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Philosophy and Einstein's Discovery of the Theories of Relativity. John D. Norton Center for Philosophy of Science and Department of History and Philosophy of Science University of Pittsburgh. CARL FRIEDRICH VON WEIZSÄCKER LECTURES UNIVERSITY OF HAMBURG June 2010.

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Philosophy and Einstein's Discovery of the Theories of Relativity

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Philosophyand Einstein'sDiscovery of theTheories of Relativity

John D. Norton

Center for Philosophy of Science and

Department of History and Philosophy of Science

University of Pittsburgh



June 2010

Carl Friedrich von Weizsäcker, The Structure of Phyics (Aufbau der Physik). From Preface, 1985.

“…the apparent distractions in my life due to politics and philosophy only slightly slowed the pace of this work.

Philosophy was indispensable for a philosophically oriented analysis of physics; attempting to understand Plato, Aristotle, Descartes, Kant, Frege, Heidegger was no distraction at all from the main topic and hence entailed no loss of time…”

Carl Friedrich von Weizsäcker, The Structure of Phyics (Aufbau der Physik). From Preface, 1985.

“When I was nineteen years old, Bohr revealed to me the philosophical dimensions of physics. He gave me what I had been looking for in physics. From him I learned to understand the influence that Socrates must have exerted of his followers…”

Carl Friedrich von Weizsäcker, The Structure of Phyics (Aufbau der Physik). From Preface, 1985.

“I have placed the three names Albert Einstein, Niels Bohr, Werner Heisenberg at the head of of the book.

Einstein was the genius of the century. The theory of relativity is his work, and it was on his account that quantum got under way. All younger workers remain under the spell cast by his insights…”

Carl Friedrich von Weizsäcker, The Structure of Phyics (Aufbau der Physik). From Preface, 1985.

“For me, the mention of these three names also carries the personal significance of admiring and affectionate remembrance. I unfortunately never met Einstein, but his name was familiar to me by time if was a schoolboy, and from decade to decade I learned better to understand his greatness.

“…I unfortunately never met Einstein…”


This moment brought to you by P. Shop.

To Einstein


criterion of reality

principle vs constructive



free creation



mathematical simplicity



This Talk


Einstein’s discovery of special relativity is decisively advanced by his reading of philosophy.



Einstein’s discovery of general relativity converts him to an advocate of an ancient epistemology.


Einstein’s Discovery of Special Relativity 1905


The Final Crisis:Reconcile the “Apparently Incompatible” …

Principle of Relativity

Equivalence of all inertial states of motion.

From ether drift experiments, observables in electrodynamics, classical mechanics.


Light Postulate

Constancy of speed of light for all inertial observers.

From Maxwell’s electrodynamics. All efforts to alter it had failed.

A later recollection…

“Today everyone knows, of course, that all attempts to clarify this paradox [of chasing the beam of light] satisfactorily were condemned to failure as long as the axiom of the absolute character of time, or of simultaneity, was rooted unrecognized in the unconscious. To recognize clearly this axiom and its arbitrary character already implies the essentials of the solution of the problem. The type of critical reasoning required for the discovery of this central point was decisively furthered, in my case, especially by the reading of David Hume’s and Ernst Mach’s philosophical writings.”

Albert Einstein, Autobiographical Notes

! Einstein did not mean Hume and Mach’s analysis of the notions of space and time specifically…

The platform observer judges the two flashes to be simultaneous and the two clocks to be properly synchronized.

The moving observer judges the A flash to happen earlier and the two clocks not to be properly synchronized.

Einstein’s analysis in his 1905 “On the Electrodynamics of Moving Bodies” (simplified):

Inertial Observers Find the Same Speed for Light

An observer in relative motion finds clocks A and B NOT to be properly synchronized. After correcting, finds light moves at c.

An observer measures the speed of a light signal with a rod and two synchronized clocks, A and B. Finds light moves at c.

The central insightA view about how concepts should be used in physical theories.

Concepts must be properly grounded in experience...

“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 insofar as they stand in a clear relation to experiences; and that experience could very well lead to the alteration of the 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.”

From a 1924 recording transcribed by Herneck in 1966.

“The concept [of simultaneity] does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case.”

