**Why the popular spacetime explanation model of the balls in the rubber blanket is misleading – and how it can be saved.**

The first photo of a black hole caused a stir in 2019. Two years earlier, the Nobel Prize in Physics had been awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne for the discovery of gravitational waves. But what is spacetime? How can we visualize it?

In both cases, it is experimental confirmation of phenomena that were predicted by general relativity over 100 years ago. Albert Einstein’s great work is also highly topical today – and one of the epitomes of physics that is difficult to understand. In order to make such complex topics understandable (what is spacetime?) to a broad audience, scientists and journalists rightly use simple analogies and models.

In the case of general relativity, the effect of gravity is often illustrated with the help of two balls in an elastic cloth, the heavier one stretching the cloth under its own weight and the lighter one rolling around the heavy one in a curved path (see Figure 1). Unfortunately, this model has little to do with the real ideas behind Einstein’s famous theory. But it can still be saved with a slight adjustment.

## What Is Spacetime?

To set the context, let’s go to the beginning of the 20th century. A few decades earlier, James Clerk Maxwell established a theory of Electrodynamics with the equations named after him and thus explained how electrical charges, currents, and electrical and magnetic fields interact.

In particular, his equations describe the propagation of electromagnetic radiation such as light or radio waves. A nagging problem for physicists, however, remains the observation that the propagation speed of these waves – i.e. The speed of light – in experiments was always independent of how the light source and the observer moved.

This cannot be explained with Maxwell’s theory alone and also contradicts our everyday experience: When we run after a departing train at the station, its speed is much lower than when we stop, relative to ours.

Albert Einstein solved this problem when he published his special theory of relativity in 1905. In it, he effectively merged space and time into space-time and described how the space and time coordinates differ from observers who are moving relative to one another.

The term space-time stems from the fact that space and time can to a certain extent swap their roles – similar to the way in which the terms front, back, right and left change subjectively to one another for an observer when he rotates in space. Acceleration can be understood as a kind of rotation in space-time, in which space and time coordinate mix.

## Why The Apple Falls From The Tree?

In the special theory of relativity, however, it remained open how gravity could be integrated into this new picture of space and time. According to Newton’s Laws, gravity is proportional to the size of two masses and decreases quadratically with the distance between them. However, according to Einstein, it is precisely this distance that cannot be clearly determined: Different observers moving at different speeds also measure different distances.

In addition, gravity acts without time delay in Newton’s theory: As soon as a mass moves, all other masses immediately feel drawn to their new position. But even this is forbidden according to Einstein’s theory since no physical effect can spread faster than the speed of light.

Einstein himself found the ingenious solution to this problem: In his general theory of relativity, he completely abolished Newton’s gravitational force and replaced it with the ability of space-time to deform. The following processes are central:

- Masses and energies bend space-time in their immediate vicinity.
- Space-time itself is elastic to a certain extent and local deformations, therefore, propagate at the speed of light. (which leads to the gravitational waves observed for the first time in 2015)
- Rays of light and objects, on which no forces act, move on so-called Geodesics (read: as straight as possible) through the curved space-time.

Unfortunately, our brains are not well equipped to imagine four-dimensional space-time – and certainly not curved. Even mathematics, which can be used to precisely calculate all these mechanisms, is not necessarily known for its simplicity.

On the other hand, at least a rough idea of the basic ideas of the general theory of relativity is necessary in order to be able to talk about phenomena such as black holes, gravitational waves, or cosmic background radiation. And so began the rise of the rubber blanket model.

## A Rubber Blanket Makes a Career

The model works as follows: One imagines space-time as a stretched elastic cloth. In the middle of it lies a heavy ball, which is supposed to represent a star with its large mass, for example, and which presses a dent in the cloth with its weight. The cloth now shows a curvature (“Like space-time!”)

Now you push a small marble onto the rubber blanket and observe that it rolls on a curved, perhaps even approximately elliptical path around the large ball in the middle (“Like a planet!”). Ergo, the star bends space-time, and the curved space-time forces the planets in elliptical orbits.

Unfortunately, this model is of limited use to learning about general relativity, and it is amazing how popular it has become. It runs through popular science books, blogs, and television programs is also used with pleasure by serious scientific organizations such as the Max Planck Society (video) and is even the subject of a specialist article in which the dynamics of rolling marbles on curved elastane tissue is examined theoretically and experimentally.

A particularly curious example was provided in 2017 by the excellent science explorer Harald Lesch in his book The Discovery of Gravitational Waves – Or Why Space-Time Is Not a Blanket. Instead of taking his own book title literally, he introduces the rubber blanket model in the traditional way and justifies the second part of the title only with the fact that space-time is extremely stiff and can be deformed much less than rubber. The latter, while undoubtedly correct, is not overly relevant to an understanding of rationale.

But what is the problem with the blanket model?

