General RelativityA2290-31 1General RelativityRelativity and AstrophysicsLecture 31Terry HerterA2290-31 General Relativity 2Outline Flickering Bulb Paradox General Relativity The Principal of Equivalence Accelerating Clocks Homework (due today/Monday) Problems 8-3, 8-25 Will hand back on Monday if you hand it in on todayPrelim Wednesday, Nov 18 Closed book and notes, will cover lectures 21 – 31 But some of this material depends on earlier lectures  Should know Will have both qualitative and quantitative questions Most equations will be provided (if needed)General RelativityA2290-31 2A2290-31 General Relativity 3Problem 6-7: Flickering Bulb Paradox  The current to a bulb is flows through a rail, a shuttle and a second rail. There is a “gap” in the rail. Both the rest length (Lo) of the gap and shuttle are the same. Will the bulb flicker? Rail Frame: Shuttle contract in direction of motion shrinks so we expect the light to flicker.  The time off will be the time to travel the gap (Lo– L) and the time for the signal to propagate (at speed vs) from B to A along the wire.soooffvLvLLt LLoABCDRail Frame2vm 2oL/oLL4/3vm 1LSuppose vs~ 1, then23/412 offt m 155.3A2290-31 General Relativity 4Flickering Bulb Paradox – 2  Shuttle Frame: It looks like we have continuous contact but the signal takes time to propagate back Let t′ = 0 be the configuration when BD contact and t′1is the time from BD contact until AC (C to reach A) de-contact Point A is moving away from event BD in this frame so that the time, t′2, for the signal to propagate back from B to A is (see above)  The time off is the difference between these two times.Shuttle Framev′sis signal speed in shuttle frame22tvLtvsvvLts2vLLvvLtosoff12tttoffABCDLoBD contactLo-L Signal travels at speed v′sfor a time t′2to get from B to A, but point A moves by a distance given by vt′2(a moving target).= v′st′2+ Lvt′2ABAvvLLto1LvABGeneral RelativityA2290-31 3A2290-31 General Relativity 5Flickering Bulb Paradox – 3  The “flicker” times in the two frames are: How are these related? We need to determine v′susing the Lorentz velocity transformation. Now: L = Lo/ and 2= 1/(1-v2) so that Factor out  and use Lo/2= (1-v2)Lo Lov2+ Lo/2= Lo ThusvLLvvLtosoffvvvvvsss1vLLvvvvvvvLtosssoff11vLLvvvvLoss211vLLvLvLooosoγ/γγ2γγ/γ/22vLLvLvLtooosooffγvLLvLtosooffoffofftt γAs we should have expected, flicker times are related by stretch factor (time dilation)γvLLvLtosooffsoooffvLvLLt soooffvLvLLt vLLvvvLtoossooffγ/1γ vLLvLvLooosoγ/-γγA2290-31 General Relativity 6General Relativity A Theory of Gravity Albert Einstein 1916 Incorporates accelerated motions into Special RelativityAlbert Einstein(1879 – 1955)General RelativityA2290-31 4A2290-31 General Relativity 7When you’re really famous …A2290-31 General Relativity 8Einstein’s Insight Newton’s law of gravity worked very well in predicting planetary motions. But Einstein wondered how gravity could be made consistent with Special Relativity. Einstein’s insight was the “Principal of Equivalence” He realized that a gravitational field would bend light rays. He also realized that Euclidean geometry would not apply.General RelativityA2290-31 5A2290-31 General Relativity 9Principle of Equivalence Gravity and acceleration due to a force are indistinguishable. In a small local environment (Must be a small enough “box”) This is the foundation of General Relativity.A2290-31 General Relativity 10Historical Background Proved a relationship between symmetries in physics and conservation principles (1915 or so).  This information was used by Einstein and is used in many areas of physics.  Noether was going to be a language teacher but became interested in mathematics.  David Hilbert and Felix Klein fought to get her on the faculty at University of Göttingen. The battle took four years, but she was finally appointed in 1919. She remained there until 1932 when the Nazis caused her dismissal because she was Jewish.  She accepted a visiting professorship at Bryn Mawr College and also lectured at the Institute for Advanced Study at Princeton. Emmy Noether(1882 – 1935)General RelativityA2290-31 6A2290-31 General Relativity 11Imagine yourself in a closed room. By the principle of equivalence you could not tell if you were on earth or in space in an accelerating rocket.GravityAcceleratingRocket shipA2290-31 General Relativity 12Now imagine yourself falling in a closed room. By the principle of equivalence you could not tell if you were falling towards earth or floating in space.FallingundergravityFloating in spaceGeneral RelativityA2290-31 7A2290-31 General Relativity 13Gravity and Time Imagine two clocks in an accelerating rocket. Clock A is in the front. Clock B is in the back. Clock A emits pulses 1 second apart. How far apart are they at Clock B?AcceleratingRocket shipABA2290-31 General Relativity 14Clocks in GRNO ACCELERATION Flash emitted from A To reach B it travels a distance d1. Since the rocket is not accelerating, we have for the next flashd2= d1 Flashes arrive one second apart.ABAB“A” emitsflash“B” receives flashd1ABABd2General RelativityA2290-31 8A2290-31 General Relativity 15Clocks in GRACCELERATING Flash emitted from A To reach B it travels a distance d1. Accelerating rocket rocket is traveling faster for next flash d2< d1 Flashes arrive less than 1 second apart.ABAB“A” emitsflash“B” receives flashd1ABABd2A2290-31 General Relativity 16Clocks in GR Clock A runs faster than clock B. The equivalence principle states Gravity and Acceleration are the same. Therefore, the same thing happens in a gravitational field! A clock on a mountain top will run faster than a clock at sea