**Lecture 9: Symmetries – On physical and aesthetic argument in the development of relativity**

Dr Richard Staley

https://www.cai.cam.ac.uk/people/richard-staley

https://www.people.hps.cam.ac.uk/index/teaching-officers/staley

Research interests: The history of the physical sciences (broadly construed) from the 19th century to the present.

https://www.linkedin.com/in/richard-staley-87b676105/?originalSubdomain=uk

**My notes from the lecture** (if they don’t make sense then it is entirely my fault)

In the 19^{th} century physics emerged as a laboratory subject.

https://en.wikipedia.org/wiki/Gustav_Fechner

Gustav Theodor Fechner (19 April 1801 – 18 November 1887) was a German philosopher, physicist and experimental psychologist. An early pioneer in experimental psychology and founder of psychophysics, he inspired many 20th century scientists and philosophers.

https://en.wikipedia.org/wiki/Hermann_von_Helmholtz

Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist who made significant contributions in several scientific fields. He explored physics and physiology etc.

Albert Einstein

https://en.wikipedia.org/wiki/Albert_Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

“On the electrodynamics of moving bodies” Annalen der Physik; 17 (1905), pg. 891”

http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19053221004

Maintaining the principle of relativity in electrodynamics requires a new theory of space and time. Einstein had identified a formal mismatch and set out to rectify this.

He brought space and time together more intimately than before which was a major first step on the way to his later General Relativity, and it was accepted quite quickly.

https://www.ias.ac.in/public/Resources/Events/Mid_Year_Meetings/16_nmukunda.pdf

In 1685 Newton in his Principia had expressed definite views on the natures of space and time. Space was the same for all observers and time flowed uniformly and was also common to all observers.

Maxwell discovered his field equations of electromagnetism around 1865, and found that they predicted the existence of electromagnetic waves with a certain speed calculable from theory. He accepted the Newtonian view of space and time, which implied a definite law for addition of velocities. This led to the conclusion: the Maxwell equations could not be valid in all inertial frames of mechanics, but only in a subset of them at rest relative to one another. In these alone the speed of light would have the calculated value of 300,000 km/sec. In all other inertial frames of mechanics the speed would be both frame and direction dependent. In brief – the principle of relativity of mechanics could not hold for electromagnetism, absolute rest would be meaningful, and departures from it could be detected by electromagnetic measurements.

Einstein seems to have approached the problem entirely in a strikingly original fashion. From around age 16 he asked himself such questions as: what does a light wave look like if I travel alongside it with light speed; what happens to my image in a mirror if I and the mirror both move at that speed? After years of puzzling over these and related questions, sometime in 1905 it suddenly struck him: time was the culprit, it was not absolute and the same for all observers!

https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments

“After seven years of reflection in vain (1898-1905), the solution came to me suddenly with the thought that our concepts and laws of space and time can only claim validity in so far as they stand in a clear relation to our experiences; and that experience could very well lead to the alteration of these concepts and laws. By a revision of the concept of simultaneity into a more malleable form, I thus arrived at the special theory of relativity”.

Einstein’s guiding idea was that a common principle of relativity had to hold for both mechanics and electromagnetism.

He also established the invariance of Maxwell’s equations under the new transformations, and dispensed with the ether as the carrier of electromagnetic waves. In accord with his postulated relativity principle he went on to modify Newton’s mechanics to fall in line with electromagnetism. His world famous formula, E = mc^{2} was a fallout of this effort.

In general terms Special Relativity rules out action at a distance and propagation speeds greater than light speed.

In 1907 Einstein was still working at the Patents office, but this didn’t stop him working on his theories

“When I was busy (in 1907) writing a summary of my work on the theory of special relativity for the Jahrbuch für Radioaktivität und Elektronik [Yearbook for Radioactivity and Electronics], I also had to try to modify the Newtonian theory of gravitation such as to fit its laws into the theory. While attempts in this direction showed the practicability of this enterprise, they did not satisfy me because they would have had to be based upon unfounded physical hypotheses. At that moment I got the happiest thought of my life in the following form: In an example worth considering, the gravitational field has a relative existence only in a manner similar to the electric field generated by magneto-electric induction. Because for an observer in free-fall from the roof of a house there is during the fall—at least in his immediate vicinity—no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, independent of their special chemical or physical nature. The observer, therefore, is justified in interpreting his state as being “at rest.

https://en.wikipedia.org/wiki/Euclidean_geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

Adrien-Marie Legendre was a mathematician sympathetic to the didactic aims of the Elements but not to its original formulations. He wrote several different versions of his Éléments de géométrie (1794) with a view to restoring Euclidean rigour in the teaching of geometry, which in his view had been corroded by texts that relied on notions of self-evidence.

https://plato.stanford.edu/entries/epistemology-geometry/

Einstein’s theory of general relativity shows that the true geometry of spacetime is not Euclidean geometry.

https://en.wikipedia.org/wiki/Ernst_Mach

Ernst Waldfried Josef Wenzel Mach (18 February 1838 – 19 February 1916) was an Austrian physicist and philosopher, noted for his contributions to physics such as study of shock waves. The ratio of one’s speed to that of sound is named the Mach number in his honour. Through his criticism of Newton’s theories of space and time, he foreshadowed Einstein’s theory of relativity.

