Special Relativity. Relativity

Transcription

1 10/17/01 Speial Relativity Leture 17 Relativity There is no absolute motion. Everything is relative. Suppose two people are alone in spae and traveling towards one another As measured by the Doppler shift! Whih one is moving? They an t tell! 1

2 10/17/01 Example: A train is moving at 65 mph relative to the traks. If the people inside the train annot see out and the trak is very smooth, they an not tell they are moving! The earth moves around the sun at 30 km/se. Can you tell? Postulates: Speial Theory of Relativity Albert Einstein (1905) 1. The speed of light is the same to all observers, irrespetive of their motion.. The laws of physis are the same everywhere no matter what the speed of the observer.

3 10/17/01 Things don t add up the way the used to. For instane, a boater throws a ball with veloity v ball (as seen by the boater): v ball v boat Standing on the shore we see the ball moving with veloity = v boat + v ball Addition of Veloities The Lorentz Transformation V1 V V (1,) VV 1 1 When V V, then : 1 V (1,) 1 3

4 10/17/01 The speed of light is onstant. But this doesn t happen with a beam of light! Flashlight v boat The veloity of light as seen from on shore is still. Results of the Theory Addition of Veloities 4

5 10/17/01 Results of the Theory Addition of Veloities SIMULTANEITY Two events simultaneous in one referene frame are not simultaneous in another frame moving relative to the first. 5

6 10/17/01 Speial Theory of Relativity Fration of Speed of Light (Relative Veloity) Length Contration Fator Mass Inrease Fator Time Dilation Fator TACHYONS There one was a lady alled Bright, Who ould travel faster than light. She went out one day, In a relative way, And ame bak the previous night! 6

7 10/17/01 Preditions of Speial Relativity Measurements of TIME LENGTH MASS depend of the veloity of the observer. With inreasing speed: Lengths shrink Masses inrease Cloks slow down How do we add veloities? The old law of Galileo and Newton was V V V TOTAL 1 If V 1 = (flashlight) and V = V boat, then V TOTAL V boat Wrong!! Can t have V >. We need a new way of adding veloities. 7

8 10/17/01 The Lorentz Transformation A new law for the addition of veloities VV TOTAL V TOTAL V1 V V1V 1 If V 1 = (flashlight) and V = V boat, then V V V V V V V V V V Another example A C B Person C sees A and B moving at 0.9. How fast does A think B is moving? V TOTAL

9 10/17/01 Simultaneity The simultaneity of events is in the eye of the beholder. v A light bulb in the enter of a high speed train flashes. An observer on the train sees the light reah the front and bak simultaneously From outside the train To the outside observer the light reahes the bak first. The bak moves towards the light and the front away. Not simultaneous! Flash goes off Flash reahes bak wall first v v 9

10 10/17/01 Preditions of speial relativity Measurement of TIME LENGTH MASS depend on the veloity of the observer. As veloity inreases - Lengths shrink (ontrat) - Masses inrease - Cloks slow down Time Dilation Consider a train moving with a veloity, v d Mirrors v Inside the train is a lok whih onsists of two mirrors separated by a distane d with a light beam bouning bak and forth. Every time the photon hits a mirror, we get a tik of the lok. 10

11 10/17/01... Time Dilation To a person sitting on the train, the time between tiks is: t = d / d What does a person outside the train see?... Time Dilation To the person outside the train, the distane between the mirrors will have hanged! d d vt If t is the time between tiks as seen by the person outside, then the mirror will have moved a distane vt. 11

12 10/17/01... Time Dilation Then or d' d v t' t' t v t' solving for t t t' 1 v / d d vt d = t and d = t So we have: t' t 1 v t = time measured by a person on the train t = time measured by a person beside the train... Time Dilation / Sine (1-v / ) < 1, the time between tiks is greater for the person beside the train. The lok on the train appears to run slower to the person outside!!! In fat, it does run slower deaying partiles take longer to deay when they move a high speeds relative to the lab. 1

13 Time (se) 10/17/01 The Triky Part?... Time Dilation On the train, a Timex must keep the same time as the photon lok. Relativity says that the person on the train an t tell she is moving. the Timex must slow down too. Same for a Rolex, a CD player, her heart beat (!), et. Otherwise physis would be different (you ould pull all the shades on the train and still tell you were moving but relative to what!) Everything slows down!!! 10 1 Time Dilation Veloity 0.6 Curve shows aging of moving objet That is, time passes more slowly At v =, the lok appears to stop!!! 13

14 Length (1 meter ruler) 10/17/01 Length Contration (Can be derived similarly) Consider now two photon loks (1 and below) oriented perpendiular to one another on the train. d 1 d v To an observer on the train d = d. What about an outside observer? 1 1 Length Contration L L 1 v / o 1/ Veloity 0.6 Lengths shrink in the diretion of motion L o = rest length, L = observed length At v =, L = 0!!! 14

15 Mass (1 kg initially) 10/17/ Mass Inrease mo m 1 v / Veloity 0.6 Mass inreases with veloity. m o = rest mass, m = observed mass At v =, the mass is infinite!!! Energy and Mass Einstein s formula E an now be written as mo E 1 v / m Problem for interstellar travel Getting hit by a dust partile would be dangerous if traveling at high speeds. 15

16 10/17/01 The Twin Paradox One twin travels away from the Earth at high speed for a long time. Returning to Earth she finds herself muh younger than her brother! How? Both twins think eah other s lok slows down, so what is going on? No warp drive The view from the roket Suppose one twin is on a roket ship traveling at 0.99 to a star 5 lyr away. She sees the distane ontrated to: L 5 1 (0.9) / lyr = 3.53 lyr She omputes her time to get there and bak as x(3.53 lyr / 0.99) = 7.13 years 16

17 10/17/01 The view from Earth The twin on Earth sees the distane as still 5 lyr, so he omputes her time to go out and bak as: x 5 lyr / 0.99 = years So he ages years, while his twin sister ages only 7.13 years!!! Faster than light? No normal matter an travel faster than the speed of light! mo m 1 v / Tahyons - Theoretial partiles that always travel faster than the speed of light Allowed in Speial Relativity Time travel paradoxes 17

18 10/17/01 What you should know about SR Postulates The speed of light is the same to all observers, irrespetive of their motion. The laws of physis are the same everywhere no matter what the speed of the observer. Some onsequenes Veloities add so that v <. Simultaneity of events depends on observer Physial Results Time dilation, length ontration, mass inrease with inreasing veloity 18