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Transcript
GTR Continued Sections 29.3 - 29.4 Professor David Marx Reminders • Lab this week LAB B1-WA Wave Addition: due by 4 p.m. Friday • Examination #3 on Thursday: – Addresses Chapters 8, 9, 11-13 only – 29 multiple-choice questions (one 2-pt question – 30 points possible (120% maximum) – Professor Khondkar Karim will proctor exam. • Don’t forget Weekly Reflection #12 due 11/18 • Remember In-class Quiz #6 moved to 11/20 Einstein, GTR, and the Universe • Einstein’s equation of general relativity: 1 Rmn - gmn R = 8p GTmn 2 • The equation required an expanding universe, so Einstein arbitrarily added a “cosmological” constant to avoid this conclusion… • Recall that galactic nebulas had yet to be discovered as galaxies, and neither had the expanding universe! Friedmann’s Universe 1 1 2 GMm • Starts with total energy: E = mv 2 r • Makes 3 simplifying assumptions: – Universe is essentially empty (on a larger scale) – Universe is isotropic (no preferred direction) – Universe is homogeneous (no preferred point) • and gets: 2E æ v ö 8p G =ç ÷ r 2 3 mR è R ø 2 Friedmann’s Universe 2 • Inserting E=mc2 and taking into account GTR, the prior equation results in: 2 2 æ v ö 8p G kc = r ç ÷ 3 R2 è Rø • where k is the curvature of the universe: – If k < 0, then universe is closed and bounded (positively curved). – If k > 0, then the universe is open and unbounded (negatively curved). – If k = 0, then the universe is flat. Friedmann’s Universe • Solving this equation, Friedmann predicted that the universe had to be expanding if it were to exist. • Einstein added the “cosmological constant” to his own field equations because he could not believe in an expanding universe. Later Einstein remarked, “That was the greatest scientific blunder that I ever made.” Hubble’s Relationship 1 • Following in the footsteps of others, Hubble determines recessional velocities from Doppler shifts. Dl v = l c Hubble’s Relationship 2 • Hubble determines distances to “nebulas” using a variety of means: – Study of variable stars whose periods are related to their intrinsic brightness – Angular diameters by type – Presence of brightest stars in a galaxy by type – Presence of globular star clusters – Presence of supernovas by type (I and II) Hubble’s Law for Galaxies • Hubble used Doppler shifts and distances to study galaxies. He found a linear relationship. v = HR Abbé Georges Lamaître • Introduces the “Big Bang” • Concludes that time since universal expansion began could be found from Hubble parameter. • v = HR; now d/v = t; • v/R = H and d/v = 1/H • The current value of H (21-23km/s/Mly) shows the age of the universe to around 13.8 billion years. Hubble’s Law is consistent with the expansion of the universe.