## The Model Does Not Fit The Theory

First: The space curvature, which is all decisive in the general theory of relativity, is indeed present in the rubber blanket – but it does not play a role at all! Even if the cloth is flat, the little marble doesn’t run in a straight line, as long as you hold it at a slight angle (as shown in figure 2). The path is then parabolic and no longer corresponds to a bound rotation around the central body.

(Figure 2) The curvature of the cloth is not critical to deflecting the marbles. They also roll-on curved tracks on a flat, but sloping surface. (Image: Helmut Linde)

What is decisive, however, is that the deviation from the rectilinear movement (i.e. Newton’s gravitational acceleration) in the model is not caused by the curvature of the cloth, but only by its local inclination in the gravitational field of the earth on which the experiment is carried out.

**And this is already the second problem: **The reason that the small marble in the cloth rolls towards the big one is precisely the gravity of the earth by which it is pulled down. The rubber blanket does nothing other than prevent the marble from falling freely and deflecting would make similar sense to describe how a steam engine works by building a large dummy out of wheels and pistons, which are then moved by a smaller steam engine. This is particularly misleading because it was Einstein’s great achievement to abolish the direct attraction between two masses and to replace it completely with the space-time curvature. A model that does not reflect this conceptual step can hardly contribute to a better understanding of the theory.

So should we ban the fairy tale of the rubber blanket from science communication once and for all? Do you have to listen to a few semesters of lectures on Riemannian geometry in order to gain a certain intuition for Einstein’s theory of gravity?

Fortunately not, because the rubber blanket can be saved with a little trick. But before that, we have to deal with what curvature actually means.

## Not All Curvatures Are Created Equal

In Asian lessons, we learned that a novel has an internal and an external plot. Not entirely dissimilar to this, there is an inner and an outer curvature in mathematics.

One speaks of external curvature when an object is embedded in a higher-dimensional space but does not run along straight lines in it – i.e. When the curvature can be seen from the outside, so to speak.

To put it simply, inner curvature means that our school geometry does not apply in a room. Internal curvature means that our school geometry does not apply in a room: triangles do not always have an angle sum of 180 degrees, the Pythagorean theorem is canceled and so on and so forth.

## Welcome to Flatland!

An example can help to better understand the two concepts of curvature: We start with a blank sheet of paper, which lies flat on the desk and therefore has neither an inner nor an outer curvature

We draw some stick figures on the paper and call them the “Flatlanders”. We imagine that these beings live in the two-dimensional world of the sheet so that all of their senses, movements, and interactions are along the plane of the paper.

The Flatlanders have never heard of three-dimensional space, and they also have no way of recognizing it or even leaving their flat world. So that the Flatlanders don’t get bored, we draw a few more triangles on the sheet, on which our two-dimensional friends can take angles and lengths measurements.

Now we roll the sheet up into a cylinder. As an object in three-dimensional space, the sheet is then obviously no longer straight – so it has an external curvature. However, the angles and side lengths of the triangles on the sheet are the same as before rolling up. The school geometry is still valid and the sheet, therefore, has no internal curvature. Even the Flatlanders do not notice that their world is now rolled up.

The situation is different with the surface of a sphere: This obviously also has an outer curvature – but in contrast to the paper cylinder it also has an inner curvature: If you draw a large triangle on a globe, its angle sum is no longer 180 degrees – so the school geometry is violated! The lowlands could also take measurements on such triangles and find that their world is curved.

Whenever “the curvature of space-time” is mentioned in the general theory of relativity, then the inner curvature is always meant. Since space-time itself is already four-dimensional (three space dimensions plus time), a surrounding space would have to be at least five-dimensional. And we cannot observe such a space any more than the Flatlanders can see the bigger picture of their two dimensions.

## A Straight Line Is Not Just a Straight Line

In the examples above, we nonchalantly ignored an important detail, namely the question of what is actually meant by a straight line in a curved space. As I said, the Flatlanders on the paper cylinder do not notice any of the outer curvatures. Therefore, in their two-dimensional world, there are of course straight lines (for example, the sides of the triangles we drew on the sheet). However, viewed from outside (i.e. from three-dimensional space) these lines are of course no longer straight – unless they happen to run parallel to the axis of the cylinder.

In mathematics such lines are called “Geodesics”. If you live in a curved space yourself, then they appear as straight lines. However, if the curved space is embedded in a higher-dimensional one, when viewed from the outside they are only as straight as possible. As already briefly indicated above, the general theory of relativity says that light and free-floating objects move on Geodesics through space-time.

We can now formulate the problem with the rubber blanket model a little more mathematically. In this, the marbles’ path is influenced by the local inclination of the rubber blanket and thus ultimately by its external curvature. What is important in the theory of relativity is the inner curvature!

## How Space-Time Becomes a Rubber Blanket

Let’s now proceed to the rescue of the rubber blanket model. First, we move the thought experiment from the cozy living room far out into the cold universe, far away from the stars and planets, to a place of complete weightlessness. So we can no longer fall into the trap of explaining gravity by gravity.

