Square

A baby sits in the bath and plays with a toy. It lifts the toy and throws it triumphantly out of the bath. It might feel even more triumphant if it knew that in lifting the toy it had defeated the efforts of the earth (which weighs about 6,000,000,000,000,000,000,000 tonnes, or 6 X 10²⁴ kilograms) to pull the toy down. Though gravitation makes it hard for us to fly, to make great buildings and bridges, and to venture into space, it is actually by far the weakest force we know, per kilogram of generating material.

That is why the baby can defeat it with ease. And a strong man can lift weights much heavier than himself. He cannot, however, throw anything fast enough escape from the gravitational field of the earth. For that he would have to generate a speed of around 40,000 km/hour, even ignoring air-resistance.

What forces are available to the baby? To simplify things we will consider the bath water instead. What is in the bath water? Let's take a 10 cm cube, or a kilogram of water. In that kilogram we have about 3.3 X 10²⁶ electrons, about 3.3 X 10²⁶ protons, and about 2.7 X 10²⁶ neutrons.

If we could remove all the protons and neutrons, leaving only the electrons, the electrical repulsion between them would be unimaginably great. No force on earth could contain it. So the forces exerted by the baby are a pale shadow of what is potentially there. These forces are the ones between molecules.

If we consider the cube of water in two halves, each containing half the electrons, we can get a rough idea of the force. The charge on one electron is about 1.6 X 10^-19 C, so the charge on each half of the kilogram is about 1.65 X 10²⁶ X 1.6 X 10^-19 C, which is about 2.6 X 10⁷ C. Now we have to multiply the two charges and divide by the square of the distance, 5 cm, or 0.05 metre, between them. This gives a big number, 2.7 X 10¹⁷. We haven't yet put in the scale factor for the electrical force, of course. The formula we need is -

Force = Q1Q2 / (4 pi e₀d²)

where e₀ is 8.85 X 10-12 C²/Nm²

and the result is 2.4 X 10²⁷ Newtons.

In the water and the baby, the number of negative electrons is precisely equal to the number of positive protons, and so this colossal force never gets a chance to manifest itself. There is a practical case where we can see a measurable result of the electrical repulsion, namely a nuclear reactor. In this device, nuclei of uranium can be made to split into two pieces, both electrically charged. The huge repulsion between the pieces sends them out with high kinetic energy, which heats the fuel. The heat is extracted using a working fluid. So it is electrical energy which is released in a reactor. We can calculate roughly the energy released by a kilogram of uranium.

Let's start with Uranium 235 and let it split into two pieces, Krypton 90 and Barium 142. This doesn't add up, so we will assume that three neutrons escaped. We now need more information. We find that the mass excesses are as follows - U235, 41 MeV, Kr90, -75 MeV, and Ba142, -78 MeV. The change in mass excess is 194 MeV per nucleus. What does this mean? The MeV is a unit of energy. What is mass excess? If you add up the mass of all the protons and neutrons in a nucleus, and subtract it from the mass of the actual nucleus, you get the mass excess. If it is positive, the nucleus weighs more than its constituents: if it is negative, the nucleus weighs less than its constituents. The energy release per kilogram is easy to calculate from these values.

1 MeV is 1.6 X 10^-13 Joule. 1 mole is 0.235 kg of uranium, and it contains 6 X 10²³ nuclei. So the energy per kilogram is 194 X 1.6 X 10^-13X 6 X 10²³ / 0.235 Joules = 8 X 10¹³ Joules = 80 million million Joules. One joule will raise 1 kg about 10 cm against the force of gravity.

So 80 mega-mega joules, or 80 tera joules is a lot of energy.

So Einstein's equation E = mc² says that we should have calibrated our weighing machines in joules.

This seems a little odd, but it happens all the time. Water, H₂O, weighs a little less than the hydrogen and oxygen which make it. But the difference is so small that it will probably never be measured. Energy and mass are different ways of saying the same thing. When we burn hydrogen to make water, energy is given out as heat, leaving less for the water, so it weighs less than the materials we started with.

The forces between the atoms and the molecules in the water and the baby are very small compared with the forces just discussed. It is impossible to understand them using classical physics, which is why an understanding of matter had to wait for the creation of quantum mechanics. All the forces between atoms are the result of interactions between electrons at the edge of atoms.

