Special and General Relativity

Transcription

1 9/16/009 Speial and General Relativity Inertial referene frame: a referene frame in whih an aeleration is the result of a fore. Examples of Inertial Referene Frames 1. This room. Experiment: Drop a ball. It aelerates downward at 9.8 m/s due to the fore of gravity.. The inside of a ar moving at onstant speed along a straight road. Repeat the experiment. Results are the same as in #1. 3. The inside of an elevator that is moving either upward or downward at onstant speed. Repeat the experiment. Results are the same as in #1 and #. 1

2 9/16/009 Examples of Non-inertial Referene Frames 1. The interior of a ar that is either speeding up, slowing down, or going around a urve. Experiment Drop a ball. If the ar is slowing down. The ball aelerates downward and towards the front of the ar. The aeleration toward the front of the ar is not due to a fore on the ball.. The inside of an elevator that is aelerating either upward or downward. Repeat the experiment. If the elevator is aelerating upward, the ball aelerates downward faster than 9.8 m/s. The additional downward aeleration is not due to a fore on the ball. Fundamental Assumptions of the Speial Theory of Relativity 1. The laws of physis have the same form in any inertial referene frame; in other words, no experiment an be done in an inertial referene frame to detet its state of motion.. The speed of light in vauum has the same value when measured by any observer, regardless of the observer s state of motion. Time Dilation as a Consequene of the Postulates Both O and O' (pronouned O prime ) are in inertial referene frames. O stands by the railroad trak and wathes O' move by at onstant veloity. Construt a simple lok in the railroad ar, onsisting of a mirror on the floor and another diretly above it on the eiling. et a pulse of light boune bak and forth between floor and eiling. Eah round trip is a tik of the lok. Referene Frame of Observer O' Moving at Constant Veloity Diretion of the Railroad Car s Constant Veloity Referene Frame of a Stationary Observer O

3 9/16/009 Time per Tik as Measured by O O' sees the light having only vertial motion. t up t down t = t up + t down t = + = the speed of light = m/s t = Time per Tik as Measured by O O sees the light moving horizontally as well as vertially. Use the Pythagorean theorem to alulate t up and t down tup = + v tup t down = t up v up = t t up = 1 v t up t down vt up vt 3

4 9/16/009 t = t up + t down t = 1 v t = t v This is the time dilation formula of speial relativity. The time per tik measured by the moving observer O' is shorter than the time per tik measured by the stationary observer O. Any lok at rest relative to O' is slower than any lok at rest relative to O. Time elapses more slowly for a moving observer. ength Contration Similar logi shows that when O measures the distane between two points on the train along the diretion of motion, he gets a smaller result than O' does. This is alled length ontration and its quantitative expression is the following equation. = v is the distane between two events as measured by someone at rest relative to the events, and ' is the distane measured by someone moving relative to them. Examples 1. Suppose that O and O' have idential meter stiks and that the speed of O' relative to O is 0.8 times the speed of light. When O measures the length of O' s meter stik, he finds that it is only 0.6 m long. ikewise, when O' measures Os meter stik, he finds that O s meter stik is only 0.6 m long.. A and db are 30 year old ldtwins. A stays on Earth while B travels in a spaeship at times the speed of light to a star 10 light years away and immediately returns after making some observations that take only a few hours. How old are the twins when B arrives bak at Earth? How far did B travel? A wathes B make a round trip of = 10 = 0 light years while moving at about 1 light year per year. For him, 0 years elapse (t = 0 years) and he is (30 + 0) = 50 years old. On the other hand, B experienes time dilation. For her, the elapsed time is t' = = 1 year. She is only 31 years old when she returns. While A wathes B make a 0 light year round trip, B finds herself traveling only ' = = 1 light year. 4

5 9/16/ Atomi Clok Experiment by C.O. Alley in 1975 t = 15 hours v = 104 m / s = 313 mph 7 Δ t = s measured. Δ = 7 t s predited by speial relativity. 4. Muon deay 6 lifetime in laboratory = 10 s 6 measured lifetime of osmi ray-generated muons = s 5. Dependene of Mass on Speed m = E= m m 0 v 6. Mass-energy Equivalene 0 Spaetime and General Relativity The examples on the previous slide illustrate the fat that the time between two events and the distanes between them are not absolute. They depend on the motion of the person who measures them. However, this doesn t mean that everything is relative. It is important to know what is absolute (independent of the observer). Physiists reognize that a ombination of spae and time alled spaetime is the same for all observers. For both Aand B, the spaetime interval between B s departure and return is zero. The spae-time of speial relativity is one in whih the shortest spaetime path between two points is a straight line. In a vauum, for example, the path of a ray of light is a straight line The presene of mass bends spae-time. The result is what we all gravity. A satellitein in orbit, for example, follows the urvature of spaetime. Physis that takes this into aount is alled general relativity. Aording to general relativity, gravity auses a time dilation similar to that whih is aused by motion. Gravity affets time by slowing it down; i.e., time elapses more slowly where gravity is strong. Gravity affets spaetime by bending it, strething it, and ompressing it. 5

