EINSTEIN’S PHYSICS

1. EINSTEIN’S PHYSICS A modern understanding Ta-Pei Cheng talk based on … Oxford Univ Press (2/ 2013) Einstein’s Physics Atoms, Quanta, and Relativity --- Derived, Explained, and Appraised

2. 2TOC ATOMIC NATURE OF MATTER Albert Einstein 1879 – 1955 1. Molecular size from classical fluids 2. The Brownian motion QUANTUM THEORY The book explains his Physics in equations 3. Blackbody radiation: From Kirchhoff to Planck 4. Einstein’s proposal of light quanta 5. Quantum theory of specific heat 6. Waves, particles, and quantum jumps 7. Bose–Einstein statistics and condensation 8. Local reality and the Einstein–Bohr debate SPECIAL RELATIVITY 9. Prelude to special relativity 10. The new kinematics and E = mc2 11. Geometric formulation of relativity Today’s talk provides without math details some highlights in historical context GENERAL RELATIVITY 12. Towards a general theory of relativity 13. Curved spacetime as a gravitational field 14. The Einstein field equation 15. Cosmology WALKING IN EINSTEIN’S STEPS 16. Internal symmetry and gauge interactions 17. The Kaluza–Klein theory and extra dimensions

3. Jean Perrin 3Atoms Molecular size & Avogadro’s numberclassical liquids with suspended particles (4/1905) U Zurich doctoral thesis:“On the determination of molecular dimensions”→ 2 equations relating P & NAto viscosity and diffusion coefficients Hydrodynamics Navier-Stokes equation, balance of osmotic and viscous forces E’s most cited publication! (11 days later) the Brownian motion paper: While thermal forces change the direction and magnitude of the velocity of a suspended particle on such a small time-scale that it cannot be measured, the overall drift of such a particle is observable quantity. Fluctuation of a particle system random walk as the prototype of discrete system A careful measurement of this zigzag motion through a simple microscope would allow us to deduce the Avogadro number! It finally convinced everyone, even the skeptics, of the reality of molecules & atoms.

4. 4Quanta 1 Blackbody Radiation (rad in thermal equilibrium) = cavity radiation Einstein, like Planck, arrived at the quantum hypothesis thru BBR 2nd law Maxwell EM radiation = a collection of oscillators u = E2, B2 ~ oscillator energy kx2 The ratio of oscillating energy tofrequencyis an adiabatic invariant ; p = u/3 Kirchhoff (1860) densities = universal functions Stefan (1878) Boltzmann (1884): Wien’s displacement law (1893): Wien’s distribution (1896): fits data well…. until IR Planck’s distribution (1900): Excellent fit of all the data Wien = high ν of Planck key: Wien 2 ->1 var What is the physics? Planck found a relation What microstate counting Wthat can lead to this S viaBoltzmann’s principle S=k lnW ? Planck was “compelled” to make the hypothesis of energy quantization

5. 5Quanta 2 Einstein’s 1905 proposal of light quanta was not a direct follow-up of Planck’s Einsteinused Planck’s calculationand invoked the equipartition theoremof stat mech to derive the Rayleigh-Jeans law noted its solid theoretical foundation and the problem of ultraviolet catastrophe showing BBR = clear challenge to classical physics Rayleigh-Jeans = the low frequency limit of the successful Planck’s distribution The high frequency limit (Wien’s distribution ) =new physics Statistical study of (BBR)wien entropy change due to volume change: (BBR)wien~ ideal gas → (BBR)wien= a gas of light quanta with energy of Einstein arrived at energy quantization independently---- cited Planck only in 2 places concentrate on

6. 6Quanta 2 An historical aside: The history of Rayleigh–Jeans law: • June 1900, Rayleigh, applying the equipartition theorem to radiation, he obtained the result of C1ν2T . Only a limit law? Intro cutoff ρ = C1ν2T exp(-C2ν/T) • October–December 1900, The Planck spectrum distribution was discovered; energy quantization proposed two months later • March 1905, Einsteincorrectly derived the R-J law noted its solid theoretical foundation and the problem of ultraviolet catastrophe • May 1905, Rayleighreturned with a derivation of C1. But missed a factor of 8 • June 1905, James Jeanscorrected Rayleigh’s error… But, explained away the incompatibility with experimental results by insisting that the observed radiation was somehow out of thermal equilibrium. • A.Pais: “It should really be called Rayleigh-Einstein-Jeans law”. “Planck’s fortunate failure”?

