DO PHYSICS ONLINE. SPECIAL RELATIVITY Frames of Reference

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1 DO PHYSICS ONLINE SPACE SPECIAL RELATIVITY Frames of Referene Spae travel Apollo 11 spaeraft: Earth Moon v ~ 40x10 3 km.h -1 Voyager spaeraft: v ~ 60x10 3 km.h -1 (no sling shot effet) Ulysses spaeraft: v ~ 80x10 3 km.h -1 (eentri orbit around Sun sling shot around Jupiter) Assume a spaeraft that an travel at speed of 100x10 3 km.h -1. Then, the time to travel to different plaes in our Universe from Earth are: Destination Distane (km) Time Mars 57x days Nepture 4360x years Proxima Centauri (loset star) 40x x10 3 years Edge of our galaxy 40x x10 6 years Andromeda (losest galaxy) x x10 9 years With our urrent tehnologies where spaeraft ould have veloities ~ 100x10 3 km.h -1 it is possible to travel within our Solar System only, the possibility for us to travel far beyond the edges of our Solar System is zero. Even if we ould travel at speeds approahing the speed of light from a Newtonian point of view, travel to plaes far beyond our Solar System would not be feasible. Travelling to our nearest galaxy Andromeda (x10 18 km = x10 6 light years) at a speed (3x10 8 m.s -1 = 1.7x10 6 km.h -1 ) would take ~x10 6 years. Newton s view of the Universe turns out not to be orret. From a Newtonian point of view time and spae are absolute quantities. A better model for the working of the Universe are Einstein s theories of Speial Relativity and General Relativity. Spae and time are interonneted and different observes an get different measurements for time intervals and lengths. Applying the ideas of Relativity, it may be possible to make journeys to distane galaxies but not with our present propulsion tehnologies for spaeraft.

2 OVERVIEW OF SPECIAL RELATIVITY Relativity is the study of the relative motions of objets. Einstein s Theory of Relativity is one of the greatest intelletual ahievements of the 0th Century. Speial Relativity, developed by Einstein in 1905, deals with systems that are moving at onstant veloity (zero aeleration) with respet to eah other. General Relativity proposed in 1916 deals with systems that are aelerating with respet to eah other. Aether model for the transmission of light Mihelson-Morley experiment: Attempt to measure the relative veloity of the Earth through the aether Other models for transmissions of light Frames of referene: non-inertial and inertial frames of referene Priniple of relativity (qualitatively and quantitatively) Analyse and interpret some of Einstein s thought experiments involving mirrors and trains and disuss the relationship between thought and reality Signifiane of Einstein s assumption of the onstany of the speed of light If is onstant then spae and time beome relative t0 Time dilation tv v 1 Relativity of Simultaneity Length ontration (length standards are defined in terms of time in ontrast to the original metre standard) v Lv L0 1 m0 Mass dilation mv v 1 Equivalene between mass and energy E m Evidene to support Einstein s theory of speial relativity Impliations of mass inrease, time dilation and length ontration for spae travel

3 FRAMES OF REFERENCE What is the trajetory of a ball thrown into the air? This is a simple question. But what is the answer? Surprisingly, there is no unique answer to this question. The answer depends upon the person making the observation of the motion of the ball. One observer laimed that the ball travelled in a straight line path up then down. Another observed laimed the ball travelled in a paraboli ar. Who is orret? Both are orret desriptions. A desription of motion depends upon the frame of referene of the observer. This is what Einstein s Theory of Speial Relativity is all about. Differene observes in different frames of referene an measure different values for length, time intervals and mass. Is it day or night? it depends on the loation of the observer. Is the house on the right or left? it depends upon the diretion the observer is faing. Whih is bigger the bear or mouse? depends upon the viewing loation up or down? For us, it has not been easy to realise that the onept up and down are relative. We are inlined to use ommon sense and asribe absolute sense to onepts.

