1General Relativity(sort of)GravityUpwardsacceleration, no gravity. Falling dueto gravityNo gravityEquivalence Principle:• Can’t tell difference between gravity & acceleration• …or between freefall & no gravity.•Soany experiment should give same answer in either case.Gravity = Curved spaceObjects follow shortest distance through curved space(-time).The equivalence principle (plus a vigorous waving of one’s hands)shows…• Curved path of light in gravitational field[CO fig. 17.10]2The equivalence principle (plus a vigorous waving of one’s hands)shows…• Gravitational Redshift• Gravitational time dilationνoEquiv. Principle Î photon frequency unchanged in free-falling lab.Why doesn’t the frequency meter see a blueshift?There must be a counteracting Gravitational Redshift:νov = velocityThis redshift is seen by the meter whichis not free-falling. The exact result: Integrate the effect out to infinite distance:Special RelativityThe Lorentz TransformationCO, pg. 90(4.16)(4.17)(4.18)(4.19)3Time Dilation in Special Relativity“Light Clock”velocity = distance / timespeed of light = D / timeAs seen by moving observerAs seen by stationary observerDirection of motionFrom the Feynman Lectures4Proper time interval along world line is smaller than distance covered along time axis!Euclidean geometry, for comparison.Figures from Taylor & Wheeler, “SpacetimePhysics”5What curves into where?Objects follow shortest distance through curved space(-time).6Paths of Objects through Curved Space-time• Geodesic = straightest possible worldline• = maximum value of ∫ds in most examples.• But actually = extremum (max or min).• Special Relativity:• Objects with no forces acting on them take straightest path through space–time.• Conservation of Energy-Momentum. • General Relativity: • Objects follow Geodesics through curved space-time.• This follows directly from the equivalence principle.• Light follows null geodesic• ds =0 at every point along path• Î ∫ds = 0• Î d(length)/dt = cDoes the Schwarzschild metric actually tell us anything?Now for some incredibly vigorous hand waving….• The orbit of a satelliteSchwarzschild metric:Assume:• Circular orbit Î dr = 0• In plane where dφ= 0• Moving at constant angular velocity ωÎ dθ= ωdtA familiar Newtonian result!Find r that makes ∆s be an
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