Space-time

An Introduction to Einstein's Theory of Gravity

Jonathan Allday

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9781351676229
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

This book, suitable for interested post-16 school pupils or undergraduates looking for a supplement to their course text, develops our modern view of space-time and its implications in the theories of gravity and cosmology. While aspects of this topic are inevitably abstract, the book seeks to ground thinking in observational and experimental evidence where possible. In addition, some of Einstein's philosophical thoughts are explored and contrasted with our modern views.

Written in an accessible yet rigorous style, Jonathan Allday, a highly accomplished writer, brings his trademark clarity and engagement to these fascinating subjects, which underpin so much of modern physics.

Features:

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• Restricted use of advanced mathematics, making the book suitable for post-16 students and undergraduates

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• Contains discussions of key modern developments in quantum gravity, and the latest developments in the field, including results from the Laser Interferometer Gravitational-Wave Observatory (LIGO)

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• Accompanied by appendices on the CRC Press website featuring detailed mathematical arguments for key derivations

1

Four Keystones

1.1 Starting Points

In developing our understanding of gravitation, we are going to be faced with some tricky concepts and fiddly mathematical ideas. Consequently, it is important to ground ourselves as and where we can in experiment and observation; otherwise, the subject becomes somewhat abstract.

There is little point in dwelling on ideas that have been rejected or replaced, but a lot of what we learn about gravity in school, or at college, needs a spring clean and setting afresh in a context that allows us to move forward. For this reason, we are going to start by focussing in on four keystones that have helped support conceptual development in the area of gravitation. I am not saying that these are the most important keystones, still less that they are the only ones. I selected them partly out of personal interest, but also as I believe there is an interlinking thread through them that illustrates our advancing experimental sophistication alongside a crucial conceptual development. The selection is as follows:

• Galileo’s supposed observation, involving the leaning tower of Pisa, that all objects fall with the same acceleration in a gravitational field, provided that air resistance can be ignored.
• Newton’s fabled observation of an apple falling that led to his theory of universal gravitation, from which he could prove that Galileo’s assertion for falling objects was correct.
• Eddington’s careful measurements during an eclipse, which helped establish Einstein’s general theory of relativity and in the process transform the conceptual foundations for our understanding of gravity. One crucial step in the development of this theory took place when Einstein was meditating on the relevance of Galileo’s observation.
• The results of the LIGO experiment, which for the first time directly detected gravitational waves, a prediction of the general theory with no Newtonian counterpart, hence showing that the general theory’s conceptual core is better suited to the description of nature.

1.2 Galileo and the Tower of Pisa

Philosophy is written in this grand book—I mean the Universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures….

G Galilei¹

As one of the key figures in the history of science, Galileo (1564–1642) is justly admired, not only for his discoveries and insights but also for his communication skills and his insistence on the value of experiment and observation in discovering natural law. He was also one of the earliest people to declare that the laws of nature are inherently mathematical, something that we take for granted now, despite the deep puzzles that it suggests. Fascinatingly, as the quote above illustrates, Galileo was alluding to the role played by geometry in the laws of nature way back in the 1600s. Given that the general theory of relativity places geometry front and centre as one of the fundamental aspects of our universe, his views turned out to be somewhat prophetic.

Galileo’s research ranged widely over topics in astronomy (Sunspots, the moons of Jupiter, phases of Venus and more), engineering (compasses, thermometers, telescopes and microscopes) and physics (the movement of objects, pendulum swings, the frequency of sound waves and of course, gravity). Most significant for us, however, are two of his key ideas:

• that the laws of physics for any observer moving at a constant speed in a straight line are the same as for any other stationary observer²;
• that objects dropped from the same height will hit the ground after the same time of flight, irrespective of their masses.

The first of these thoughts (known as Galileo’s principle of relativity) connects with one of Einstein’s central aims: to show how the laws of physics could take the same mathematical form, irrespective of the choice of co-ordinate system selected for their expression. Galileo, on the other hand, was concerned to dissolve any objections to the Copernican view that the Earth orbited the Sun. His relativity principle ensured that the Earth’s orbital motion (and its rotation for that matter) did not directly impact on people’s daily lives. In one of his writings, Galileo discusses the fate of a collection of butterflies trapped below deck on a ship in gentle (unaccelerated) motion “nor will it ever happen that they [the butterflies] are concentrated towards the stern, as if tired out from keeping up with the course of the ship”.³

The second thought nudged Einstein towards the radical conceptual jump that replaced the notion of gravitational force with that of curved space-time.

In Galileo’s day, people generally accepted that our world is formed from four elements: earth, air, fire and water – a philosophical view that dated back to the ancient Greeks. In a perfect world, these elements would exist in pure form as layers, like an onion, from the earth element at the centre through water, air and fire at the outside (Figure 1.1).

FIGURE 1.1 A 1524 representation of the universe, according to Aristotle’s cosmology. The water and earth elements (drawn as continents and oceans) are at the centre, immediately surrounded by regions of air and fire. Then, in their respective spheres come the Moon, Mercury, Venus, The Sun, Mars, Jupiter and Saturn. Finally, we have the firmament with its stars and astrological groupings. (Image credit: Public Domain, and Edward Grant, ‘Celestial Orbs in the Latin Middle Ages’, Isis, Vol. 78, No. 2. [June 1987], pp. 152–173 [PD-US].)

However, due to some event (which Christian theology later associated with the Fall), the elements had become mixed up so that none of the objects we actually come across exist in pure elemental form. However, manipulating things, for example, dropping them, can show their innate nature. Equally, setting fire to something releases the fire element to rise to its outer layer. The impermanence and decay that we see in the world, according to this philosophy, also arises due to the mixing of the elements.

Working in this context, Aristotle proposed that objects fall at a speed related to their mass, so that heavier objects, imbued with more of the earth element, fell faster due to their greater impulsion to return to their natural position in the hierarchy of elements.

Galileo is supposed to have refuted this by dropping different masses from the top of the leaning tower at Pisa (Figure 1.2): a story that seems to have grown from a biography by one of Galileo’s pupils, Vincenzo Viviani.⁴

As Galileo himself never wrote about this demonstration, most historians doubt th...

Table of contents

• Cover
• Half Title
• Dedication
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Thanks
• Author
• Introduction
• 1 Four Keystones
• 2 The Road to Relativity
• 3 The Theory of Special Relativity
• 4 Space-time
• 5 Mass, Energy and Dust
• 6 Generalising Relativity
• 7 The Theory of General Relativity
• 8 Weak Field Gravitation
• 9 Space-time in the General Theory
• 10 Black Holes
• 11 Gravitational Waves
• 12 Cosmology
• 13 Quantum Considerations
• Bibliography
• Index