7. 7Quanta 3 The quantum idea Einstein vs Planck • Einstein 1905: as P’s W-calculation unreliable… E’s quantum “in opposition” to P’s quantum • Einstein: • the quantum idea must represent new physics; proposed photoelectric effect as test. Planck 1900: is only a formal relation, not physical (radiation not inherently quantum: only during transmission, packets of energy, somehow) 1906 Einstein came in agreement with Planck’s. Also, gave a new derivation of Planck’s law It clearly explained why energy quantization can cure ultraviolet catastrophe The new physics must be applicable beyond BBR: quantum theory of specific heat Einstein’s photon idea was strongly resisted by the physics community for many years because it conflicted with the known evidence for the wave nature of light (Millikan 1916): “Einstein’s photoelectric equation . . . cannot in my judgment be looked upon at present as resting upon any sort of a satisfactory theoretical foundation”, even though “it actually represents very accurately the behavior” of the photoelectric effect”. Planck did not accept Einstein’s photon for at least 10 years

8. 8Quanta 4 Photon carries energy + momentum 1st time stated: quanta carried by point-like particles “point of view of Newtonian emission theory” (1909) Light quanta = particles ? Wave-Particle Duality: a deep riddle

9. 9 Quanta 5 1916–17, Einstein used Bohr’s quantum jump idea to construct a microscopic theory of radiation–matter interaction: absorption and emission of photons (A and B coefficients); he showed how Planck’s spectral distribution followed. The central novelty and lasting feature is the introduction of probability in quantum dynamics Looking beyond Einstein: His discoveries in quantum theory: Wave/particle nature of light and quantum jumps Modern quantum mechanics : states = vectors in Hilbert space (superposition) observables = operators (commutation relations) can all be accounted for in the framework of quantum field theory Classical radiation field = collection of oscillators Quantum radiation field = collection of quantum oscillators • A firm mathematical foundation for Einstein’s photon idea • Quantum jumps naturally accounted for by ladder operators The picture of interactions broadened QFT description: Interaction can change not only motion, but also allows for emission and absorption of radiation →creation and annihilation of particles

10. 10Quanta6 The riddle of wave–particle duality in radiation fluctuation elegantly resolved in QFT forgotten history “three-man paper” of (Born, Heisenberg, and Jordan 1926): The same calculation of fluctuation of a system of waves, but replacing classical field by operators Alas, Einstein never accepted this beautiful resolution as he never accepted the new framework of quantum mechanics Was Einstein just too set-in-his-ways to appreciate the new advances ?

11. 11Quanta 7 Local reality & the Einstein-Bohr debate Orthodox interpretation of QM (Niels Bohr & co): the attributes of a physical object (position, momentum, spin, etc.) can be assigned only when they have been measured. Local realist viewpoint of reality (Einstein,…): a physical object has definite attributes whether they have been measured or not. …. QM is an incomplete theory The orthodox view (measurement actually produces an object’s property) the measurement of one part of an entangled quantum state would instantaneously produce the value of another part, no matter how far the two parts have been separated. Einstein, Podolsky & Rosen (1935) : a thought experiment highlighting this “spooky action-at-a-distance” feature ; the discussion and debate of “EPR paradox” have illuminated some of the fundamental issues related to the meaning of QM • Bell’s theorem (1964) : these seemingly philosophical questions could lead to observable results. The experimental vindication of the orthodox interpretation has sharpened our appreciation of the nonlocal features of quantum mechanics. Einstein’s criticism allowed a better understanding of the meaning of QM. • Nevertheless, the counter-intuitive picture of objective reality as offered by QM still troubles many, leaving one to wonder whether quantum mechanics is ultimately a complete theory

12. 12SR 1 Special Relativity Maxwell’s equations: EM wave – c Contradict relativity? 2 inertial frames x’ = x - vt get velocity add’n rule u’ = u - v The then-accepted interpretation: Max eqns valid only in the rest-frame of ether Q: How should EM be described for sources and observers moving with respect to the ether-frame? “The electrodynamics of a moving body” 1895 Lorentz’s theory (a particular dynamics theoryof ether/matter) could account all observation stellar aberration, Fizeau’s expt… to O(v/c) [ + a math construct ‘local time’] Michelson-Morley null result @ O(v2/c2)  length contraction Lorentz transformation Maxwell ‘covariant’ to all orders (1904) Einstein’s very different approach ..