A. Einstein, Relativity, §8

“…an illustration which Einstein offered in discussion. Suppose somebody uses the word ‘hunchback.’ If this concept is to have any clear meaning, there must be some way of finding out whether or not a man has a hunched back. If I could conceive of no possibility of reaching such a decision, the word would have no real meaning for me.”

To Wertheimer in 1916 interview.

…so we may purge a priori (absoluteness of simultaneity) from our concepts.

“The illusion which prevailed prior to the enunciation of the theory of relativity--that, from the point of view of experience the meaning of simultaneity in relation to spatially distant events and, consequently, that the meaning of physical time is a priori clear--this illusion had its origin in the fact that in our everyday experience we can neglect the time of propagation of light. We are accustomed on this account to fail to differentiate between "simultaneously seen" and "simultaneously happening"; and, as a result, the difference between time and local time is blurred.

The lack of definiteness which, from the point of view of its empirical significance, adheres to the notion of time in classical mechanics was veiled by the axiomatic representation of space and time as given independently of our sense experiences. Such a use of notions--independent of the empirical basis

to which they owe their existence--does not necessarily damage science. One may, however, easily be led into the error of believing that these

notions, whose origin is forgotten, are logically necessary and therefore unalterable, and this error may constitute a serious danger to the progress of science.”

Einstein, “Physics and Reality,” 1936.

…and Einstein credits Hume and Mach

“Today everyone knows, of course, that all attempts to clarify this paradox [of chasing the beam of light] satisfactorily were condemned to failure as long as the axiom of the absolute character of time, or of simultaneity, was rooted unrecognized in the unconscious. To recognize clearly this axiom and its arbitrary character already implies the essentials of the solution of the problem. The type of critical reasoning required for the discovery of this central point was decisively furthered, in my case, especially by the reading of David Hume’s and Ernst Mach’s philosophical writings.”

Albert Einstein, Autobiographical Notes


“Science is, according to Mach, nothing but the comparison and orderly arrangement of factually given contents of consciousness, in accord with certain gradually acquired points of view and methods….

…concepts have meaning only in so far as they can be found in things, just as they are the points of view according to which these things are organized. (Analysis of concepts)”

Empirical grounding of concepts

“Concepts that have proven useful in ordering things can easily gain authority over us such that we forget their worldly origin and take them as immutably given. They are then rather rubber-stamped as a ‘necessity of thought’ and an ‘a priori given,’ etc. Such errors often make the path of scientific progress impassable for a long time…”

Dangers of a priori

Einstein attributes this view of concepts to Mach

from Einstein’s obituary

for Mach, 1916

Quotes of Mach’s reanalysis of judgments of time

(as expressions of dependence upon pendulum oscillations or the Earth’s position);

of motion; Newton’s bucket.


It is not improbable that Mach would have hit upon relativity theory if, in the time that he was of young and fresh spirit, physicists would already have been moved by the question of the meaning of the constancy of the speed of light. In this absence of this stimulation, which follows from Maxwell-Lorentz electrodynamics, even Mach’s critical urge did not suffice to arouse a feeling for the necessity of a definition of simultaneity for spatially distant events.

Einstein sees link to simultaneity

Einstein attributes this view of concepts to Mach

from Einstein’s obituary

for Mach, 1916


on the known reading list of Einstein’s Olympia Academy

“…and still much more Hume…”

“Your exposition is also quite right that positivism suggested rel. theory, without requiring it. Also you have correctly seen that this line of thought was of great influence on my efforts and indeed E. Mach and still much more Hume, whose treatise on understanding I studied with eagerness and admiration shortly before finding relativity theory.”

Einstein to Moritz Schlick, Dec 14 1915

“treatise on understanding” =

“A Treatise of Human Nature”?

or “An Enquiry concerning Human Understanding”?

Dependence of concepts on experience

“…all our simple ideas proceed either mediately or immediately, from their correspondent impressions.

This then is the first principle I establish in the science of human nature…”

Book 1, Part 1, Section 1.

“As ‘tis from the disposition of visible and tangible objects we receive the idea of space, so from the succession of ideas and impressions we form the idea of time, nor is it possible for time alone ever to make its appearance, or be taken notice of by the mind.

