Introduction

The determination of physical constants and the definition of the units with which they are measured is a specialised and, to many, hidden branch of science.

A quantity with dimensions is one whose value must be expressed relative to one or more standard units. In the spirit of the rest of the book, this section is based around the International System of units (SI). This system uses seven base units (the number is somewhat arbitrary), such as the kilogram and the second, and defines their magnitudes in terms of physical laws or, in the case of the kilogram, an object called the “international prototype of the kilogram” kept in Paris. For convenience there are also a number of derived standards, such as the volt, which are defined as set combinations of the basic seven. Most of the physical observables we regard as being in some sense fundamental, such as the charge on an electron, are now known to a relative standard uncertainty, ur, of less than 10–7. The least well determined is the Newtonian constant of gravitation, presently standing at a rather lamentable ur of 1.5 – 10–3, and the best is the Rydberg constant (ur = 7.6 – 10–12). The dimensionless electron g-factor, representing twice the magnetic moment of an electron measured in Bohr magnetons, is now known to a relative uncertainty of only 4.1 – 10–12.

No matter which base units are used, physical quantities are expressed as the product of a numerical value and a unit. These two components have more-or-less equal standing and can be manipulated by following the usual rules of algebra.

Introduction Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the motion they produce), and statics (the way forces combine to produce equilibrium). We will take the phrase dynamics and mechanics to encompass all the above, although it clearly does not!

To some extent this is because the equations governing the motion of matter include some of our oldest insights into the physical world and are consequentially steeped in tradition. One of the more delightful, or for some annoying, facets of this is the occasional use of arcane vocabulary in the description of motion. The epitome must be what Goldstein calls “the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode lying in the invariable plane, describing “Poinsot's construction” – a method of visualising the free motion of a spinning rigid body. Despite this, dynamics and mechanics, including fluid mechanics, is arguably the most practically applicable of all the branches of physics.

Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical apparatus that has found uses in other fields. Most notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind much of quantum mechanics.

ORIGIN OF LIFE ON EARTH

How did life originate on Earth? There are various theories, most of which fall into the four classes: special creation theories, spontaneous creation theories, panspermia theories, and biochemical theories.

Special creation theories

The belief that life originated as a supernatural event is the metaphysical theory of special creation. It has numerous mythic variations. Most recorded myths distinguish between nonliving and living things. Often the nonliving world comes first, the living world follows, and creation is thus a twofold act. Catastrophe theories of the eighteenth and nineteenth centuries elaborated on such beliefs and proposed that many acts of creation had occurred in the past. After a catastrophe had destroyed the terrestrial environment, newly created life arose in more evolved forms, and evolution occurred supernaturally, not naturally. Organic life even in its rudest forms was thought to be composed of substances fundamentally different from those of nonliving things. To this day we speak of organic and inorganic chemistry, although this distinction is now a matter of convenience only, and organic chemistry deals mostly with the numerous compounds containing carbon atoms. It came as a shock when the chemist Friedrich Wöhler in 1828 first made urea (a simple organic substance) from inorganic chemicals. Subsequent developments showed that chemicals are interchangeable between inorganic and organic things, thereby unifying the living and nonliving worlds at the atomic and molecular levels.

OUR GALAXY

Milky Way

Our Galaxy, an enormous system of clouds of glowing gas and 100 billion stars, is also known as the Milky Way. Light takes 100 000 years to cross the Galaxy from side to side, and the center of the Galaxy lies in the constellation of Sagittarius, obscured from view by clouds of dusty gas that drift among the stars. Far from the center of the Galaxy is our own star the Sun.

The disk and halo

The Galaxy consists of two basic components: disk and halo (see Figures 6.1 and 6.2). The Milky Way is actually our panoramic view of the disk that has a diameter of about 100 000 light years and a thickness of about one-twentieth, or less, of the diameter. The disk is composed of stars and interstellar gas, and contains over half the visible mass of the Galaxy. The gas amounts to one-tenth of the matter in the disk, and the dust amounts to about 1 percent or more of the mass of the gas. The disk of stars, gas, and dust rotates about the center, or nucleus, of the Galaxy like a giant carousel.

STATIC NEWTONIAN UNIVERSE

Until the 20th century everybody believed that the universe is naturally static: not expanding and not contracting. Even Albert Einstein, after the discovery of general relativity, continued to hold this belief for several years.

In the late 17th century, belief in a static order remained unshaken when Newton advanced the theory of universal gravity. In response to a question in a letter from the young clergyman Richard Bentley (Chapter 3), Newton wrote in reply that in an infinite universe it would be impossible for all matter to fall together and form a single large mass, but “some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one to another throughout all that infinite space.”

The Newtonian theory of universal gravity, in which all bodies attract one another, reinforced the growing belief that the universe must be edgeless and therefore infinite. For if the universe were finite and bounded by a cosmic edge, it would have a center of gravity, and the attraction between its parts would cause it, said Newton, to “fall down into the middle of the whole space, and there compose one great spherical mass.” This argument led him finally to abandon the finite Stoic cosmos in favor of the infinite Atomist universe.

STATIC UNIVERSES

Before the twentieth century most Europeans and people of European descent believed the universe was created only a few thousand years ago, or at most a few hundred thousand years. Some people in the 18th and 19th centuries, more radical in outlook, thought the static Newtonian universe was in a steady state – everything remaining eternally unchanged – and the stars would shine endlessly. The realization in the late 19th century that stars have finite energy resources brought to an end the idea of a perpetually unchanging cosmos.

