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MIT Course 8.033, Fall 2007, Study Guide Prof. Enectal´ ı Figueroa-Feliciano La

MIT Course 8.033, Fall 2007, Study Guide Prof. Enectal´ ı Figueroa-Feliciano Last revised August 25, 2007. Math • Be able to do basic calculations with matrices (add, multiply, transpose, etc.). • Be able to solve variational calculus problems with the Euler-Lagrange equation. Special relativity basics & kine- matics Be able to solve problems that involve converting between diﬀerent intertial frames. Key concepts: • Invariance/symmetry under a transformation (translation, rotation, Galilean, Lorentz, etc.) • Aether, Michaelson-Morley experiment • The two postulates of special relativity • Event, inertial frame • Lorentz transformation, transforming a 4-vector • c, β, γ • 4-vectors: spacetime 4-vector, velocity 4-vector, wave 4-vector • Time dilation, length contraction, relativity of simultaneity, spacetime • Invariants: rest length, proper time interval • Event separations: spacelike, timelike & null • Velocity addition, aberration, Doppler shift • Numerical value of speed of light. Breadth • Key experimental evidence for special relativ- ity • Galilieo, Newton, Maxwell, Michaelson & Mor- ley, Lorentz, Emmy Noether, Einstein (if you can’t remember their key achievement(s) dis- cussed in class, Google them – same for the people listed below) Special relativity dynamics Be able to solve problems that involve acceleration and force. Examples: Rocket problems, particle accelerators. Key concepts: • Mass-energy uniﬁcation • 4-vectors: momentum 4-vector, acceleration 4-vector, force 4-vector • Rest mass invariant Particle physics Depth Be able to solve particle physics problems (colli- sions etc.) using energy-momentum conservation and these key concepts: • Conservation laws: energy-momentum, charge, lepton number, etc. • Rest mass, rest energy, kinetic energy, total energy, binding energy Breadth Basic familiarity with this: • Particles: electron, muon, proton, neutron, pion, neutrino, photon, graviton, nucleus, atom, hydrogen, deuterium, helium, carbon, iron, ion, molecule, alpha particle, gamma ray, the six quarks, the six leptons, baryon, fermion, boson, antiparticles • Ballpark masses of electron, proton, neutron. 1 • Terminology: atomic number, atomic weight, isotope, mass excess, • The four fundamental interactions, chemical reactions, nuclear reactions, elementary par- ticle reactions • Beta decay, Compton scattering, recoil, M¨ ossbauer eﬀect, Pound Rebka experiment • Particle accelerators: linear accelerator, cir- cular accelerator, ﬁxed-target experiment vs. col- liding beam experiment • Fusion in stars, Eddington • Qualitative knowledge of some key unsolved problems in particle physics Electromagnetism Be able to solve problems involving the Lorentz force law and transforming the electric and mag- netic ﬁelds between frames. • Lorentz force law • Lorentz transforming the electromagnetic ﬁeld • Current 4-vector • Electromagnetic ﬁeld from moving charge, re- tarded position • Concept: What’s meant by “E implies B” General Relativity Basics Be able to solve problems involving particle motion in various metrics (Minkowski, Newtonian, FRW and Schwarzschild). Depth • General concept of a metric, how to work with it. • How a metric transforms when you change coordinates. • For a spatial metric (2D, 3D), how to com- pute length of a given curve. • For a spacetime metric (4D), deﬁnition of time- like, spacelike and null curves through space- time. • How to compute ageing along a timelike curve. • How to compute proper length of a spacelike curve. • Deﬁnition of geodesic. • How do compute geodesics given a metric. • How to compute the trajectory of a massive particle. • How compute the trajectory of a photon. • How to compute gravitational redshift. Breadth • Deﬁnition of weak and strong equivalence prin- ciple. • Eﬀects of spatial curvature (on angles, “par- allel lines”). Eﬀects of spacetime curvature (can’t set up global inertial frame; no coordi- nate transformation gives Minkowski metric). • Experimental evidence for general relativity. Cosmology Depth • The FRW metric • Interpretation of FRW metric (spatial curva- ture, expansion, comoving objects, geodesics, cosmological redshift) • The Friedmann equation, how its solution de- pends qualitatively on the cosmologigical pa- rameters. • How ρ depends on a for various de density components. • Cosmological parameters: Ωγ, Ωm, Ωk, ΩΛ, Ωb, Ωd, h, their meaning • Age of the Universe, qualitative dependence on cosmological parameters. Breadth • Evidence for Big Bang • Rough timeline for the history of the Uni- verse (planck time, inﬂation, nucleosynthesis, recombination, ﬁrst stars, typical galaxy for- mation, formation of Earth, now, death of Sun, ultimate cosmic fate) and rough quali- tative understanding of these stages. 2 • Rough values of key cosmic distance scales (size of Earth’s orbit, size of solar system, distance to nearest star, size of Milky Way galaxy, distance to Andromeda galaxy, size of observable universe). • Rough measured values of the above-mentioned cosmological parameters. • Qualitative observational knowledge about the metric of our Universe from Science article (curvature, topology, expansion history, ﬂuc- tuation growth, black holes). • Qualitative knowledge of some key unsolved problems in Cosmology • Friedmann, Gamow, Hubble, Penzias & Wil- son Schwarzschild metric & black holes Depth Be able to solve problems using the Schwarzschild metric. • The Schwarzschild metric • Interpretation of the Schwarzschild metric (t- coordinate, r-coordinate, shell coordinates, grav- itational redshift, event horizon, Schwarzschild radius, event horizon) • Geodesics of the Schwarzschild metric: radial and angular, stable and unstable circular or- bits, computing general geodesics using the eﬀective potential, computing weak light de- ﬂection and Mercury perihelion shift, tidal forces • Deﬁnition of a black hole • Gravitational lensing, Einstein rings Breadth • Evidence that General Relativity is correct • Evidence that black holes exist (both stellar mass and supermassive) • How black holes probably form • No-hair theorem: the three properties of a black hole • Singularity • Hawking radiation • Falling into a black hole: what it feels like and what it looks like from afar • River model of black holes, sense in which nothing special happens at the event horizon • Time travel: possibilities for going forward and backward, wormholes • Taylor-Hulse binary neutron star system • Shapiro time delay • The GPS satellite system • Schwarzschild, Wheeler, Hawking • Qualitative knowledge of some key unsolved problems in general relativity 3 uploads/Finance/ guide - 2023-05-29T214827.440.pdf