a force to keep the object’s path in spacetime vertical (i.e. at a constant position). This could be, for example, a force such as the one your chair is exerting on you as you read this! The chair is constantly pushing you away from a straightest line path through spacetime. Setting aside air friction, a falling object experiences no forces at all and follows a straightest line path through curved spacetime. This is how general relativity uses the curvature of spacetime to explain gravity.

We said we could measure the “warping of space” in terms of the amount of space measured “inside” a region. Is there something similar that shows what we mean by “warping of time”? Such a warping of time implies a different amount of time within a region — the rate at which clocks run changes with distance from a massive body. This is observed to be true and extremely well-tested.

In fact, almost all our everyday experiences of gravity (i.e. throwing a ball or being stuck to the Earth) are due to the warping of time in general relativity and not the warping of space. The warping of time gives us essentially Newton’s gravity, while the effect of curvature of space is quite small and makes a difference only in extreme cases, such as the discrepancy in the precession of Mercury’s orbit.

Second, have students explore a demonstration of the stretchy fabric in the ‘bent up’ orientation. Discuss how the curvature of space deflects the paths on the surface of the fabric, in contrast to the force description of Newtonian gravity.

Third, explicitly state the limitations of the analogy ― only two dimensions of space are being shown and time is being completely ignored. Also, the fabric is being bent into a make-believe dimension that doesn’t exist but helps us visualize curvature. Curvature does not actually require such an extra dimension.

Finally, if time permits, introduce an alternative visualization — the spacetime diagram. This visualization is also limited in that it only shows one or two dimensions of space, but it can help us better comprehend the effects through time, which produce gravity as we usually experience it.

For more information and hands-on activities for students, download Perimeter Institute’s new Black Hole resource.