We have seen that fission of a heavy nucleus unleashes electrical energy. At the other end of the mass spectrum we can calculate the effect of nuclear energy. If we could take four protons from the hydrogen in water, and combine them to form a helium nucleus, we would get a release of energy. The mass excesses are 4 X 7.3 MeV - 2.4 MeV, giving about 27 MeV of energy per helium nucleus. This less than for fission, but helium nuclei have only about 1/60th of the mass of uranium ones, so in fact fusion is very efficient Unfortunately the electrical repulsion that was helpful with fission is now a very great hindrance. To get the particles close enough for the nuclear forces to operate, the particles have to be given a high energy.

The only known way is to achieve a high temperature, perhaps 100 million degrees. This of course leads to immense difficulty in holding the material together. It is normally done with cunningly designed magnetic fields. If people could fuse hydrogen to make helium we would have a very good supply of energy using the baby's bath water.. The problem is that there is little sign of anyone being able to build equipment that will produce more energy than is put in, let alone being able to get the excess energy out in a usable manner.

We can work out the energy per kilogram of hydrogen, in the same way as we did for uranium. It will be (235 / 4) X (27 / 194) times as much, ie about 8 times as much as for uranium, about 640 million million Joules.

If uranium weighs more than its constituents, it must have more energy, and so it must be unstable. And so it is. But that is not the whole story. In order to break up into its constituents, it has to do so by a physically possible process. If there isn't one, it won't happen.

That is why there can be so many chemical elements, and not just the ones with the most stable nuclei. A boulder in the crater of a volcano would have a lot less energy if it were at the foot of the mountain, but it hasn't a way of climbing over the crater wall to get there. This is discussed elsewhere in this web-site (Tunnelling)

So we can see that inside matter there are potentially enormous forces at work. Working inwards, atoms are held together by forces between electrons at the edge of the atoms. This enables them to form molecules, crystals, or amorphous substances. The electrons are held in place by their attraction for the protons in the nucleus. These protons all repel each other as we already saw, so something must hold them together against that repulsion. This something is the nuclear force, or strong interaction, between the various protons and neutrons. It is called strong because it is stronger than other forces.

In the uranium nucleus, all the protons repel each other, because the electrical force has a long range, like gravity. But the nuclear force has a very short range, and only those particles that are close to each other can interact. In a nucleus, each particle probably has no more than about twelve close neighbours, giving about 0.5 X 12 X 235 = 1410 lots of force, whereas there are ninety-two repelling protons, all affecting each other, making roughly 0.5 X 92 X 91 = 4186 lots of force. This calculation is really much too crude.

To understand the nuclear force we need to know what the protons and neutrons are made of. It seems that they are made of quarks, whose size, if any, has yet to be determined. Each particle behaves as if it were made of three quarks. Why can't we take them out and examine them? It turns out that the force behaves in a peculiar manner. The further apart two quarks are, the greater the force between them. It is as if there was an elastic string between them. It therefore requires energy to pull them apart.

The energy builds up so rapidly that before they can be separated very far, there is enough energy to create a new particle. What happens is as if the elastic breaks, and the free ends acquire a quark and an anti-quark respectively. So free quarks never appear.

In collisions at high energy, jets of secondary particles emerge, which are believed to correspond to the original quarks.

How big is the force between quarks? If we imagine that pulling the quarks apart by one proton radius generates enough energy to make the smallest particle, a pion, it might give us a very rough idea. The mass of a pion is about 140 MeV/C², and the distance is about 10^-15 metre. If we write energy = 0.5 X force X distance, we get 140 MeV /10^-15 = force. ! MeV is 1.6 X 10^-13 Joule. So the force is about 22400 Newtons.

10 Newtons is about the weight of a kilogram, so this force is the equivalent of about 2000 kg, or 2 tonnes. This is a very rough calculation, but it does suggest that the force between quarks is very strong. Imagine a bungee or a wire breaking under a force of two tonnes. The sound emitted would be very loud. With a quark pair the energy would be emitted in the form of pions or other particles. However, the energy would me much smaller, because the energy is related to the distance as well as to the force.

It is believed that during the first microsecond of the big bang, the temperature and the density were great enough for a quark-gluon plasma to be formed. In such a state, the quarks and gluons behave as if they had real and separate existence, just as in a normal plasma, ions and electrons are observed, as never in a normal substance. Gluons are the particles presumed to hold quarks together. Like quarks, they have never been observed directly.

Recent high energy collisions of particles at CERN, Meyrin, in Switzerland, give results which appear to be consistent with the formation of a quark-gluon plasma.

