Physicists have been studying Newtonian mechanics ever since the great scientist published his book Principia in 1687. But some people still struggle with the book’s content. In certain circumstances the laws of motion, as laid down by Newton, appear to contradict our everyday experience—and some people continue to favour their everyday experience over the findings of science. (This possibly explains why, even today, there are those who believe the Earth is flat: it looks flat so it must be flat.)

A famous example of a basic misunderstanding of Newton’s laws of motion occurred in January 1920, when a

New York Times editorial ridiculed the rocket pioneer Robert H. Goddard for daring to suggest a rocket might one day reach the Moon. The editorial stated:

That Professor Goddard, with his ‘chair’ in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react—to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools.

The

New York Times eventually published a good-natured, if belated, retraction. In July 1969, the day after the Apollo 11 rocket was on its way to the Moon, the editors wrote:

Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error.

In “A tale of negative gravity” Frank Stockton, who was writing in 1884 , also gets Newtonian mechanics wrong—but he has more excuse than those nameless editors of the New York Times . It turns out negative gravity is a rather subtle concept.

When explaining how to overcome the force of gravity Stockton’s inventor first presents some handwaving nonsense involving centrifugal force. Immediately afterwards, however, he defines what he means by negative gravity:

Positive gravity attracts everything toward the centre of the earth. Negative gravity, therefore, would be that power which repels everything from the centre of the earth, just as the negative pole of a magnet repels the needle, while the positive pole attracts it.

This is the key notion: common sense suggests negative gravity would manifest itself as a repulsion, with two masses repelling each other just as two similar magnetic poles repel each other. If we could control the repulsion then we could float—as did the couple in Stockton’s tale and the people illustrated in an 1878

Punch cartoon wearing antigravity underwear; see Fig.

9.1 . (Perhaps Stockton got his idea for this story from reading

Punch ?) But it’s not that simple.

Fig. 9.1 This cartoon, one of several linked cartoons appearing in the 9 December 1878 issue of Punch, was drawn to illustrate an irreverent prediction of the inventions Edison might unveil in the forthcoming year (Credit: Public domain)

Newton declared that any two masses experience a mutual gravitational force, the strength of which is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. No variables other than mass and distance are involved. It’s of no importance what the masses consist of, what colour they are, whether they are soft or hard—the gravitational force between two masses depends solely on the quantity of mass and the distance between the masses.

So imagine two masses, floating freely somewhere out in empty space. There will be an attractive gravitational force between them. If the distance between the masses increases by a factor of 2 then the force between them decreases by a factor of 2 × 2 = 4. If the distance between the masses increases by a factor of 3 then the force between them decreases by a factor of 3 × 3 = 9. And so on. Note, however, that the force is always one of attraction . The force depends upon the inverse square of the spatial separation between masses, so changing the distance between the masses changes the magnitude of the force but not its direction. (If the distance between the masses was made imaginary then the gravitational force between them would indeed change sign. But imaginary distance makes no physical sense. One can mutter the words, but the concept is meaningless.) So how can we change the direction of the force, and produce a repulsion rather than an attraction?

Well, a negative number multiplied by a positive number produces a negative number. So if one of the masses is negative and one is positive then the product of the masses will be negative. The force will be one of repulsion rather than attraction. And a repulsive force is what Stockton imagined when he described negative gravity.

Let’s ignore, for the moment, the question of whether it’s possible for a “negative mass” to exist and consider instead the question: would a negative mass behave in the way envisaged by Stockton in his story?

Suppose the inventor in Stockton’s story was holding a mass of −1 kg. Then the magnitude of the gravitational force acting on the mass due to Earth’s mass would be exactly the same as occurs on a mass of +1 kg; the direction of the force, however, would point away from the centre of the Earth. Repulsion rather than attraction. But how would that repulsion manifest itself? Well, Newton argued that a force acting on a mass causes the mass to accelerate. Earth’s attractive gravitational force acting on a positive mass causes the mass to accelerate towards the centre of the Earth. In other words, the mass falls. A repulsive force acting on a negative mass causes … the mass to accelerate towards the centre of the Earth. In other words, the mass falls! The situation Stockton describes involves two changes of sign: the direction of the gravitational force flips, but so does the response of the mass to an applied force. The effects cancel!

