The Most Important Single Work in the Physical Sciences

Last year the world of science celebrated the 300th anniversary of Isaac Newton’s Principia. Trinity College, Cambridge (UK) celebrated the event with a Newton Tercentenary Conference last summer. One result of the conference is the book 300 Years of Gravitation (Cambridge University Press, 1988), edited by Stephen Hawking and Werner Israel. The book contains 16 review papers by leading researchers in cosmology, relativity and particle physics. In his pref- ace to the book, excerpted below, Hawking discusses the principal issues Newton raised in his work and some of the effects the Principia has had on science and society. The Philosophiae Naturalis Principia Mathematica by Isaac Newton, first published in Latin in 1687, is probably the most important single work ever published in the physical sciences. Its significance is equalled in the biological sciences only by The Origin of Species by Charles Darwin. The original impulse which caused Newton to write the Principia was a question from Edmond Halley as to whether the elliptical orbits of the planets could be accounted for on the hypothesis of an inverse square force directed towards the Sun. This was something that Newton had worked out some years earlier but had not published, like most of his work on mathematics and physics. However, Halley’s challenge, and the desire to refute the suggestions of others such as Hooke and Descartes, spurred Newton to try to write a proper account of this result. In order to do so he had first to develop a Theory of Mechanics and the mathematical techniques needed to analyze this theory. The old Aristotelian idea had been that rest was the natural state of a body and that a body moved only if driven by some cause. However, Newton substituted for this his First Law: ‘Every body continues in its state of rest, or of uniform motion in a straight line, unless it is acted upon by a force.’ Newton’s Second Law stated that the acceleration, or rate of change of velocity, was proportional to the force. These laws formalized the Theory of Mechanics that Galileo had implicitly assumed though not explicitly stated. In the case of a force that was constant in magnitude and direction, like the gravitational force near the surface of the Earth, Galileo had shown that they implied that the distance moved by a body initially at rest would be proportional to the square of the time elapsed. However, the question of the motion of the planets was much more difficult because the force on them varied both in magnitude and direction. Newton seems to have solved the problem ‘originally using what we would now call differential calculus, which he had invented along with integral calculus. However, for some reason he did not think that calculus, or ‘the theory of fluxions,’ as he called it, was suitable for publication in the Principia. He therefore developed a new method to show that a body that was attracted towards a fixed point by a force proportional to the inverse square of the disance would obey the laws that Kepler had discovered by observation. Newton’s failure to publish his theory of fluxions led Leibnitz to discover calculus independently and to a bitter dispute about priority. Not only did Newton explain the elliptical orbits of the planets, but he also showed that the same force of gravity was responsible for the orbits of the satellites of Jupiter and Saturn, the orbit of the Moon about the Earth, the tides in the sea and the fall of an apple off a tree. In other words, Gravity is Universal: every body in the universe attracts every other body with a force that is proportional to its mass and inversely proportional to the square of the distance between them. By Newton’s Third Law, which states that action and reaction are equal and opposite, every body is similarly attracted towards every other body. It was natural to suppose that this universality might extend to light which was first shown to travel at a finite velocity by Roemer in 1676. Roemer observed that there were apparent discrepancies between the timing of the orbits of the moons of Jupiter and the predictions of Kepler’s theory. These discrepancies disappeared if he assumed that light travels at a finite velocity and therefore takes a variable amount of time to cross the variable distance between Jupiter and the Earth. If light travels at a finite velocity and if it is affected by gravity in the same way as planets or cannonballs, then one can imagine a situation in which one has a body which is so massive and compact that light cannot escape from it but gets dragged back by gravity. The possible existence of such objects was pointed out by Michell in 1783 and by Laplace in 1799 but not much notice was taken at the time. Indeed, Laplace removed the suggestion from the later editions of his book Exposition du Systême du Monde. However, we now believe that these objects, which we call black holes, are the final stage of evolution of massive stars and that matter falling into them provides the power for compact X-ray objects and the radio and optical emission that we observe from quasars, very compact and luminous sources that we can see even at the other side of the universe. Another consequence of the universality of gravity that was not properly appreciated until much later was that it was inconsistent with the prevailing idea that the universe was infinite in extent and filled with a distribution of stars with a density that was more or less constant in space and time. If the stars were at rest with respect