We float out here next to a stretched rubber blanket and notice that the large, heavy ball makes no move to make a dent in it due to the lack of gravity. So we have to force the ball into the cloth to stretch it.

We replace the little marble with a toy car that can drive on the rubber blanket. A little double-sided tape around the tires will keep the car from floating away. If we give it a nudge and it rolls along with the blanket. We actually see the effect of the inner curvature. As long as the car is far away from the ball and is therefore in the flat part of the blanket, it moves in a straight line. However, as soon as it passes close to the sphere, it is deflected towards the sphere due to the curvature (!) Of the surface (see Figure 3).

(Figure 3) The toy car drives across the rubber sheet as straight as possible – and is actually distracted by its curvature. (Image: Helmut Linde)

That the curvature is actually at work here can be seen as follows. When a car drives straight ahead, its left and right wheels turn at the same speed. In a curve, on the other hand, the outer wheels turn faster than the inner ones because they have to cover a longer distance.

If our car now drives past the sphere without changing its direction of travel, the inner wheels turn more frequently: You have to drive through the deeper dent and therefore cover a longer distance (see Figure 4).

(Figure 4) If the car did not change its direction of travel, the left wheels would have to cover a longer distance (green) than the right ones (yellow), because the left ones drive deeper into the hollow. In a car driving straight ahead, however, all wheels cover the same distance. What looks like a (preferably) straight line in the surrounding three-dimensional space is therefore a curve in the two-dimensional world of the rubber blanket! (Image: Helmut Linde)

So a path without changing the direction of travel is indeed a straight line in the surrounding three-dimensional space (apart from the inevitable deviation when driving through the dent) – but in the world of the rubber blanket, it is a curve. In order to drive straight ahead in the rubber blanket – i.e. On a Geodesic – the car must turn while driving in such a way that the rotation compensates for the differences between the paths of the left and right wheels caused by the curvature.

## Small Change – Big Impact

Some may dismiss swapping marbles for toy cars as splitting hairs, especially since the curved paths in both cases have so far looked similar. But this is by no means the case.

The difference between the two models and thus also the difference between the inner and outer curvature becomes particularly clear if we make another modification. We turn the rubber blanket over. The dent becomes a hill (see figure 5).

(Figure 5) You can see the difference between a car and a marble most clearly when you turn the rubber sheet inside out – the hollow becomes a hill. This has no effect on the deflection of the car, as it follows the inner curvature of the cloth, which does not change when it is turned inside out. The ball, on the other hand, is now pushed outwards, since its path depends on the embedding of the cloth in the outer three-dimensional space and the direction of its weight. (Image: Helmut Linde)

If we do this experiment again on Earth, we will see a major impact on the marble’s orbit. It will now be repelled by the large ball. However, nothing changes in the path of the car. Because the car follows the inner curvature of the cloth and it remains unchanged, no matter in which direction the dent is pointing. (If you are not convinced: the argumentation is the same as in Figure 4).

## Lowland/Flatland Newton Explains Gravity

If the path of the car in the two-dimensional world of the rubber blanket is a straight line and only appears as a curve from the outside, how do the Flatlanders actually notice that there is gravity at all?

You notice it when you compare the paths of several cars. Two cars that start parallel to each other at a certain distance will go in different directions after a while. A flatland Isaac Newton would say: A force was exerted on the cars and distracted them. In reality, however, there is no force at all – both cars were simply going straight in their respective areas of curved space.

Here a key statement of the general theory of relativity becomes clear. Gravity is not a force – but a purely geometric effect that influences the paths of all bodies just like that of rays of light.

## Ultimately, Only Math Helps

Of course, the car model isn’t perfect either: In Einstein’s general theory of relativity, space-time curves in particular, while in the model only space – i.e. The elastic cloth – curves.

This is also related to the restriction that the car cannot move in ellipses around the sphere, as the planets do around the sun. The path of the car is more likely to a comet which enters the solar system, is deflected, and then leaves it again. In order to get beautiful elliptical orbits, you not only have to bend space, but space and time together – and for that you need mathematics.

*Refecrences*

Rainer Weiss (German Scientist – Wikipedia)

Nobel Prize Organization (Nobel Prize in Physics 2017)

ScienceBlogs.de (Gravity visualized: The experiment with the rubber blanket)

Britannica (James Clerk Maxwell, Scottish mathematician, and physicist)

Maxwell’s Equations (Maxwell’s Equations)

Khan Academy (Newtonian path–time diagram)

Cornell University (arxiv.org, Circular orbits on a warped spandex fabric)

Wikipedia (Riemannian geometry)

Good Reads (Flatland: A Romance of Many Dimensions)

Helmut Linde (Research Gate: Helmut Linde Covestro Deutschland AG PhD)