Download Epilogue from the Twentieth Century

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-
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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.