to each other at one time, the gravitational attraction of each star for the others would cause them to start to fall together and the density would go up with time. Alternatively, the stars could be moving away from each other. In that case the density would decrease with time. Gravity would act to slow down the recession of stars from one another and might eventually stop the expansion of the universe and convert it into a contraction. Newton was aware of this problem but he argued that, in an infinite distribution of stars, the force on a star caused by the attraction of other stars to one side of it (say, the right side) would be almost exactly balanced by the force of attraction of the stars to the left of the star. The net force on the star would therefore be small and the system of stars could continue to exist at more or less constant distances from each other. However, there is a flaw in this argument. The force of attraction produced by the stars to the right of the given star would be infinite because, although the force produced by each star to the right gets smaller the further away the star is from the given star, The number of stars at a given distance gets larger the further one is from the given star. Similarly, the force of attraction produced by the stars to the left would be infinite. When one subtracts infinity from infinity, it is well known that one can get any answer one wants. We now know that the universal attractive nature of gravity is inconsistent with a static infinite universe: the universe has to be either expanding or contracting. This was not realized, however, until in the 1920s observations of distant galaxies revealed that the universe is expanding. It was a great missed opportunity for theoretical physics: Newton could have predicted the expansion of the universe. In the Prinicpia Newton introduced the concepts of Absolute Time and Absolute Space. By Absolute Time he meant that there was a quantity called time which would be measured by all properly constructed clocks, regardless of their motion. This was an idea that nobody questioned until the advent of the Theory of Relativity at the beginning of this century. Much more controversial was Newton’s advocacy of Absolute Space. Newton argued that in the case of circular motion there was a preferred state of rest in which there were no centrifugal forces. He therefore claimed that by analogy there ought to be a preferred state of Absolute Rest with respect to linear motion, though he admitted that it would be difficult to determine this state of rest by observations. He suggested, as a hypothesis, that this state of rest coincided with the center of gravity of the solar system. In fact Newton’s Laws of Motion are the same in all frames of reference moving uniformly in straight lines though they are not the same in rotating frames. Thus the laws do not determine a state of rest and so do not support the concept of Absolute Space, though they are not inconsistent with it. They do, however, require the concept of Absolute Time. In other words, in Newton’s theory one can determine whether two events at different points in space occur at the same time but one cannot tell whether events at different times occur at the same point of space: that depends on the frame of reference and would be different in two frames that were moving relative to each other. The laws presented in the Principia remained the accepted theories of mechanics and gravity for more than two hundred years. Even today they are the basis of nearly all practical calculations. It is only in very extreme situations that one has to take into account the modifications introduced by the Special and General Theories of Relativity formulated by Albert Einstein in 1905 and 1915 respectively. In the Theory of Relativity one has to abandon the concept of Absolute Time as well as that of Absolute Space. The time measured by a clock now depends on its velocity and one can no longer determine whether events at different points of space occur at the same time: again this depends on the motion of the frame of reference. The philosophical implications of this change from Absolute Space and Time to Relative Space and Time have been profound but have often not been properly appreciated. Lenin wrote a pamphlet attacking Relativity because he thought it was a threat to the absolute system of Hegel and Marx. It is only in the last 30 years that the study of relativity has become respectable in the Soviet Union. Other people in the West react in a similar way. Einstein was accused of undermining moral standards by suggesting that everything was relative. However, this attack and that of Lenin were based on a misunderstanding of the Theory of Relativity: it is a mathematical model of space and time. It makes no statement about how human affairs should be conducted or organized. Einstein is the only figure in the physical sciences with a stature that can be compared with Newton. Both Newton and Einstein put forward a theory of mechanics and a theory of gravity but Einstein was able to base General Relativity on the mathematical theory of curved spaces that had been constructed by Reimann while Newton had to develop his own mathematical machinery. It is therefore appropriate to acclaim Newton as the greatest figure in mathematical physics and the Principia as his greatest achievement. Hawking is Lucasian Professor of Mathematics at the University of Cambridge CB3 9EW, UK.

The Most Important Single Work in the Physical Sciences

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