1Phy107 Fall 2006 1ExamHour Exam 2: Wednesday, October 25th• In-class, covering waves, electromagnetism, and relativity• Twenty multiple-choice questions• Will cover: Chapters 8, 9 10 and 11Lecture material• You should bring– 1 page notes, written single sided– #2 Pencil and a Calculator– Review Monday October 23rd– Review test will be available online on MondayPhy107 Fall 2006 2From before• The total energy of a particle is dependent on it’skinetic energy and its mass.• Even when the particle is not moving it has energy.• Mass is another form of energy– Moreover, energy can show up as mass.– The energy to put all the protons together in a nucleusgives the nucleus more mass!E =moc2, or E = mc2E =moc2= KErel+ moc2Phy107 Fall 2006 3Space/Time - Energy/Momentum• Relativity mixes up space and time - also energy andmomentum– When converting from one inertial frame to another– In the time dilation and length contraction formulastime is in the length formula and length is in the time volumethrough the velocity (length/time)– In the total energy formula momentum(or kinetic energy) and massenergy are related• There are combinations of space/time and energy/momentum thatobservers in any inertial frame will measure the as the same– For energy and momentum this invariant says that all observers canagree on mass an object has when it’s at rest!E2 c2p2= moc2()2x2 c2t2Phy107 Fall 2006 4Observing from a new frame• In relativity these eventswill look different inreference frame movingat some velocity• The new referenceframe can berepresented as sameevents along differentcoordinate axes• A graphical way ofshowing that length andtime are contracted orexpanded.Coordinates inoriginal frameCoordinates innew framect’x’xctNew frame movingrelative to originalPhy107 Fall 2006 5The Equivalence Principle• Led Einstein to postulate theEquivalence PrincipleClip fromEinstein NovaspecialPhy107 Fall 2006 6Equivalence principleAccelerating referenceframes areindistinguishablefrom a gravitationalforce2Phy107 Fall 2006 7Try some experimentsConstant velocityCannot do any experiment to distinguishaccelerating frame from gravitational fieldt=0 t=tot=2toFloor accelerates upward to meet ballConstant accel.t=0 t=tot=2toPhy107 Fall 2006 8Light follows the same pathPath of light beam inour frameVelocity = vt=0Velocity = v+atot=toPath of light beamin acceleratingframeVelocity = v+2atot=2toPhy107 Fall 2006 9Is light bent by gravity?• If we can’t distinguish an acceleratingreference frame from gravity…• and light bends in an accelerating referenceframe…• then light must bend in a gravitational fieldBut light doesn’t have any mass.How can gravity affect light?Maybe we are confusedabout what a straight line isPhy107 Fall 2006 10Which of these is a straight line?A. AB. BC. CD. All of themABCPhy107 Fall 2006 11Straight is shortest distance• They are the shortest distances determined bywrapping string around a globe. On a globe, theyare called ‘great circles’. In general, geodesics.• This can be a general definition of straight,and is in fact an intuitive one on curved surfaces• It is the one Einstein used for the path of allobjects in curved space-time• The confusion comes in when you don’t know youare on a curved surface.Phy107 Fall 2006 12Mass and curvature• General relativity saysthat any mass will givespace-time a curvature• Motion of objects inspace-time is determinedby that curvature• Similar distortions tothose we saw when wedrew graphs in specialrelativity3Phy107 Fall 2006 13Idea behind geometric theory• Matter bends space and time.• Bending on a two-dimensionalsurface is characterized by theradius of curvature: r• Einstein deduced that 1/r2 isproportional to the the localenergy and momentum density• The proportionality constant is• where G is Newton's constant8Gc2Phy107 Fall 2006 14A test of General Relativity• Can test to see if the path of light appearscurved to us• Local massive object is the sun• Can observe apparent position of stars withand without the sun• But need to block glare from sunPhy107 Fall 2006 15Eddington and the Total Eclipse of 1919Apparent position of starActual positionof starMeasure this angle to beabout 1.75 arcsecondsPhy107 Fall 2006 16Eddington’s Eclipse Expedition 1919• Eddington, British astronomer, wentto Principe Island in the Gulf ofGuinea to observe solar eclipse.• After months of drought, it waspouring rain on the day of the eclipse• Clouds parted just in time, they tookphotographic plates showing thelocation of stars near the sun.• Analysis of the photographs back inthe UK produced a deflection inagreement with the GR predictionPhy107 Fall 2006 17Space is Curved• Einstein said to picture gravity as a warp in space• Kepler’s Laws canall be explained bymovement aroundthese “puckers”• Everything moving isaffected, regardlessof massPhy107 Fall 2006 18Other Consequences of GR• Time dilation from gravity effects• Gravitational Radiation!– Created when big gravity sources are movedaround quickly– Similar to the electromagnetic waves that werecaused by moving electron charges quickly• Black Holes• Expanding Universe (although Einstein missedthe chance to predict it!)4Phy107 Fall 2006 19Gravitational time dilation• Gravity warps both space and time!• At 10,000 km above the Earth’s surface, aclock should run 4.5 parts in 1010faster than one on the Earth• Comparing timing pulses from atomicoscillator clocks confirms the gravitationaltime dilation in 1976 to within 0.01%.• Corrections are now standard in thesynchronizing satellites• This correction needed in addition to thespecial relativity correction for GPSPhy107 Fall 2006 20Gravitational radiation• When a mass is moved, the curvature ofspace-time changes• If a mass is oscillated, ripples of space-timecurvature carry the signal• Gravitational radiation carries energy andmomentum and wiggles mass in its pathPhy107 Fall 2006 21Evidence for gravity waves• In 1974, Joseph Taylor and his student RussellHulse discovered a binary neutron star systemlosing energy as expected from gravitationalradiationPhy107 Fall 2006 22Direct detection of gravity wavesLIGO is a collection of large laser interferometers searchingfor gravity waves generated by exploding stars or collidingblack holesPhy107 Fall 2006 23The big bang• In 1929 Observation of nearby and far awaygalaxies indicate
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