13. 13SR 2 Special Relativity Einstein’s very different approach .. The magnet-conductor thought expt constructive theory vs theory of principle Relativity = a symmetry in physics Physics unchanged under some transformation Case I: moving charge in B (ether frame)Lorentz force (per unit charge) How to reconcile (Galilean) relativity u’ = u - v with the constancy of c? Resolution: simultaneity is relative Time is not absolute, but frame dependent Relation among inertial frames Correctly given by Lorentz transformation, with Galilean transformation as low v/c approx Case II: changing B induces an Evia Faraday’s law, resulting exactly the same force. yet such diff descriptions ▪ Invoke the principle of relativity This equality can be understood naturally as two cases have the same relative motion ▪ Dispense with ether

14. 14SR 3 Special Relativity 1905 The new kinematics allows for an simple derivation of the Lorentz transformation. All unfamiliar features follow from . time dilation, length contraction, etc. From “no absolute time” to the complete theory in five weeks 10yr Transformation rule for EM fields, radiation energy,.. Lorentz force law from Max field equations Work-energy theorem to mass-energy equivalence E = mc2

15. 15SR 4 Special Relativity Geometric formulation • Even simpler perspective • Hermann Minkowski (1907) • Essence of SR: • time is on an equal footing as space. • To bring out this, unite them in a single math structure, spacetime • Emphasizes the invariance of the theory: c → s • s = a spacetime length (c as the conversion factor) • Lorentz-transformation = rotation → SR features • 4-tensor equations are automatically relativistic SR: The arena of physics is the 4D spacetime Einstein was initially not impressed, calling it “superfluous learnedness” .. until he tried to formulate General relativity (non-inertial frames) = Field theory of gravitation Gravity = structure of spacetime SR = flat spacetime GR = curved spacetime

16. 16GR 1 The Equivalence Principle (1907)played a key role in the formulation ofgeneral theory of relativity Why does GR principle automatically bring gravity into consideration? How is gravity related to spacetime? “Gravity disappears in a free fall frame” starting from Galileo Remarkable empirical observation All objects fall with the same acceleration a ↑ = ↓ g Einstein: “My happiest thought” accelerated frame = inertial frame w/ gravity EP as the handle of going from SR to GR From mechanicsto electromagnetism… → light deflection by gravity, time dilationwith such considerations...Einstein proposed a geometric theory of gravitation in 1912 gravitational field = warped spacetime Note: A curved space being locally flat, EP incorporated in GR gravity theory in a fundamental way.

17. 17GR 2 metric tensor [gμν]= rela. grav. potential gravitational field = warped spacetime Source particle Field Test particle Eqn of motion Field eqn Metric = gravi pot Curvature = tidal forces The Einstein equation 10 coupled PDEs solution = [gμν] Source particle Curved spacetime Test particle energy momentum tensor Geodesic Eqn Einstein field eqn curvature tensor = nonlinear 2nd derivatives of[gμν] 1915 Newton’s constant In the limit of test particles moving withnon-relativistic velocity ina staticandweak grav field Einstein → Newton (1/r 2 law explained!) ienew realms of gravity

18. 18GR 3 GR = field theory of gravitation 3 classical tests Grav redshift Bending of light Precession of planet orbit In relativity, space-dep → time-dep, GR → gravitational wave Indirect, but convincing, evidence thru decade-long observation of Hulse-Taylor binary pulse system Black Holes = full power and glory of GR Gravity so strong that even light cannot escape Role of space and time is reversed: lightcones tip over across the horizon Alas, Einstein never believed the reality of BH

19. 19 cosmo Cosmology In order to produce a static universe he found a way to introduce a grav repulsion in the form of the cosmology constant Λ Easier to interpret it as a vacuum energy: constant density and negative pressure → repulsion that increases w/ distance. – significant only on cosmological scale (Einstein 1917) The 1st paper on modern cosmology The universe = a phys system the constituent elements being galaxies Gravity the only relevant interaction GR = natural framework for cosmology Spatial homogeneity & isotropy (the cosmological principle) → Robertson-Walker metric : k, a(t) Λ = a great discovery key ingredient of modern cosmology Inflation theory of the big bang: a large Λ→ the universe underwent an explosive superluminal expansion in the earliest mo Einstein equation derivatives Expanding Universe Λ = dark energy → the U’s expansion to accelerate in the present epoch GRprovide theframework ! Still, Einstein missed the chance of its prediction before the discovery in late 1920’s The concordant ΛCDM cosmology