…time cannot make its appearance to the mind either alone, or attended with a steady unchangeable object, but is always discover’d by some perceivable succession of changeable objects.”

Application to time

Book 1, Part II, Section III.

Hume, A Treatise of Human Nature

Inapplicability of concept without corresponding experience

“I know there are some who pretend, that the idea of duration is applicable in a proper sense to objects, which are perfectly unchangeable…But to be convinced of its falsehood we need but reflect on the foregoing conclusion, that the idea of duration is always deriv’d from a succession of changeable objects, and can never be convey’d to the mind by any thing stedfast and unchangeable…

…Ideas always represent the objects or impressions from which they are deriv’d, and can never without a fiction represent or be appl’d to any other…”

Book 1, Part II, Section III.

This mode of analysis is applied throughout the Treatise.

We have no idea of substance beyond the collection of particular qualities. We have no idea of causation beyond contiguity and succession--no necessary connection.

Hume, A Treatise of Human Nature

“I see [Mach’s] weakness in this, that he more or less believed science to consist in a mere “ordering” of empirical “material”; that is to say, he did not recognize the freely constructive element in the formation of concepts. In a way he thought that theories arose through discoveries and not through inventions. He even went so far that he regarded “sensations” not only as material which has to be investigated, but, as it were, as the building blocks of the real world…”

Einstein to Besso on 6. Jan. 1948

“Hume saw clearly that certain concepts, as for example that of causality, cannot be deduced from the material of experience by logical methods.”

Einstein, Autobiographical Notes

WhyHume more than Mach?

A conjecture: Einstein thought that Mach (but not Hume) denied the freedom of creation of concepts exercised by Einstein when he introduced a new definition of distant simultaneity in 1905?

Einstein on…

How to Use Concepts in Physical Theories

Concepts must be properly grounded in experience, else they fail to represent the physically real and are fictional. (From Hume and Mach)

Concepts without proper physical grounding need not be abjured (contrary to Mach and Hume). They can be retained in a physical theory as long as their arbitrary character is recognized and in a way that does not unwittingly introduce false presumptions.

The breakthrough in Einstein’s discovery of special relativity came when he applied this view to the traditional concept of the simultaneity of distant events.

Einstein’s Discovery of General Relativity: 1907-1915


Based on physical principles with evident empirical support.

Principle of relativity. Conservation of energy.

Special weight to secure cases of clear physical meaning.

Newtonian limit. Static gravitational fields in GR.

Exploit formal (usually mathematical) properties of emerging theory.

Covariance principles. Group structure.

Theory construction via mathematical theorems.

Geometrical methods assure automatic covariance.

Physical naturalness.

Extreme case: thought experiments direct theory choice.

Formal naturalness.

Extreme case: choose mathematically simplest law.

Physical versus Formal



Considerable overlap. Often both are the same inferences in different guises.

Einstein’s early distain for higher mathematics in physics

Special relativity, light quantum use only calculus of many variables.

Marked reluctance to adopt Minkowski’s four-dimensional methods. He does not use them until 1912.

Quip: “I can hardly understant Laue’s book” [1911 textbook on special relativity that used Minkowski’s methods].

Four-dimensional methods disparaged as “superfluous learnedness.”

…is condemned by Einstein for its purely formal basis.

“…at the first moment (for 14 days) I too was totally “bluffed” by the beauty and simplicity of its formulas.” (To Besso)

“[it] has been created out of thin air, i.e. out of nothing by considerations of mathematical beauty, and is completely untenable.” (To Besso)

“totally untenable” (To Ehrenfest)

“incorrect is every respect” (To Lorentz)

“totally unacceptable” (To Wien)

“totally untenable” (To Zangger)

Abraham’s 1912 theory of gravity…

Abraham’s theory is the simplest mathematically delivered by four-dimensional methods.


where c=c(F)

…Einstein’s idea!

“I am now working exclusively on the gravitation problem and believe that I can overcome all difficulties with the help of a mathematician friend of mine here [Marcel Grossmann]. But one thing is certain: never before in my life have I toiled any where near as much, and I have gained enormous respect for mathematics, whose more subtle parts I considered until now, in my ignorance, as pure luxury. Compared with this problem, the original theory of relativity is child's play.”