The force between the protons and neutrons in a nucleus is rather analogous to the force between atoms. It is not as strong as the quark force.

We have seen that there are three forces at work in matter, the quark force, the electrical force, and the gravitational force, the last being extremely weak compared with the others.

But gravity has one attribute that can compensate for its weakness. It is always attractive. Electrical forces can be attractive or repulsive, and there are both positive and negative particles, apparently in about equal numbers. So they tend to cancel out. But in principle, every bit of matter in the universe attracts every other. If a mass is large enough, gravity can be very powerful indeed.

In fact, gravity can do what we cannot do. It can control a thermonuclear reaction, even at 100 million degrees, by holding a mass like the sun into a glowing ball. If a star has more than a certain mass, then during the time when it starts to run down, the pressure of the gas will no longer be able to resist gravity, and it will shrink into a dwarf state.

A further state of collapse forms a neutron star. Neutrons are normally heavier than protons, and every neutron decays into a proton, an electron, and an anti-neutrino, in a time measured in minutes. After an hour, not many are left. But in a collapsed star, the repulsion of the electrons raises their energy. If it is high enough, remembering that mass is energy, the electron and proton can be effectively than a neutron. What happens then is that the electrons and protons combine to form neutrons and neutrinos. The neutrinos escape, leaving a neutron star. This is amazingly dense. because it is like a gigantic atomic nucleus. In ordinary matter, every atom comprises mainly empty space. In a neutron star there is very little space between the particles.

Yet another state of collapse is possible, producing a black hole. From inside the event horizon of a black hole, nothing can escape. Although a baby can defy gravity on earth, a black hole represents the triumph of gravity on the large scale. But according to Hawking, gravity does not have the last laugh. Radiation can be emitted from the vicinity of a black hole, which in theory can slowly dissipate its mass. (Click here for more on black holes.)

Everyone knows that even light cannot escape from a black hole. How does this work? Before 1900, people believed that Newton's theory of gravitation was a good description. It stated that every mass is surrounded by a field, which generates a force on any other object in it. If the two masses are M₁ and M₂, and the distance is R, the force is given by the formula -

F = G X M_! X M₂ / R² , where G is apparently a universal constant.

Another property of a mass is that it resists acceleration by a force . This property is called inertia. The formula for acceleration is A = F / M.

Now the masses in the two formulas are strictly proportional. It doesn't matter what an object is made of - solid, liquid or gas, metal or insulator, dense or light. Nobody has ever found a variation from the formulas.

When two things appear to be the same, it is natural to wonder if they might actually be the same. One example is the speed of electromagnetic radiation, calculated by James Clerk Maxwell, and found to be close to the measured speed of light. A much later one occurred when apparently two types of meson seemed to have the same properties, except for one which could not be the same for both. The resolution of this destroyed a conservation law. Actually it wasn't a law, because nobody had fully tested it - it was more like folklore. (Click Kaons to find out more.)

In 1900, nobody knew why inertial mass and gravitational mass were apparently the same, but after 1905, when he had produced the special theory of relativity, Einstein began to work on this problem. In special relativity he had resolved the conflict between Maxwell's electromagnetic theory and classical mechanics. By placing all non-accelerating frames of reference on the same footing, he reasoned that all measurements should lead to the same laws of physics. Not the same measured values, but the same formulas relating them. In this theory there is in fact one measurement that does remain the same, namely the speed of light. So it isn't exactly a measurement, more a property of space-time.

The natural thing to do when something has been worked out is to generalise it, and relativity was no exception. Einstein thought about a person in a lift, in the case where the cable breaks and the automatic brakes don't work. He realised that the free-falling person wouldn't feel any effects of gravity. He started to think of a way of finding some equivalence between acceleration and gravity.

This type of thought experiment is very useful, though of course you have to subject any conclusions to experimental test. If you see a theory or a formula, you can ask things like - "What if the is very small, or very big?" - "What if two things are equal, or one is very much bigger than the other?" - etc. Then you can look at the consequences and see whether they make sense.

It is obvious that you can't equate acceleration and gravitation just like that. Gravity points inwards all around the earth, and it's quiet clear that there isn't an inwardly accelerating frame of reference. What Einstein did was to equate things locally, and then integrate the small steps over a large volume. For this he had to find the right mathematical tool, which turned out to be tensor calculus. He then had to apply it in the right way. His brilliant intuition told him what the right type of answer should be, and gave him the motivation to struggle with the mathematics.

