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Epilogue from the Twentieth Century Epilogue from the Twentieth Century 75 that the Earth really goes around the Sun, not the Sun around the Earth, has any objective validity there must be some important physical property, expressible in precise mathematical terms, which emerges in the heliocentric picture but not in a geocentric one. What can this property be? Consider the well-known Newtonian equation mass X acceleration = force. AT THE BEGINNING OF CHAPTER I IT WAS STATED that we can take either the Earth or the Sun, or any other point for that matter, as the centre of the solar system. This is certainly so for the purely kinematical problem of describing the planetary motions. It is also possible to take any point as the centre even in dynamics, although a recognition of this freedom of choice had to await the present century. Scientists of the nineteenth century felt the heliocentric theory to be established when they determined the first stellar parallaxes. The positions of nearby stars were found to undergo annual oscillations which were taken as reflections of the Earth's annual motion around the Sun. But, kinematically speaking, we can always give to the stars epicyclic motions similar to the ones we found in Chapter IV for the planets. Indeed, if we wish to consider the Earth to be at rest, it will be necessary to give an annual epicyclic motion to every object in the distant universe, as well as to the planets of the solar system. We cannot dismiss such a procedure simply on the grounds of inconvenience or absurdity. If our feeling The mass which appears here for a particular body is intended to be always the same, independent of where the body is situated or of how it is moving. Suppose we describe the position of a body as a function of time in some given reference frame, and suppose we know the mass. Then provided we also have explicit knowledge of the force acting on the body, Newton's equation gives us the acceleration. Determining the motion is from there on a mathematical problem onlyin technical terms we have to integrate the above equation. This procedure, which forms the basis of Newtonian mechanics, fails unless we know the force explicitly. In the Newtonian theory of the planetary motions, the theory leading to the basic ellipse from which we worked in Chapter IV, the force is taken to be given by the well-known inverse square law: Two masses m1, m2 , distant r apart, attract each other with a force Nicolaus Copernicus where G is a numerical constant. The force is directed along the line joining the bodies. N ow comes the critical question: In what frame of reference is this law considered to operate? In the solar system we cannot consider the inverse square law to operate both in the situation in which the Sun is taken as the centre and in which the Earth is taken as centre, because Newton's equation would then lead to contradictory results. We should find a planet following a different orbit according to which centre we chose, and a body cannot follow two paths (at any rate not in classical physics). It follows that in order to use the inverse square law in a constructive way we' must make a definite choice of centre. The situation which now turns out is that to obtain results agreeing with observation it is the Sun which must be chosen as centre. If the Earth were chosen instead, some law of force other than the inverse square law would be needed to give motions agreeing with observation. Although this argument was believed in the nineteenth century to be a satisfactory justification of the heliocentric theory, there were causes for disquiet if one looked into it a little more carefully. When we seek to improve the accuracy of calculation by including mutual gravitational interactions between planets we findagain in Qrder to calculate correctly-that the centre of Epilogue from the Twentieth Century 77 the solar system must be placed at an abstract point known as the 'centre of mass' which is displaced quite appreciably from the centre of the Sun. And if we imagine a star to pass moderately close to the solar system, in order to calculate the perturbing effect correctly, again using the inverse square rule, it would be essential to use a 'centre of mass' which included the star. The 'centre' in this case could lie even farther away from the centre of the Sun. It appears then that the 'centre' to be used for any set of bodies depends on the way in which the local system is considered to be isolated from the universe as a whole. If a new body is added from outside to the set, or if a body is taken away, the' centre' changes. A similar circumstance was already present throughout our calculations, when we regarded angles as being measured with respect to a 'fixed direction', it being implied that distant stars had directions that were 'fixed' in this sense. If we make a calculation, using both Newton's equation and the inverse square law, but measuring angles with respect to a direction that rotates with respect to the distant universe, things go very badly wrong. Newton was fully aware that his system of dynamics would only work correctly provided the 'fixed directions' in the theory were chosen in a suitable way. His reference to the well-known rotating bucket experiment was intended to illustrate this point. It is dear therefore that in order to define the appropriate 'centre' of the local system in a useful way, and Nicolaus Copernicus Epilogue from the Twentieth Century in order to define 'fixed directions' relative to which angles are to be measured, we must take account of the relation of the local system to the universe outside. It seems that the local laws of force only take simple forms when the centre is unaccelerated with respect to a frame of reference determined by the universe in the large, and when the fixed directions do not rotate with respect to the distant universe. From this point of view we can compare the heliocentric and geocentric theories of the solar system in an unequivocal way. We ask: centric picture. The relation of the two pictures is reduced to a mere coordinate transformation and it is the main tenet of the Einstein theory that any two ways of looking at the world which are related to each other by a coordinate transformation are entirely equivalent from a physical point of view. Moreover, the method of calculating the effect of gravitation is changed in the Einstein theory to a form which applies equally to all such related ways of expressing a problem. It may still happen that it is easier to work through the details of a particular problem with respect to one coordinate system rather than another, but no special physical merit is to be adduced from such a circumstance. Indeed, from a mathematical point of view, the problem of the planetary motions certainly continues to be easier to grapple with in the heliocentric picture. The simplification of such a picture shows itself in the Einstein theory through boundary conditions which are impressed on the spacetime structure at a large distance from the Sun--which is to say in terms of the control impressed by the universe in the large. So we come back full circle to what was said at the beginning of this book. Today we cannot say that the Copernican theory is 'right' and the Ptolemaic theory 'wrong' in any meaningful physical sense. The two theories, when improved by adding terms involving the square and higher powers of the eccentricities of the planetary orbits, are physically equivalent to one another. What we can say, however, is that we would Is it the Sun that is accelerated with respect to the universe, or is it the Earth? Neglecting small effects, the answer is that the Earth is accelerated, not the Sun. Hence we must use the heliocentric theory if we wish to take advantage of simple rules for the local forces. But this is not to say that we cannot use the geocentric theory if we are willing to. use more complex rules for the forces. The present discussion has been formulated from the standpoint of the Newtonian theory, which is not well suited to problems concerning the universe in the large. We might hope therefore that the Einstein theory, which is well suited to such problems, would throw more light on the matter. But instead of adding further support to the heliocentric picture of the planetary motions the Einstein theory goes in the opposite direction, giving increased respectability to the geo- 79 80 Nicolaus Copernicus hardly have come to recognize that this is so if scientists over four centuries or more had not elected to follow the Copernican point of view. The Ptolemaic system would have proved sterile because it would have been too hard to make progress that way.