https://en.wikiquote.org/wiki/The_Science_of_Mechanics

The Science of Mechanics: A Critical and Historical Exposition of its Principles is an 1893 Thomas J. McCormick translation of the second German edition of Ernst Mach’s original 1883 Die Mechanik in ihrer Entwickelung (Mechanics and Its Evolution). It is not a treatise upon the application of the principles of mechanics. Its aim was to clear up ideas, expose the real significance of the matter, and get rid of metaphysical obscurities. The little mathematics it contains is merely secondary to that purpose. Mechanics was treated, not as a branch of mathematics, but as a physical science. Its reader was expected to acquire therein, some enlightenment on the principles of mechanics: how they were ascertained, their sources of the origin, and how far they might be regarded as permanent. Mach maintained that all of this, the positive and physical essence of mechanics, was buried and concealed beneath a mass of technical considerations in the extant treatises of his time. He believed that principles must be based on experience/fact.

Mach described the first assumption Archimedes invoked when discussing the law of the lever that magnitudes of equal weight at an equal distance from the point of support are in equilibrium.

The above diagram shows how Mach analysed the idea further. “One could think that independent of all experience (according to the principle of sufficient reason) it is self-evident that the symmetry of the entire arrangement gives no ground for the bar to rotate in one rather than another direction”.

“We must not be surprised… that… all physicists of the last century saw in classical mechanics a firm and final foundation for all of physics, yes, indeed, for all natural science, and that they never grew tired in their attempts to base Maxwell’s theory of electromagnetism, which, in the meantime, was slowly beginning to win out, upon mechanics as well. Even Maxwell and H. Hertz, who in retrospect appear as those who demolished the faith in mechanics as the final basis of all physical thinking, in their conscious thinking adhered throughout to mechanics as the secure basis of physics. It was Ernst Mach who, in his History of Mechanics, shook this dogmatic faith; this book exercised a profound influence upon me in this regard while I was a student. I see Mach’s greatness in his incorruptible scepticism and independence; in my younger years, however, Mach’s epistemological position also influenced me very greatly, a position which today appears to be essentially untenable.”

Albert Einstein, Autobiographical Notes (1946) Tr. Paul Arthur Schilpp.

“My attention was drawn to Ernst Mach’s Science of Mechanics by my friend Besso while a student, around the year 1897. The book exerted a deep and persisting impression upon me… owing to its physical orientation toward fundamental concepts and fundamental laws.”

Albert Einstein, Letter to Carl Seelig (April 8, 1952) as quoted by Gerald Holton, Thematic Origins of Scientific Thought: Kepler to Einstein (1973)

In Beiträge zur Analyse der Empfindungen (1886; Contributions to the Analysis of the Sensations, 1897), Mach advanced the concept that all knowledge is derived from sensation; thus, phenomena under scientific investigation can be understood only in terms of experiences, or “sensations,” present in the observation of the phenomena. This view leads to the position that no statement in natural science is admissible unless it is empirically verifiable. Mach’s exceptionally rigorous criteria of verifiability led him to reject such metaphysical concepts as absolute time and space, and prepared the way for the Einstein relativity theory.

Mach also proposed the physical principle, known as Mach’s principle that inertia (the tendency of a body at rest to remain at rest and of a body in motion to continue in motion in the same direction) results from a relationship of that object with all the rest of the matter in the universe. Inertia, Mach argued, applies only as a function of the interaction between one body and other bodies in the universe, even at enormous distances. Mach’s inertial theories also were cited by Einstein as one of the inspirations for his theories of relativity.

https://www.britannica.com/biography/Ernst-Mach#ref59576

The Universe of General Relativity

In many of his explanations of general relativity, Einstein offered two principles as the foundation of the theory. The first was that gravitation has to be seen as an effect of the structure of space-time. The second demanded that all reference systems, not just inertial ones, should be treated on an equal footing.

One of his most extensive expositions of general relativity was the unpublished paper “Fundamental ideas and methods of the theory of relativity”. In it Einstein offers a new formulation and argument for general relativity which is considerably less well known than his previous statements. In section 15 of the original form of the paper, he first introduces the principle of equivalence and compares the relation of gravitation and inertia it postulates to the fact that in special relativity electric and magnetic fields have only relative existence, depending on the state of motion of the observer.

https://en.wikipedia.org/wiki/Equivalence_principle

In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein’s observation that the gravitational “force” as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

Fearful Symmetry: The Search for Beauty in Modern Physics; A. Zee; Princeton University Press (1986) https://philpapers.org/rec/ZEEFST

http://euro-math-soc.eu/review/fearful-symmetry-search-beauty-modern-physics

A. Zee, a distinguished physicist and skilful expositor, tells the exciting story of how today’s theoretical physicists are following Einstein in their search for the beauty and simplicity of Nature.

The Principle of Relativity https://books.google.co.uk/books?id=yECokhzsJYIC&pg=PA111&redir_esc=y&hl=en#v=onepage&q&f=false