Forces of Nature. Andrew Cohen

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СКАЧАТЬ in Newton’s equation (here).

      In more sophisticated language, we can say that Newton’s law of gravitation possesses a rotational symmetry. By that, we mean that it gives the same results for the gravitational force between any two objects regardless of their orientation. This is an example of what physicists and mathematicians mean when they speak of the symmetries of an equation or law of Nature, and it means that the calculation for the maximum height of a mountain at any place on the Earth’s surface must give the same answer irrespective of the position of the mountain because Newton’s law of gravitation is symmetric under rotations. The symmetry of the law of gravity is reflected in the symmetry of the objects it forms. Gravity smooths mountains democratically, symmetrically, with the result that lumps of matter with a gravitational pull strong enough to overcome the rigidity of the substance out of which they are made end up being spherical. This is the reason why the Earth is spherically symmetric.

      There is a deep idea lurking here that lies at the heart of modern theoretical physics. Thinking of things in terms of symmetry is extremely powerful, and perhaps fundamental. Consider the possibility that the laws of Nature possess certain symmetries, which are the fundamental properties of the Universe. This would be reflected in the physical objects they create. For example, imagine a Universe in which only laws of Nature that are symmetric under rotations through 90 degrees are allowed. In such a Universe, objects that remain the same under rotations through 90 degrees are created; cubes exist but spheres are forbidden. This isn’t quite as crazy as it sounds. As far as we can tell, our Universe does possess a set of extremely restrictive symmetries, and the subatomic particles that exist and the forces that act between them are determined by these underlying symmetries.⁶ In fact, all of the laws of Nature we regard as fundamental today can be understood by thinking in this way. There is certainly a strong case to be made that Nature’s symmetries can be regarded as truly fundamental. The Nobel Prize-winning physicist Steven Weinberg wrote, ‘I would like to suggest something here that I am not really certain about but which is at least a possibility: that specifying the symmetry group of Nature may be all we need to say about the physical world, beyond the principles of quantum mechanics.’ Nobel laureate Philip Anderson wrote, ‘It is only slightly overstating the case to say that physics is the study of symmetry.’ Nobel laureate David Gross wrote, ‘Indeed, it is hard to imagine that much progress could have been made in deducing the laws of Nature without the existence of certain symmetries … Today we realise that symmetry principles are even more powerful – they dictate the form of the laws of Nature.’ The complexity we perceive when casually glancing at the Universe masks the underlying symmetries, and it is one of the goals of modern theoretical physics to strip away the complexity and reveal the underlying simplicity and symmetry of the laws of Nature.

      Returning to the task in hand, this reasoning leads to a prediction about the size and shape of planets and moons that can be checked: they should be spherical if they are large enough, and therefore massive enough, for their gravitational pull to overcome the structural strength of the rock out of which they are made. The strength of rock is ultimately related to the strength of the force of Nature that holds the constituents of rock together – molecules of silicon dioxide, iron and so on. This is the electromagnetic force; what other force could it be? There are only four forces, and the two nuclear forces are confined within the atomic nuclei themselves. Big things like planets are shaped by the interplay between gravity, trying to squash them into spheres; and electromagnetism, trying to resist the squashing. We can perform a calculation to estimate the minimum size that a lump of matter must be to form into a near-spherical shape by equating the weight of a mass of rock near its surface to the structural strength of the rocks below.⁷ Our answer is approximately 600 kilometres.

      We can check this by direct observation of the Solar System. Phobos fits with our prediction; with a mean radius of just over 11 kilometres and a mass of only 10¹⁶ kilograms, the gravitational force at its surface is far too weak to overcome the rigidity of rock and act to flatten the surface and sculpt Phobos into a sphere. At around 550 kilometres across, the asteroid Pallas is the largest known non-spherical object. Saturn’s moon, Mimas, with a radius of just under 200 kilometres, is the smallest known body in the Solar System that is spherical. It is made mostly of ice, which is much easier to deform than rock – this is why it is so small and still round. Our estimate is certainly in the right ballpark.

      As an important aside, ‘back-of-the-envelope’ estimates such as these are very important in physics; they tell us that we are on the right track, without overcomplicating things unnecessarily. We could have refined our calculation by taking into account the different compositions of different objects, and by computing the gravitational pull at different depths more carefully. We could even have tried to use General Relativity instead of Newton’s laws, but we wouldn’t have learnt a lot by doing so. Learning what to ignore and what to include is part and parcel of becoming a professional scientist – one might call it physical intuition. There is no precise size above which a body will be spherical; the limit depends on the object’s composition; a mixture of rock and ice is easier to deform than solid rock. As a general rule, any icy moon over 400 kilometres in diameter will be a sphere. Objects made of rock need to be larger, because the gravitational force needed to deform rock is greater. If a rocky moon has an internal heat source, perhaps as a result of the presence of large amounts of radioactive material in its core or tidal heating, the body is easier to deform and may be spherical at a smaller size than would be expected for a less active object. The Solar System is full of examples of this interplay of rigidity and gravity in action, but, very roughly, we have deduced that anything with a radius in excess of a few hundred kilometres must be spherical because the gravitational forces will overcome the strength of the rock.

      When gravity wins, the shape of the objects it creates reflects the underlying symmetry of the physical law, and this is why large single objects such as planets and stars are always spherical.

      At larger scales, however, things change. Our nearest galactic neighbour, Andromeda, contains around 400 billion stars, formed and bound together by gravity. It is disc-shaped, not spherical. The Solar System itself is also a disc and not a sphere. Why?

      Why are there discs as well as spheres in the Universe?

      We argued above that planets and large moons are spheres because, if the gravitational forces are large enough to overcome the electromagnetic forces that keep matter rigid, the underlying symmetry of the gravitational force is made manifest in the objects it creates. Because there is no special direction in Newton’s Law of Gravitation, there will be no special direction in the objects that it creates. This is not entirely true, however, even for planets, because they spin.

      Our planet turns on its axis once every 24 hours. The spin axis marks out a special direction, which means that all points on the Earth are not the same. Someone standing on Earth’s Equator is rotating at a speed of 1670 km/hour, whilst someone in Minnesota is rotating at a speed of 1180 km/hour. The spherical symmetry is broken – the two points are different. As we’ll see in Chapter Two, this difference leads to observable effects such as the rotation of storm systems and the deflection of artillery shells in flight. It also leads to a very slight flattening of the Earth – the equatorial circumference is 40,075 kilometres and the polar circumference is 40,008 kilometres. The Earth is not spherical, but an oblate spheroid, because of its spin. If it were spinning faster, the Earth would be more oblate. When our Solar System formed, the spin – or more correctly angular momentum – was ‘exported’ outwards from the newly forming Sun, primarily through collisions and magnetic interactions in the protoplanetary disc, resulting in the system becoming flattened into a disc.

      The transfer of spin outwards from the centre also results in the flattening of some galaxies; for example, Andromeda. Globular star clusters, such as the spectacular Messier 80, remained spherical because they were too diffuse for angular momentum to be transferred outwards. The shapes of objects that are bound together СКАЧАТЬ