6 9/16/009 oal Inertial Referene Frames A loal inertial referene frame is one that is falling freely and is small enough for tidal effets to be negligible. A planet or some other massive objet 3 a = g 1 is far from any massive objet and is aelerating. It is not a loal inertial referene frame Experiments in will show the effets of gravity. It is not a loal inertial referene frame. 1 a =-gg 3 is falling freely. It is a loal inertial referene frame if it is small enough that no experiment in 3 an show the effets of gravity. The Equivalene Priniple If is at rest in a region where there is a uniform gravitational field g and ' is far from any massive bodies but is undergoing an aeleration g, idential experiments in the two laboratories will give idential results. A loally uniform gravitational field is equivalent to a uniform aeleration No experiment onduted inside a suffiiently small laboratory an distinguish between a uniform gravitational field and a uniform aeleration of that laboratory. A a = -g B A g ' B aser beam from A to B in '. AB urved due to aeleration. aser beam from A to B in. Same result as in '. The effet of gravity is to bend spaetime. In a freely falling laboratory, AB would be a straight line. 6

7 9/16/009 The Bending of the Path of a ight Ray as it Passes a Massive Objet (Star, Neutron Star, Blak Hole) E C S The diagram shows Earth (E), a massive objet (C), and a distant star (S). C ould be a star, neutron star, or blak hole. The solid green line shows the path of a light ray from S. Instead of being a straight line as it would if C were not present, it is urved beause C bends the spae-time in its neighborhood. From Earth, the star appears to be loated in the diretion indiated by the dotted line. Animations showing what an astronaut would see as he approahes a neutron star or blak hole an be found at Double Einstein Rings A foreground galaxy ( about 3 billion light years away) bends spae around it to form a gravitational lens. As light from two more remote galaxies travels toward us, it follows the urvature of this spae to form the Einstein rings shown here. 7

8 9/16/009 Gravitational Time Dilation Time elapses more slowly in regions where gravity is strong than in regions where it is weak. Speifially, time elapses more slowly near a ompat stellar objet than it does far from it. This is expressed quantitatively by the equation at the right. t = t R s r t is the time interval between two events that take plae at distane r from the enter of the ompat objet as measured by an observer very far from the ompat objet. t' is the time interval between the same events as measured by an observer O' at distane r from the enter of the ompat objet. R s = the Shwartzshild radius = (3 km) M M = the mass of the ompat objet in solar masses. An indestrutible astronaut O' falling into a blak hole would find himself very quikly rossing the event horizon (time t'). As we wathed him, aording to the time dilation formula, we would never see him arrive at the event horizon beause as r approahes R s, t approahes infinity. Gravitational Red Shift Suppose that O' (the observer near the ompat objet sends light toward O at a frequeny f'. Beause of time dilation, the time between the peaks of the eletromagneti wave reeived by O is longer than the time between peaks of the wave when they were sent by O'. In other words, the period of the eletromagneti wave sent by O' is inreased as it travels to O. Sine frequeny is inversely proportional to the period, the frequeny of the wave reeived by O is lower than the frequeny sent by O' and the wavelength is longer. This is alled a gravitational redshift. The equation that desribes this redshift is λ= λ = λ. R s r wavelength of the light emitted near the ompat objet λ= wavelength reeived at Earth. Example If O' emits red light of wavelength 700 nm towards us, then when he is R s from the event horizon, the light we reeive would be infrared with a wavelength of 980 nm. 8

9 9/16/009 Confirmation of General Relativity Perihelion Preession Planet Observed Exess Rltiiti Relativisti Preession Predition Merury 43.11± Venus 8.4 ± Earth 5.0 ± Iarus 9.8 ± Defletion of starlight passing near the sun: Predition of general relativity: Best measurement: 1.66 ±