Einstein to Sommerfeld, October 1912

Sommerfeld: edited Minkowski’s papers and wrote introductory papers on four-dimensional methods.

General relativity begins to turn the tide

In 1912, Einstein began work on the precursor to general relativity, the “Entwurf” theory of 1913 with the mathematical assistance of Marcel Grossmann, who introduced Einstein to Ricci and Levi-Civita’s “absolute differential calculus” (now called tensor calculus).

The Einstein equations!

Gik = k (Tik – (1/2) gik T)

Gik = 0 source free case

Ricci tensor Gik is first contraction of Riemann curvature tensor Riklm

(Yes--the notation is non-standard.)

Einstein and Grossmann’s “Entwurf…” 1913

Complete framework of general theory of relativity. Gravity as curvature of spacetime geometry.

One thing is missing…

Riemann curvature tensor

“Christoffel’s four-index-symbol”

Its first contraction as the unique tensor candidate for inclusion is gravitational field equations.

“But it turns out that this tensor does not reduce to the [Newtonian] Dj in the special case of an infinitely weak, static gravitational field.”

The “Einstein Equations” are approached…

Einstein and Grossman present gravitational field equations that are not generally covariant and have no evident geometrical meaning.

Einstein expected the physical and formal/mathematical approaches to give the same result.

When he erroneously thought they did not, he chose the physical approach over the formal and selected equations that would torment him for over two years.

Einstein worked from both ends.

Einstein’s “Zurich Notebook”

A notebook of calculation Einstein kept while he worked on the “Entwurf” theory with Grossmann.

Inside the cover…

Einstein writes the spacetime metric for the first time as

ds2 = S Glm dxl dxm

Glmsoon becomes glm

Importing of special case of his 1907-1912 theory in which a variable c is the gravitational potential.

First attempts at gravitational field equations based on physical reasoning of 1907-1912 theory.

Einstein connects gravity and curvature of spacetime.

p. 39L

Equations of motion for a speck of dust (geodesic)

Expressions for energy-momentum density and four-force density for a cloud of dust.

Combine: energy-momentum conservation for dust

Rate of accumulation energy-momentum

Force density

The physical approach to energy-momentum conservation…


Check: form

It should be 0 or a four-vector.

It vanishes!


…and the formal approach to energy-momentum conservation.

Is the conservation law

of the form


Einstein writes the Riemann curvature tensor for the first time… with Grossmann’s help.

First contraction formed.

To recover Newtonian limit, three terms “should have vanished.”

The formal approach to the gravitational field equations

Following pages: Einstein shows how to select coordinate systems so that they do vanish.

p. 14L

“Special case [of the 1907-1912 theory] apparently incorrect”

Failure of the formal approach

Einstein finds multiple problems with the gravitational field equations based on the Riemann curvature tensor.

p. 21R

“Entwurf” gravitational field equations

Derived from a purely physical approach. Energy-momentum conservation.

pp. 26L-R

Einstein’s short-lived methodological moral of 1914

The physical approach is superior to the formal approach.

“At the moment I do not especially feel like working, for I had to struggle horribly to discover what I described above. The general theory of invariants was only an impediment. The direct route proved to be the only feasible one. It is just difficult to understand why I had to grope around for so long before I found what was so near at hand.”

Einstein to Besso, March 1914

David Hilbert in Göttingen applies formal methods to general field equations for Einstein’s theory

... and Einstein knows it.

Communications to the Göttingen Academy:

Einstein snatches triumph from near disaster: Fall 1915.

Einstein realizes his “Entwurf” field equations are wrong and returns to seek generally covariant equations.

Communications to the Prussian Academy:

Nov. 4 Almost generally covariant field equations

Nov. 11 Almost generally covariant field equations

Nov. 18 Explanation of Mercury’s perihelion motion

Nov. 26 Einstein equations

Nov. 20 Something very close to Einstein’s equations

Einstein’s new methodological moral

Triumph of formal methods over physical considerations.

