Chapter 35. Special Theory of Relativity (1905)

Transcription

1 Chapter 35 Speial Theory of Relatiity (1905) 1. Postulates of the Speial Theory of Relatiity: A. The laws of physis are the same in all oordinate systems either at rest or moing at onstant eloity with respet to one another. B. The speed of light in auum has the same alue of 300,000 km/se regardless of the eloity of the obserer or the eloity of the soure emitting the light. All obserers who measure the speed of light will find that it has the same alue. In atual experiments, sientists hae always shown that the speed of light is always, no matter how fast we trael. Einstein s two postulates of the Speial Theory of Relatiity lead to: 1. slowing down of time in moing loks, and. ontration of length in referene frames moing with onstant eloity relatie to an obserer. 1

2 The first of Einstein s postulates of the Speial Theory of Relatiity implies that any physis experiment performed in a laboratory at rest must gie the same results when performed in a laboratory moing at onstant speed past the first one.!it is impossible to detet absolute motion!!! An inertial frame of referene is one for whih Newton s laws applies. An inertial frame of referene moes at onstant eloity.. Consequenes of the Postulates of the Speial Theory of Relatiity A. Simultaneity Two eents, whih are simultaneous when obsered from one frame of referene, are generally not simultaneous when obsered from a seond frame of

3 referene, whih is moing relatie to the first. Time is not uniersal as Newton thought. From the point of iew of the obserer who traels with the ompartment, light from the sourse traels equal distanes to both ends of the ompartment and therefore strikes boths ends simultaneously. The eents of light striking the front and bak ends of the ompartment are not simultaneous from the point of iew of an obserer in a different frame of referene. Beause of the ship s motion, light that strikes the bak of the ompartment doesn t hae as far to go and strikes sooner than light strikes the front of the ompartment. 3

4 B. Time Dilation The time interals between two eents as measured by two people on different frames of referenes are not always the same. Here we wish to derie a quantitatie relationship between different time interals in different oordinate systems. Consider an obserer in frame S!moing with onstant eloity to the right (along the ommon x!and x axes) relatie to an obserer in frame S. Let s ompare the time interals between two eents as measured by the two obserers. Let eent #1 be the emission of a flash of light from a light soure at point O " in the frame of referene S!. Eent # is when the flash of light returns to point O " in frame S!after haing been refleted from a mirror a distane d away. The person at rest in frame S sees the two eents our at two different points in spae, while the person in frame S! sees the eents our at the same point in spae. (From Young & Freedman, 13 th edition): 4

5 The proper time is defined as the time interal measured in the frame in whih the two eents ourred in the same plae. In the example aboe the proper time is " t # $ "t o. The result Einstein obtained is!t =!t o 1 " Suppose that in the aboe thought experiment, "t o =1 se and = 50% = 0.5. Then what is! t? The result is that! t = 1.15 se. Hene time has been dilated or expanded! The obserer inside the train (frame S!) laims that 1 se has passed but the obserer outside the train (frame S) laims that 1.15 se has passed! Thus the obserer outside the train (frame S) laims that the lok of the obserer in the train (frame S!) is running slow. See the results tabulated below for other alues of the relatie eloities: 5

6 "t o!t!t = o 1 " << 1 se 1 se = (50%) 1 se 1.15 se = (87%) 1 se se = (99.5%) 1 se 10 se Note: 1. The proper time interal is always the shorter time interal between the two eents.. A lok in motion runs more slowly than an idential stationary lok. 3. Note that time is absolute, i.e., "t o = "t only when <<. That s when both obserers will agree that the time interal between the two eents are the same. Sine we humans moe at eloities relatie to eah other (and see objets moing) with <<, then we always agree that the time interals are the same without any time dilation taking plae. Thus time dilation is a onept that is weird, anomalous, and alien, and ounter-intuitie to us. 4. Aording to the speial theory of relatiity, time an beat at different rates, depending on how fast one is moing through spae!!! 6

7 5. Time dilation has been onfirmed experimentally: using partile aelerators in 1971 when atomi loks were flown around the world in opposite diretions in ommerial jet flights. C. Length Contration The distane measured between two points depends on the frame of referene of the obserer. In general, the distane between two points measured by two people on different frames of referene are not the same. Consider measurement of the length of a rod. Let L o The length of a rod when the rod is at rest. This is also known as the rest length or the proper length of the rod. L The length of the same rod when it is moing at a eloity relatie to the obserer making the measurement. 7

8 The Lorentz ontration formula that relates the two measured lengths of the rod is L = Lo 1 Let s suppose that in a thought experiment, the rest length of a rod is L o = 1 meter. Thus, for different eloities between the two frames of referene, the length L of the rod measured by an obserer moing with respet to the rod at a eloity will be different. 8

9 In fat, L o L = Lo 1 << 1 meter 1 meter = (50%) 1 meter 0.87 meter = (87%) 1 meter 0.5 meter = 50 m = (99.5%) 1 meter 0.1 meter = 1 m Comments: 1. The rest length of an objet is longer than the length measured in any moing frame where the objet might be.. The length ontration takes plae only along the diretion of motion. 3. Lorentz thought that length ontration was a result of the atual matter ontrating. Einstein said it was a result of the spae itself ontrating. 4. Distane is absolute only when <<. That s when both obserers will agree that the length of the rod was L = L o = 1meter. We humans moe at eloities relatie to eah other (and see objets moing) with 9

10 <<, then we always agree that the distane between two points in spae is always the same, without any length ontration taking plae. Thus length ontration is a onept that is ounterintuitie to us. 3. Relatiisti Momentum In order to hae linear momentum onseration in all inertial referene frames at all speeds, then linear momentum must be defined for a partile of rest mass m o moing at speed as p 1 m o = m rel where the relatiisti mass of the partile is defined as m rel 1 m o Comments: 1. As the speed of an objet approahes the speed of light, the linear momentum of the objet approahes 10

11 infinity (beause the denominator approahes zero). No objet that ontains mass an be aelerated to the speed of light!. One ould say that as the speed of an objet approahes the speed of light, its mass approahes infinity! The faster the objet moes, the more massie it beomes, the more inertia it has, the more fore is required to aelerate it further loser to the speed of light. By the time the objet moes ery lose to the speed of light, its mass will be almost infinite thus requiring an infinite fore to push it so it moes at the speed of light. Sine you ll neer hae aess to an infinite fore, the partile will neer reah the speed of light! 4. Mass and Energy The total relatiisti energy E of a partile of rest mass m o moing with speed is E = m rel = m o 1 If the partile is at rest, its rest energy is Eo = mo. 11

12 This fundamental equation says that mass and energy are equialent. This is alled the unifiation of mass and energy. Beause of the fator = (3x10 8 ) = 9x10 16, een a small amount of mass m has an enormous energy ontent. What is the energy equialent of 1 kg of matter? E = m E = (1) (3x10 8 ) E = 9x10 16 Joules 5. The Correspondene Priniple The new theory (Einstein s theory of speial relatiity) must agree with the old theory (Newton s theory) where the old theory gies orret results. Note that t to = t = t o, if << 1 1

13 L = Lo 1 L = L o, if << p mo p = m o, if << 1 In essene, aording to the speial theory of relatiity, loks tik at different rates depending on how fast they are moing. Einstein onsidered time as a fourth dimension, meaning that time is intrinsially linked with moement in spae. Thus spae and time must be treated as two aspets of the same thing: spae-time. 13