The effects of negative mass are far from intuitive.

Imagine you are the inventor in Stockton’s story, and you have a kilogram of this negative mass material on the end of a string. You pull

down on the string and feel a tug

upwards , just as if you were holding a helium balloon. But if the string breaks the negative mass falls

down , in exactly the same way as a positive mass falls. Or imagine a +1 kg mass and a −1 kg mass floating freely out in space. The negative mass experiences a negative gravitational force, but because of the way it responds to a force it accelerates towards the positive mass; the positive mass experiences a negative gravitational force and therefore accelerates

away from the negative mass. Starting from rest the two masses self-accelerate along the line joining them, with the negative mass chasing the positive mass (see Fig.

9.2 ). This isn’t what Stockton envisaged when he wrote about negative gravity, but it’s what Newton’s equations tell us will happen—if such a thing as negative mass exists.

Fig. 9.2 A sphere made of negative-mass material will repel a positive-mass spaceship, which will attract the negative-mass sphere … which means the two objects accelerate away together (Credit: Author’s own work)

Let’s return to the question: is it possible to have a negative mass? Or is this two-word phrase as meaningless as “imaginary distance”?

Well, there’s absolutely no evidence that any material in the universe possesses negative mass. Unlike the case of electric charge, which can be positive or negative, mass seems to come in only one variety: positive. That’s why the forces of electromagnetism and gravity have such very different impacts on the universe. Electromagnetism is overwhelmingly more powerful than gravity, but when large amounts of matter are involved electric charges tend to cancel each other out. Gravity is a puny force but, because mass can pile upon mass without cancelling, the effects build up—with the result that, on the largest scales, gravity is the force that shapes the universe. If there were large amounts of negative matter in the universe then we’d soon know about it.

Are there any other hopes for antigravity? “What about antimatter?”, I’m sure some people will say. “Doesn’t antimatter possess negative mass?” Well, no, it doesn’t. Compared to a normal particle, an antimatter particle does indeed have various charges reversed. The electron, for example, has electric charge −1 while its antimatter counterpart, the positron, has electric charge +1. However, although the electron and positron possess opposite electric charges, they possess the same mass: experiments show that they fall in Earth’s gravity at the same rate. It’s also worth mentioning that astronomers can point to evidence that matter and antimatter react in the same way to gravitational fields. SN1987A, a supernova in the Large Magellanic Cloud, created countless neutrinos and antineutrinos. These particles travelled 160,000 years to reach Earth, and the gravitational field of the Galaxy bent them from a straight-line path. This gravitationally-induced bending added five months to the travel time—and yet neutrinos and antineutrinos arrived at the same time. From this we can conclude that those cosmic neutrinos and antineutrinos “fell” in those gravitational wells at the same rate, to a precision of about one part in a million. Antimatter doesn’t give us a route to antigravity.

What about relativity? We know Newton’s theory of gravity has been superseded by Einstein’s theory of general relativity. Does relativity permit antigravity?

In most cases Einstein’s theory reduces to Newton’s theory and so the Newtonian arguments against negative mass apply to relativity too. If it were possible to collect large amounts of negative mass then Einstein’s theory opens up the possibility of exotic entities such as wormholes and warp drives. But the possibility seems remote.

The conclusion is almost inevitable: we are no more likely to achieve antigravity than we are to achieve faster then light travel. The universe just isn’t set up in a way that would permit it.

The story in Chapter 10 examines a phenomenon that seems even more bizarre than antigravity—but that might, after a fashion, be possible.