“I had already taken into consideration the only possible generally covariant equations, which now prove to be the right ones, three years ago with my friend Grossmann. Only with heavy hearts did we detach ourselves from them, since the physical discussion had apparently shown their incompatibility with Newton's law.”

“Hardly anyone who has truly understood it can resist the charm of this theory; it signifies a real triumph of the method of the general differential calculus, founded by Gauss, Riemann, Christoffel, Ricci and Levi-Civita.”

Einstein to Hilbert Nov 18, 1915

“This time the most obvious was correct; however Grossmann and I believed that the conservation laws would not be satisfied and that Newton's law would not come out in the first approximation.”

Communication to Prussian Academy of Nov. 4, 1915

Einstein to Besso, Dec. 10, 1915

“If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter?

I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena.

Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.”

Einstein’s manifesto of June 10, 1933

Herbert Spenser Lecture, "On the Methods of Theoretical Physics," University of Oxford


“The reciprocal relationship of epistemology and science is of noteworthy kind. They are dependent upon each other.

Epistemology without contact with science becomes an empty scheme.

Science without epistemology is -- insofar as it is thinkable at all -- primitive and muddled.”

Einstein, “Autobiographical Notes--Remarks Concerning the Essays Brought together in this Co-operative Volume." p. 683

Read all about it…


Einstein Recalls the Decisive Moment

“Why are these two things inconsistent with each other? I felt that I was facing an extremely difficult problem. I suspected that Lorentz’s ideas had to be modified somehow, but spent almost a year on fruitless thoughts. And I felt that was puzzle not to be easily solved.

But a friend of mine living in living in Bern (Switzerland) [Michele Besso] helped me by chance. One beautiful day, I visited him and said to him: ‘I presently have a problem that I have been totally unable to solve. Today I have brought this “struggle” with me.’ We then had extensive discussions, and suddenly I realized the solution. The very next day, I visited him again and immediately said to him: ‘Thanks to you, I have completely solved my problem.”

My solution actually concerned the concept of time. Namely, time cannot be absolutely defined by itself, and there is an unbreakable connection between time and signal velocity. Using this idea, I could now resolve the great difficulty that I previously felt.

After I had this inspiration, it took only five weeks to complete what is now known as the special theory of relativity.”

From a lecture given in Kyoto, Dec. 14, 1922. Notes by Jun Ishiwara; translation Akira Ukawa; revised John Stachel.

Physical approach illustrated

Based on Einstein’s 1905 magnet-conductor thought experiment.

Principle of relativity requires that the electromagnetic field manifests as different mixtures of magnetic field B and electric field E according to motion of observer.

Lorentz transformation

Hyperbolic rotation in spacetime mixes E’s and B’s

Mixed magnetic and electric field

Pure magnetic field

Sign and coordinate conventions after Pauli, Theory of Relativity, p. 78.

Formal approach illustrated

Write Maxwell’s equations using four-vector and six-vector (now antisymmetric second rank tensor) quantities and operators of Minkowski’s 1908 spacetime, geometrical approach.

Satisfaction of the principle of relativity is automatic.

Frame dependence of decomposition of electromagnetic field is a consequence of spacetime geometry.


Five dimensional        

unified field 1922-41

Distant parallelism 1928

Bivector fields 1932-33

Unified field via non-       

symmetric connection 1925- 1955

Evolution of Einstein’s approaches

1902-1904 statistical physics

1905 Brownian motion

1905 Light quantum

1905 Special relativity

1906 Specific heats

1909 Wave particle duality

1907-1915 General relativity

1916 A and B coefficients

1917 Relativististic cosmology

1924-25 Bose-Einstein


1935 EPR


Einstein’s search for unified field theory

“I have learned something else from the theory of gravitation:

no collection of empirical facts however comprehensive can ever lead to the setting up of such complicated equations [as non-linear field equations of the unified field]. A theory can be tested by experience, but there is no way from experience to the construction of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition that determines the equations completely or almost completely. Once one has obtained those sufficiently strong formal conditions, one requires only little knowledge of facts for the construction of the theory; in the case of the equations of gravitation it is the four-dimensionality and the symmetric tensor as expression for the structure of space that, together with the invariance with respect to the continuous transformation group, determine the equations all but completely.”

Autobiographical Notes, 1946

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