# Did Gravity have to be an inverse square law?

How fundamental is the fact that gravity is proportional to $1/r^2$ versus a more general $1/r^n$ or something more complicated like $f(r) = e^{-r}$? Is this solution unique? Does it have any special properties other forces would not have?

This was settled in a publication in to the Royal Society in 1684 by Newton. The idea that the force might be $1/r^2$ was not new.  This had been proposed by Hooke, at the time the resident experimenter at the recently formed Royal Society. Hooke postulated an inverse square law but did not prove it, lacking the mathematical insight of Newton. He was always a bit bitter about not being recognized for his contribution in postulating the correct formula and was combative in his exchanges with Newton. It is possible that Newton’s “If I have seen further it is by standing on the shoulders of giants…” was partly to acknowledge Hooke. It’s also possible this was a bit of a dig at Hooke who was a tall, somewhat stooped gentleman. 1

To understand what was really done by Newton it helps to pause and think about what he had to work with. The most precious thing was accurate data – the positions of planets measured carefully over sustained periods of time. This data measured the shape of the planets orbits and their speed at different points of the orbit. Kepler had done a lot of data collection and in 1609 using this information he was able to see patterns that helped him determine relationships in the data. Kepler’s first law was the radical notion that planets moved in ellipses (not circles) and the second was the bodies moved such that the line from the body to the center swept out an equal area in equal times.

So what do you think? Is that in itself enough to determine what force gravity is? Enter Newton.

Newton proved a number of things about generic forces acting on a line between two bodies. In particular he established that for such a force the motion will be in a plane and there will be a constant of motion (angular momentum) that is conserved. He also demonstrated that the shape of the path of one body around the other is the same as each body around their common centre of mass. He showed that for a general force a body will sweep out equal areas in equal times. This fact alone does not distinguish between forces. It is true for any central force.

How about the path of the orbit? Does that uniquely determine the force? Halley (of Halley’s comet fame) had the same question and in 1684 he went to Cambridge to see Newton. He asked what shape a force of $1/r^2$ would produce. Newton’s responded that it would be an ellipse and that he had proved it. Halley was amazed and asked for the proof, leaving Newton mumbling “Gee, I know that used to be around here somewhere…”. Newton promised to provide it and several months later he provided it and a lot more in his note The Motion of Gyrating Bodies (De Motu Corporum in Gyrum). This was read to the Royal Society on Dec. 10, 1684. This then sparked Newton into writing the Principia which was then published in 1687. (Halley had to foot the bill for the publication, since the Royal Society had “bet big” on a a previous publication The History of Fishes that turned out to be a flop and consumed all their funds. At one point the society paid Robert Hooke in copies of the book.)

Principia is simply amazing. Any one of the theorems in it would have given Newton a place in history but the book has dozens and dozens of ground breaking results. The way mathematics was written at the time and the terminology used makes a translation of this work pretty tough going, it’s an improvement on trying to read the original latin but not by much. Fortunately there is a marvellous book by Chadrasekhar Newton’s Principia for the Common Reader. It is a masterful book from a physicist whose ability to do difficult calculations is astounding. (His The Mathematical Theory of Black Holes (Oxford Classic Texts in the Physical Sciences)
is perhaps the most densely packed book of equations I ever studied.) The term “common reader” is perhaps a bit generous to the reader, since there are differential equation at about a second year undergrad level.

Newton did WAY more than just figure out the law of gravity. He looked at central forces in general, proved that a spiral would be the path due to a force of $1/r^3$. (Depending on the specific type of spiral the force could be slightly different. For example the spiral $r = k \theta$ is of the form $f(r)\propto \frac{1}{r^3} + \frac{2 k^2}{r^5}$ 2)

Newtonian gravity does have one special characteristic. The orbits are closed. After one orbit the object returns to exactly where it started. That is the path is an ellipse and not a “spirograph” type pattern. This characteristic also occurs for a force of $f(r)\propto r^2$. No other forces have this characteristic, so in a way $1/r^2$ is unique.

Note I was careful to say “Newtonian gravity”. Actual gravity is best described by Einstein’s general relativity. One of the drivers for revising our view of gravity was the fact the the orbit of Mercury was not closed. It’s point of closest approach to the sun shifts around very slightly over time (even after all the effects of the other planets are taken into account). As an approximation this can be modelled as a slightly different central force. The careful measurements of the orbits in the solar system (and increasingly the orbits of binaries and exoplanets) are still scrutinized for signs that our understanding of gravity is correct. (See e.g. http://arxiv.org/pdf/gr-qc/0302048.pdf) Another reason for re-thinking gravity is that a central classical force has a a kind of spooky action at a distance built in. If a mass doubles then that effect is immediately felt everywhere in the universe and that would be hard to reconcile with the idea that nothing moves faster than light. In general relativity the changes in the curvature propagate with finite speed so there is a time delay (although you can play games with the equations in some cases using negative energy density to make changes go at faster than the speed of light 3)

Notes:

1. Hooke was an interesting figure who did a great deal and is probably a bit under-appreciated; see The Forgotten Genius: The Biography Of Robert Hooke 1635-1703
.
2. This comes from a book I used to bridge the gap between my engineering undergrad education and “real” physics. Introduction to Classical Mechanics by Atam P. Arya (first edition, 1990). Example 7.1
3. I first bumped into this in graduate school in a paper:M. Alcubierre, “The Warp Drive: Hyper-fast Travel within General Relativity”, Class. Quantum Grav., 11, pp.L73-L77, 1994. This is still an active area of investigation; see http://arxiv.org/pdf/1202.5708.pdf

## 3 thoughts on “Did Gravity have to be an inverse square law?”

1. AJG

Thanks for the Mondays morning venture into the world of physics. I had to pull a few books off the shelf to see if I could return the favour, but alas, it will require more time than I have to spend at the moment. I wanted to see if I could use some of Moffat’s arguments to influence your observations. A well written and researched post. Thanks
..FG..

1. musgrave.peter@gmail.com Post author

Glad you enjoyed it. I’ve been meaning to catch up on the latest in alternative gravity (Moffat is one of many with alternative theories). The careful measurement of orbits is one way to get some insight into this. I’ve been meaning to read Was Einstein Right? 2nd Edition: Putting General Relativity To The Test which is a more approachable version of Will’s excellent reference on the subject. I did once work my way through that in grad. school.

2. akb48 dvd

Thank you for helping out, good information. “If you would convince a man that he does wrong, do right. Men will believe what they see.” by Henry David Thoreau.