The result of the calculations was that space-time is curved in the vicinity of a mass. Presumably Einstein had already decided this, and "merely" had to find the mathematics to make a workable theory.

What this means is that the rules of geometry that work for flat space, such as Pythagoras' theorem, give slightly wrong answers. Amazingly, having made the constancy of the speed of light a basic assumption in special relativity, Einstein now said that light travels more slowly near a mass. This effect is utterly undetectable except near a very large mass, such as a star. In such a region the wave-fronts of light wheel round very slightly, as light nearer moves a little more slowly than light further away. During the eclipse of 1919, measurements were taken that were deemed to confirm the theory.

More recently, photographs appear to have shown lensing of light by stars, producing multiple images of very distant galaxies.

On a curling-rink or a snooker table, the natural path between two points is a straight line. In curved space time we have seen that it isn't. Similarly, the shape and structure of every large object is greatly influenced by gravity. Walls are generally vertical, and floors generally horizontal. With the advent of concrete and steel, the dominance of horizontal and vertical became almost absolute. The influence of movements like De Stijl, including the artist Mondrian, and also of architects such as Le Corbusier and Mies van der Rohe must have contributed to this. So it became a square world, as a result of our living in a region of curved space-time.

In earlier times, the vertical and the horizontal were, of course, dominant, but they were often tempered by the presence of sloping roofs, curved or sloping shapes over windows, doors and porches, helical staircases, and other means of breaking the tyranny of the rectangle. This is not to say that no modern rectangular buildings are elegant or attractive, nor that highly abstract art cannot be beautiful - it a question of balance.

To see what objects can be like in a region which is almost free of gravity, we need to look at the world of the very small, as in insects and other arthropods, or at the aquatic world, as in jelly fish and plankton, or in space, as in satellites. In all the these cases the effects of gravity can be small, allowing great freedom of design and movement. We may envy the birds, as we watch the bee-eater swooping gracefully, the kestrel hovering, the buzzard soaring, or the peregrine stooping.

In fact, of course, their designs are optimised for lightness and aerodynamic efficiency. They have not escaped gravity. It has played a large part in their evolution. The clumsiness of the swift and the wandering albatross on the ground show just how extremely they are adapted for flight, and how badly when their weight is on the ground. At the opposite end of the spectrum from Mondrian - Miro and Kandinsky made many pictures containing forms that seem to have the weightless character of small water-creatures.

The influence of gravity, that is, curved space-time, can be seen in language. We speak of an upright or upstanding character, right-angles, level-headed, on the level, etc. Upper and lower can denote more than physical position. Perhaps Mr Slope in the Barchester novels was so named by Trollope to point to his character. Even a building can be more famous for leaning than if it were straight.

Words or phrases like High Court, high-key, high season, high spirits, high spot, high school, high table, highway, moral high ground, top of the table, top of one's voice, on top of the world, top dog, top-flight, top-level, top-notch, topping, the tops, up-beat, upgrade, uphold, uplift, upswing, and upturn, generally denote positive thoughts or importance, while the corresponding down phrases are generally negative, as in down-beat low-life, and low cunning.

Super- and its derivatives often denote quality or excellence, though "super" in Latin meant upper. Opposite to super- is sub-, used to denote inferiority. Superior and inferior are used for many other ideas besides vertical position.

We have tall story and tall order, and tall used to have the meaning excellent, which has nothing to do with height.

Haughty is derived from the Latin "altus" meaning high. In French there are haute couture, haute cuisine and haute école.

We think of high and low numbers - this is not inevitable - we could have counted "sideways".

Then there are words like overbearing, underdog, and many other over- and under- words. Perhaps it is not being too over-confident to suggest that we should understand the presence of many words in our languages as the result of physical properties of the surroundings.

Important offices in a building are often at or near the top, even though height no longer offers any benefit in terms of defence or ease of command or communcation from a high view-point.

Grave and gravity, weight and weighty, and gravitas, are often applied in a figurative sense also.

What other natural forces have influenced language? We may speak of an electric atmosphere, electric blue, or personal magnetism. Electricity and magnetism, like gravity, are long range forces, and are noticeable by us. The other basic forces, the weak and strong force, have such minute range that any effects we might detect are very indirect. They are unlikely to affect general speech for this reaso.

Only recently, with the exploitation of nuclear energy, have phrases such as "critical mass" and "meltdown" come into common usage as non-technical terms.

The ultimate example of the effect of space-time curvature on light is of course a black hole.