20. General relativity curved spacetime with moving basis vectors spacetime –dependent metric [g] = [g(x)] general coord transf = spacetime dependent Local symmetry 20 sym Einstein and the symmetry principle Before Einstein, symmetries were generally regarded as mathematical curiosities of great value to crystallographers, but hardly worthy to be included among the fundamental laws of physics. We now understand that a symmetry principle is not only an organizational device, but also a method to discover new dynamics. Differentiation results in a non-tensor Rotation symmetry Must replace by covariant differentiation Special relativity = Lorentz transformation 4-tensor eqns are auto relativistic Tensors have def transf property Tensor equations are automatically rotational symmetric. . . Spacetime-independent Global symmetry symmetry → dynamics

21. 21gauge1 Gauge principle: Regard ψ transf as more basic, as it can be gotten by changing U(1) from global to be local. brings in the compensating field A , the gauge field Einstein & unified field theory the last 30 years of his life , strong conviction: GR + ED → solving the quantum mystery? Was not directly fruitful, but his insight had fundamental influence on effort by others: Gauge theories and KK unification, etc. But both made sense only in modern QM Gauge invariance of electrodynamics E, B → A, Φ invariant under in quantum mechanics must + wf transf U(1) local transformation Given A , Maxwell derived by SR+gaugeie the simplest Electrodynamics as a gauge interaction Gauge principle can be used to extend consideration to other interactions History: Inspired by Einstein’s geometric GR 1919 H Weyl attempt GR+ED unification via Local scale symmetry [g’ (x)]= λ(x)[g(x)] Calling it eichinvarianz 1926 V Fock, after the advent of QM, discovered phase transf of ψ(x) F London: drop “i” is just Weyl transf Weyl still kept the name: gauge transf Transformation in the internal charge space “changing particle label” Such local symmetry in a charge space is now called gauge symmetry

22. 22gauge2 Straightforward extension of QED ? Particle physics Quantization and renormalization of Yang-Mills th extremely difficult. Furthermore, the truly relevant degrees of freedom for strongly interacting particle are hidden (quark confinement). Special relativity, photons, & Bose-Einstein statistics = key elements But Einstein did not work directly on any particle phys theory. Yet, the influence of his ideas had been of paramount importance to the successful creation of the Standard Model of particle physics The applicability of gauge sym to weak int was doubted because the symmetry itself is hidden (spontaneous sym breaking due to Higgs mech) 1970’s renaissance of QFT → SM’s triumph The Standard Model is a good example of a theory of principle: the gauge symmetry principle → dynamics, as well asa constructive theory : discoveries of * quarks and leptons, * the sym groups of SU(2)xU(1) & SU(3) follow from trial-and-error theoretical prepositions and experimental checks Symmetry principle as the guiding light. ED is a gauge interaction based on abelian (commutative) transf. 1954 CN Yang + R Mills extend it to non-abelian (non-commutative) Much richer, nonlinear theory, can describestrong & weak interactions SM is formulated in the framework of QM Holy grail of modern unification = [GR + QM]

23. 23KK Q: What is the charge space? What’s the origin of gauge symmetry? Kaluza-Klein theory unification of GR+Maxwell *Gauge transf = coord transf in extra D Internal charge space = extra D 1919Th Kaluza : 5D GR • extra dimension w/ a particular geometry [g]kk • GR5kk= GR4 + ED4 The Kaluza-Klein miracle! In physics , even a miracle requires an explanation 1926 O Klein explained in modern QM Foreshadowed modern unification theories. GR + SM • the compactified space = multi-dimensional • *Compactified extra D → a tower of KK states • the decoupling of heavy particles • simplifies the metric to [g]kk Einstein’s influence lives on!

24. 24 Summary Summary of a summary The fundamental nature of Einstein’s contribution illustrated by Planck unit system h -- c -- gN form an unit system of mass/length/time Natural units, not human construct Dimensions of a fundamental theory i.e. quantum gravity (GR + QM) Fundamental nature of these constants shown as conversion factors connecting disparate phenomena • All due to Einstein’s • essential contribution ! h: Wave & Particle c: Space & Time gN: Mass/energy & Geometry (QT) (SR) (GR)

25. These PowerPoint slides are posted@ www.umsl.edu/~chengt/einstein.html