What is the origin of the rubber sheet analogy?

Question

I am trying to track down the first use of the 'bowling ball on a rubber sheet' analogy to explain spacetime curvature in general relativity. I have found a lot of secondary sources that give contradictory citations but no primary sources. ChatGPT generated a lot of fake references. Some modern papers (from the 1990s and early 2000s) even refer to Einstein in 1915 or Eddington in 1919 but these do not link to primary sources and are not very credible. I searched through the Einstein Papers and couldn't find anything. It seems likely to have originated in the mid 1950s-1960s by either John Wheeler or Robert Dicke but I would really like to find the first citation. I know it was before 1967 as Asimov published a short story using the idea in that year (unless he invented it which seems unlikely). Does anyone know?

Answer 1

Beware of ChatGPT, it can easily generate fake references for historical materials. A while ago, I asked about the origin of the concept of vector "norm". Not only it provided a fake German reference but also quoted the relevant text in German. All imaginary! In short, don't use this experimental interface for science for the time being.

Anyway, coming to the main query. It appears from this article, (Our Elastic Spacetime: Black Holes and Gravitational Waves: A new computer program shows that the old analogy of spacetime as a rubber sheet is remarkably valid by Larry L. Smarr and William H. Press in American Scientist, 1978, 66, 72-79), that Arthur Eddington proposed this main theme in 1928 in his book written for the public.

It is not exactly what we see in planetarium shows, but the elements seem to be there. The book is titled The nature of the physical world (1928). On page 127 he writes:

I should like to show you in a general way how it is possible for a law controlling the curvature of empty space to determine the tracks of particles without being supplemented by any other conditions. Two "particles" in the four-dimensional world are shown in Fig. 5 , namely yourself and myself. We are not empty space so there is no limit to the kind of curvature entering into our composition; in fact our unusual sort of curvature is what distinguishes us from empty space. We are, so to speak, ridges in the four-dimensional world where it is gathered into a pucker. The pure mathematician in his unflattering language would describe us as "singularities". These two non-empty ridges are joined by empty space, which must be free from those kinds of curvature described by the ten principal coefficients. Now it is common experience that if we introduce local puckers into the material of a garment, the remainder has a certain obstinacy and will not lie as smoothly as we might wish. You will realise the possibility that, given two ridges as in Fig. 5, it may be impossible to join them by an intervening valley without the illegal kind of curvature. That turns out to be the case.

Two perfectly straight ridges alone in the world cannot be properlyjoined by empty space and therefore they cannot occur alone. But if they bend a little towards one another the connecting region can lie smoothly and satisfy the law of curvature. If they bend too much the illegal puckering reappears. The law of gravitation is a fastidious tailor who will not tolerate wrinkles (except of a limited approved type) in the main area of the garment; so that the seams are required to take courses which will not cause wrinkles. You and I have to submit to this and so our tracks curve towards each other. An onlooker will make the comment that here is an illustration of the law that two massive bodies attract each other.

Answer 2

A rubber sheet analogy for curved spacetime was used very soon after Einstein's publication of General Relativity. The classic "bowling ball" version may have taken a few years to evolve, but was already in circulation by 1925, and an essentially equivalent but perhaps less clear "worms-bullet" analogy was published in 1920.

Early massive central weight rubber sheet analogies

Donald Menzel very clearly describes the classic heavy ball on a rubber sheet analogy in his article on "What is gravitation?" on page 223 of "Science and Invention for July, 1925"

Let us make our plane out of tightly stretched rubber (see Fig. 4). A small virtually weightless marble will move across it in a straight line, the path of least resistance. If a heavy iron ball is now placed in the center, the plane will be "warped" and the path of least resistance will no longer be straight, but curved about the object, and if there were no friction, it would continue to revolve about it forever. This is a vivid picture of the case of the solar system. The planets do not revolve about the sun because its constraining force keeps them from flying away, but rather because space is bent and they follow their own inclination. Einstein called the natural path of any object a geodesic.