Static Einstein universe

In 1917, Einstein contrived an ingenious static universe using his recently developed theory of general relativity. In this universe, as in all universes we discuss, all places are alike and matter is distributed with uniform density.

Space and time in the new theory of general relativity had at last been awakened from the dead and become active participants in the world at large. Einstein, believing the universe to be static, tranquilized spacetime with a counteracting agent. In his 1917 paper, “Cosmological considerations on the general theory of relativity,” he wrote, “I shall conduct the reader over the road I have myself traveled, rather a rough and winding road, because otherwise I cannot hope that he will take much interest in the result at the end of the journey. The conclusion that I shall arrive at is that the field equations of gravitation that I have championed hitherto still need a slight modification.”

After Newton, astronomical advances in observation and theory were at first slow. Better telescopes had yet to be developed, photography and spectroscopy introduced into astronomy, and the chemical compositions and radial velocities of stars and nebulae determined. The puzzling nature of the nebulae had yet to be resolved, nebulae in the Galaxy to be distinguished from extragalactic nebulae, distance indicators to be found and calibrated, globular clusters to be identified as systems of stars lying in and on the outskirts of the Galaxy, and the confusing obscuration of starlight caused by interstellar absorption to be recognized. All this would be accomplished and accompanied by continual debate over controversial issues from the time of Newton to the time of Einstein during the eighteenth, nineteenth, and early twentieth centuries.

THE DISTANT STARS

Light travel time

We look out from Earth and see the Sun, planets, and stars at great distances (see Figure 5.1). The Sun, our nearest star, is at distance 150 million kilometers or 93 million miles. Kilometers and miles, suitable units for measuring distances on the Earth's surface, are much too small for the measurement of astronomical distances (see Table 5.1).

Almost all information from outer space comes to us in the form of light and other kinds of radiation that travel at the speed 300 000 kilometers per second (see Table 5.2). Light from the Sun takes 500 seconds to reach the Earth, and we see the Sun as it was 500 seconds ago. We say the Sun is at distance 500 light seconds. The time taken by light to travel from a distant body is called the light travel time. Light travel time is an attractive way of measuring large distances and has the advantage that we know immediately how far we look back into the past when referring to a distant body. A star 10 light years away (almost 100 trillion kilometers) is seen now as it was 10 years ago. Always, when looking out in space, we look back in time.

THE LOCATION PRINCIPLE

The Greeks developed the “two-sphere” universe that endured for 2000 years and consisted of a spherical Earth surrounded by a distant spherical surface (the sphere of stars) studded with celestial points of light. This geocentric picture was finally overthrown by the Copernican revolution in the sixteenth century and replaced by the heliocentric picture with the Sun at the center of the cosmos. The sphere of stars remained intact. But revolutions, once begun, do not readily stop, and by the seventeenth century the heliocentric picture had also been overthrown. Out of the turmoil of the revolution emerged an infinite and centerless universe that ever since has had a checkered history. In the eighteenth century the idea arose of a hierarchical universe of many centers, and in the nineteenth came the idea of a one-island universe – the Galaxy – in which the Sun had central location. Once again, in the twentieth century, we have the centerless universe.

EUCLIDEAN GEOMETRY

In the ancient Delta civilizations, geometry was the art of land measurement, and indispensable in the construction of such mammoth works as the Great Pyramid of Giza and Stonehenge. Geometry at first consisted of trial-by-error and rule-of-thumb methods. According to the sacred Rhind Papyrus, the Egyptians of 1800 bc used for π, the ratio of the circumference and the diameter of a circle, the value (16/9)2 =3.1605, as compared with its more exact value 3.1416. The Babylonians of 2000 bc and the Chinese of 300 bc used the rule that the circumference of a circle is three times its diameter, and this value for π is found in Hebraic scripture. The Greeks, in their thorough fashion, developed geometry into a science that climaxed in the axiomatic and definitive treatment presented by Euclid at the Museum in Alexandria in the third century bc.

Axioms

The axiomatic method starts with a set of self-consistent propositions (called postulates or axioms), which are often the simplest and most obvious truths, and examines their logical consequences. Suppose that we wish to persuade someone that statement S is true. We might try to show that this statement follows logically from another statement R that the person already accepts. But if the person is unconvinced of the truth of R, then we must try to show that R follows logically from yet another statement Q.

PRINCIPLE OF EQUIVALENCE

Gravitational and inertial forces produce effects that are indistinguishable – this is the principle of equivalence. It serves as an essential stepping-stone to the theory of general relativity, and makes a basic connection between motion and gravity. It leads to a second stepping-stone: the realization that geometry and gravity have much in common. Then, in an inspired leap across the gulf of non-Euclidean geometry, we enter a country into which comparatively few explorers have ventured. No person entering the third millenium may claim to have a liberal education who has not glimpsed, however briefly, the universe of general relativity.

An inertial force, such as centrifugal force, exists when a body is accelerated. We recall from Newtonian theory that when a body is in free fall, and hence moves freely in space under the influence of gravity, it follows a path of such a kind that the sum of the inertial and gravitational forces is zero. With items of knowledge such as these, sufficient to land men on the Moon, we have made our first step toward the theory of general relativity.