1. 1 Einstein’s Methods John D. Norton Department of History and Philosophy of Science University of Pittsburgh

2. 2 What can we know of how Einstein made his discoveries? 1905 Special relativity 1905 Light quantum 1905 Atoms/Brownian motion 1906 Specific heats 1909 Wave particle duality 1907-1915 General relativity 1916 Gravitational waves 1916 “A and B” coefficients 1917 Relativistic cosmology 1924-25 Bose Einstein statistics …and more Inscrutable flashes of insight or methodical exploration? This talk. Einstein thought a great deal about his methods. They changed almost completely in his lifetime. Through a remarkable manuscript, we can look over Einstein’s shoulder and watch the struggle unfold.

3. 3 The view of Einstein’s work on general relativity as driven by the tension of physical and formal ways of thinking was developed in a collaborative research group working on Einstein’s Zurich Notebook of 1912-1913 at the Max Planck Institut für Bildungsforschung and then at the Max Planck Institut für Wissenschaftgeschichte. Michel Janssen, John Norton, Jürgen Renn, Tilman Sauer, John Stachel, et al. General Relativity in the Making: Einstein's Zurich Notebook. Vol. 1 The Genesis of General Relativity: Documents and Interpretation. Vol. 2 The Genesis of General Relativity. Vol. 3 Vol.4. Dordrecht: Kluwer. available now

4. 4 E xploit formal (usually mathematical) properties of emerging theory. Covariance principles. Group structure. T heory construction via mathematical theorems. Geometrical methods assure automatic covariance. Formal naturalness. E xtreme case: choose mathematically simplest law. B ased on physical principles with evident empirical support. Principle of relativity. Conservation of energy. S pecial weight to secure cases of clear physical meaning. Newtonian limit. Static gravitational fields in GR. Physical naturalness. E xtreme case: thought experiments direct theory choice. Physical versus Formal approach Considerable overlap. Often both are the same inferences in different guises.

5. 5 Physical approach illustrated 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. Based on Einstein’s 1905 magnet-conductor thought experiment.

6. 6 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. Sign and coordinate conventions after Pauli, Theory of Relativity, p. 78. Lorentz transformation Hyperbolic rotation in spacetime mixes E’s and B’s Pure magnetic field Mixed magnetic and electric field Frame dependence of decomposition of electromagnetic field is a consequence of spacetime geometry.

7. 7 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 statistics 1935 EPR Physical Formal Five dimensional unified field 1922-41 Distant parallelism 1928 Bivector fields 1932-33 Unified field via non- symmetric connection 1925- 1955

8. 8 Einstein’s early distain for higher mathematics in physics S pecial relativity, light quantum use only calculus of many variables. M arked reluctance to adopt Minkowski’s four-dimensional methods. He does not use them until 1912. Q uip: “I can hardly understant Laue’s book” [1911 textbook on special relativity that used Minkowski’s methods]. F our-dimensional methods disparaged as “superfluous learnedness.”

9. 9 Abraham’s 1912 theory of gravity… Abraham’s theory is the simplest mathematically delivered by four- dimensional methods. u=ict where c=c  ) …Einstein’s idea! …is condemned by Einstein for its purely formal basis. “[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) “…at the first moment (for 14 days) I too was totally “bluffed” by the beauty and simplicity of its formulas.” (To Besso)

10. 10 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). “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.

11. 11 Einstein and Grossmann’s “Entwurf…” 1913 Complete framework of general theory of relativity. Gravity as curvature of spacetime geometry. One thing is missing… The Einstein equations! G ik = k (T ik – (1/2) g ik T) G ik = 0 source free case Ricci tensor G ik is first contraction of Riemann curvature tensor R iklm (Yes--the notation is non-standard.)

12. 12 The “Einstein Equations” are approached… Einstein and Grossman present gravitational field equations that are not generally covariant and have no evident geometrical meaning. 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]  in the special case of an infinitely weak, static gravitational field.”

13. 13 Einstein’s “Zurich Notebook” A notebook of calculation Einstein kept while he worked on the “Entwurf” theory with Grossmann. Einstein worked from both ends. 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.

14. 14 Inside the cover…

15. 15 Einstein connects gravity and curvature of spacetime. E instein writes the spacetime metric for the first time as ds 2 =  G  dx dx  G   soon becomes g  p. 39L I mporting of special case of his 1907-1912 theory in which a variable c is the gravitational potential. F irst attempts at gravitational field equations based on physical reasoning of 1907-1912 theory.

16. 16 The physical approach to energy-momentum conservation… 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 p.5R

17. 17 …and the formal approach to energy-momentum conservation. Is the conservation law of the form Check: form It should be 0 or a four-vector. It vanishes! Stimmt! p.5R

18. 18 The formal approach to the gravitational field equations Following pages: Einstein shows how to select coordinate systems so that they do vanish. p. 14L 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.”

19. 19 Failure of the formal approach Einstein finds multiple problems with the gravitational field equations based on the Riemann curvature tensor. “Special case [of the 1907-1912 theory] apparently incorrect” p. 21R

20. 20 “Entwurf” gravitational field equations Derived from a purely physical approach. Energy-momentum conservation. pp. 26L-R

21. 21 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

22. 22 Einstein snatches triumph from near disaster: Fall 1915. David Hilbert in Göttingen applies formal methods to general field equations for Einstein’s theory... and Einstein knows it. Communications to the Göttingen 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 Einstein realizes his “Entwurf” field equations are wrong and returns to seek generally covariant equations. Communications to the Prussian Academy: Nov. 20 Something very close to Einstein’s equations

23. 23 Einstein’s new methodological moral Triumph of formal methods over physical considerations. “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.” Communication to Prussian Academy of Nov. 4, 1915 “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.” 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.” Einstein to Besso, Dec. 10, 1915

24. 24 Hesitations “Except for the agreement with reality, it is in any case a grand intellectual achievement.” Einstein to Hermann Weyl, Apr. 6, 1918, on Weyl’s mathematically most natural, geometric unification of gravity and electromagnetism “It seems to me that you overrate the value of formal points of view. These may be valuable when an already found truth needs to be formulated, but fail always as heuristic aids.” Einstein to Felix Klein, 1917, on the conformal invariance of Maxwell’s equations

25. 25 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 “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? Herbert Spenser Lecture, "On the Methods of Theoretical Physics," University of Oxford 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.

26. 26 “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.” Einstein’s search for unified field theory Autobiographical Notes, 1946

27. 27 A concluding puzzle "If you want to find out anything from the theoretical physicists about the methods they use, I advise you to stick closely to one principle: don't listen to their words, fix your attention on their deeds." E instein’s manifesto begins: Why does he say this?! To lessen the shock of the extraordinary view he is about to present that so fully contradicts the then present mainstream of philosophical thought? Or to induce you to look at what physicists--Einstein and others--actually do so you come to reject the mainstream in favor of Einstein’s view?