On page 74 of his 1920 popular science book, "Easy lessons in Einstein; a discussion of the more intelligible features of the theory of relativity", Edwin Slossen (Harcourt, Brace and Howe, 1920) uses a bullet for the central weight and worms to map out the geodesics.

How a heavy object can alter space relations may be seen from this simple illustration: Stretch a sheet of rubber over a hoop like a drumhead. It is now level and flat and if parallel lines are drawn across it in two directions so as to divide it up into squares like a checkerboard all these lines are straight and equidistant and all the squares are of equal size.

A row of worms, starting in an even rank and crawling along the parallel lines across the drumhead, would keep even all the way. Now lay a bullet on the center of the drumhead. The rubber sags down and stretches, most in the middle, least at the edges. The "parallel" lines are no longer equidistant. The squares are no longer equal. The lines are no longer of the same length. If now we repeat our worm race we shall find that those worms following lines close to the weight have to go down hill and up again and so travel a greater distance to traverse the same number of squares than those following lines nearer the edge which lie comparatively flat and are nearly as short as before. Consequently the worms will be slowed up in proportion to their nearness to the center and the row of their heads will be swung around at an angle to their former frontage.

Other early rubber sheet analogies, but without the heavy ball:

Nicholas Rashevsky uses a rubber sheet analogy in a November 1924 Scientific American article titled "The Possibility of Other Kinds of Space", but only talks about statically stretched spacetime.

… if instead of the sheet of paper we take a sheet of rubber. The rubber will stretch. … some parts of the original rubber plane have had their dimensions altered and this alteration of the dimensions has occurred in different degree in different parts of the rubber sheet. … we will find that the figures on the rubber no longer obey the rules of Euclid's plane geometry.

In his General Theory of Relativity, Einstein was led to assume that the space of the real universe is, in fact, non-Euclidean. We have seen in this article that this is quite possible.

Even earlier, Joseph Ames briefly mentions distorting world lines on rubber sheets in a 1919 American Physical Society Presidential Address on "Einstein's Law of Gravitation. (For a non-paywalled version see page 238 of "Contemporary Science", a 1921 book edited by Benjamin Harrow.)

I can draw a line on a sheet of paper or of rubber and by bending and stretching the sheet, I can make the line assume a great variety of shapes; each of these new shapes is a picture of a suitable transformation. Now, consider world-lines in our four dimensional space. The complete record of all our knowledge is a series of sequences of intersections of such lines. By analogy I can draw in ordinary space a great number of intersecting lines on a sheet of rubber; I can then bend and deform the sheet to please myself ; by so doing I do not introduce any new intersections nor do I alter in the least the sequence of intersections.

Einstein's 1917 "Cloth" analogy

I can't find any explicit use of a rubber sheet analogy by Einstein himself in those early years, but he came close in a March 1917 letter to Willem de Sitter where he wrote:

I compare the space to a cloth floating (at rest) in the air, a certain part of which we can observe. This part is slightly curved similarly to a small section of a sphere’s surface. We philosophize on how we must construe the continuation of the cloth so that an equilibrium is reached in its tangential tension, whether it is fastened in position at the edges, extends infinitely, or has a finite size and is a closed unit.

Earliest physical manifestation/construction

Although the question is about the concept, I'll note that the earliest documented physical construction of the analogy I've found so far is "The Gravity Field Simulator built by Victor Showater at Ohio State University and published in 1963. This was a natural rubber sheet stretched on a 36 inch square frame with a 9 pound shot put ball in the middle and small steel balls as orbiting "planets".

Physics professors and teachers do not, however, always publish their demonstrations, so it is very possible earlier versions were built elsewhere.

Answer 3

In the 12 December 1919 Monthly Notices of the Royal Astronomical Society there is starting on page 96 a "Discussion of the Theory of Relativity" with multiple speakers and in the portion where Eddington is speaking, he says:

Imagine this blackboard to be a sheet of rubber ; distort it until these irregular meshes become neat uniform squares...