4 In the Theory of Speial Relative both spae and time are relative onepts and this idea is in onflit with our notions of spae and time based upon ommon sense. But what is meant by time and spae being relative? Consider observes A and B in two inertial frames of referene. They both make length interval and time interval measurements of the same event. Observer A reords the length interval as 13 and the time interval 456 while observer B measures the length interval as 31 and the time interval as 656. Who is orret? Both an be orret beause their measurements on the same event depend upon the relative veloity between the two inertial frames of referene. So to start out study of Speial Relativity we need to define learly the frame of referene for an observer. A system, almost always a oordinate system, used to desribe the position, and also the veloity and aeleration, of an objet is alled a frame of referene. If you are in a room, very most likely you use the walls, floor, and eiling of that room as a frame of referene to judge the motion of objets in the room. These walls and the floor and the eiling form planes that lie parallel to an x, y, z oordinate system whih one might imagine in the room. It is espeially easy to see this x, y, z oordinate system loated with its origin at one orner where the floor and two walls meet. For example, if a ball rolled aross the room, you would know that it was moving beause you would see its position hange relative to the floor or walls. One would, very most likely, believe that the walls of the room were standing still, and, sine the ball was hanging position against this bakdrop of still walls, one would think that the ball was in motion. In that way we would use the frame of referene of the room to determine the motion of the ball. In a like manner, if one was in an automobile, and a ball was rolling around on the floor, one would use the frame of referene of the interior of the automobile to judge the motion of the ball. The interior of the automobile is, after all, quite a bit like a small room. Although its walls, eiling, and floor are not as flat or retilinear as that of a normal room, one ould still imagine an x, y, z oordinate axes attahed to it somewhere, most likely the floor, and one ould use this x, y, z oordinate axes to loate the ball. Now, let us think of the automobile simply as a room that might be standing still or that may be in motion. When it is in motion, it might be travelling at a steady speed down a straight path; it might be speeding up or slowing down; it might be going around a orner. Therefore, a frame of referene an be in motion, and that motion an be of several different types. Anyone who has ever taken a ride in a ar has been in a moving frame of referene and has experiened the several different types of motion through whih this frame an move. Atually, frames of referenes are lassified into two types depending upon how they are moving. Those two types are alled inertial and non-inertial frames of referene. Inertial Frames of Referene An inertial frame of referene has a onstant veloity. That is, it is moving at a onstant speed in a straight line, or it is standing still. Understand that when something is standing still, it has a onstant veloity. Its veloity is onstantly zero meters per seond. To say that the veloity of a frame of referene is onstant is the same as saying that the frame is not aelerating. So, we ould define an inertial frame of referene to be a oordinate system whih is not aelerating. Suh a onstant veloity frame of referene is alled an inertial frame beause the law of inertia holds in it. That is, an objet whose position is judged from this frame will tend to resist hanges in its veloity; it obeys the law of inertia. An objet within this frame will not spontaneously hange its veloity. An objet within this frame will only hange its veloity if an atual non-zero net fore is applied to it. There are several ways to desribe an inertial frame. Here are a few desriptions: An inertial frame of referene is a frame of referene with onstant veloity. An inertial frame of referene is a non-aelerating frame of referene. An inertial frame of referene is a frame of referene in whih the law of inertia holds. An inertial frame of referene is a frame of referene in whih Newton's laws of motion hold. In an inertial frame of referene no fititious fores arise.

5 Non-Inertial Frames of Referene A non-inertial frame of referene does not have a onstant veloity. It is aelerating. There are several ways to imagine this motion: The frame ould be travelling in a straight line, but be speeding up or slowing down. The frame ould be travelling along a urved path at a steady speed. The frame ould be travelling along a urved path and also speeding up or slowing down. Suh an aelerating frame of referene is alled a non-inertial frame beause the law of inertia does not hold in it. That is, an objet whose position is judged from this frame will seem to spontaneously hange its veloity with no apparent non-zero net fore ating upon it. This ompletely violates the law of inertia and Newton's laws of motion, sine these laws laim that the only way an objet an hange its veloity is if an atual non-zero net fore is applied to the objet. Objets just do not start to move about here and there all on their own. There are several ways to desribe a non-inertial frame. Here are a few desriptions: A non-inertial frame of referene is a frame of referene with a hanging veloity. The veloity of a frame will hange if the frame speeds up, or slows down, or travels in a urved path. A non-inertial frame of referene is an aelerating frame of referene. A non-inertial frame of referene is a frame of referene in whih the law of inertia does not hold. A non-inertial frame of referene is a frame of referene in whih Newton's laws of motion do not hold. In a non-inertial frame of referene fititious fores arise. Anient Greek philosopher Aristotle - it was obvious that objets would assume a preferred state of rest unless some external fore propelled them - onepts of absolute spae and absolute time that is that both spae and time exist in their own right, independently of eah other and of other material things - possible to assign absolute values of position and time to events. Aristotle s work was held in suh high regard that it remained basially unhallenged until the end of the sixteenth entury, when Galileo showed that it was inorret. Galileo - motion must be relative motion involves displaements of objets relative to some referene system Priniple of Galilean Relativity: the laws of mehanis are the same for a body at rest and a body moving at onstant veloity. Isaa Newton - Laws of Motion and his Law of Universal Gravitation - only possible to determine the relative veloity of one referene frame with respet to another and not the absolute veloity of either frame - no preferred or absolute referene frame exists. The Priniple of Newtonian Relativity: the laws of mehanis must be the same in all inertial referene frames. Due to Galileo and Newton, the onept of absolute spae beame redundant sine there ould be no absolute referene frame with respet to whih mehanial measurements ould be made. However, Galileo and Newton retained the onept of absolute time, or the ability to establish that two events that happened at different loations ourred at the same time - if an observer in one referene frame observed two events at different loations as ourring simultaneously, then all observers in all referene frames would agree that the events were simultaneous. The Newtonian onept of the struture of spae and time remained unhallenged until the development of the eletromagneti theory in the nineteenth entury, prinipally by Mihael Faraday and James Clerk Maxwell. Maxwell showed that eletromagneti waves in a vauum ought to propagate at a speed of = 3x10 8 m.s -1 (the speed of light). To 19th Century physiists this presented a problem. If EM waves were supposed to propagate at this fixed speed, what was this speed measured relative to? How ould you measure it relative to a vauum? Newton had done away with the idea of an absolute referene frame. We now have to do away with the onept of absolute time. Note: An event is something that happens independently of the frame of referene, for example, a flash of lightning. An event an be haraterized in a Cartesian referene frame by stating its (x, y, z and t) oordinates.

6 AETHER MODEL FOR THE TRANSMISSION OF LIGHT Fig. 1. Classial piture for the speed of light. The speed of light is relative to the motion of the observer, and so the speed of light is +v or -v. But this is not orret. The orret answer, is that the person will measure the speed of light to be the onstant value and it does not matter how fast or slower they are approahing or reeding from the light beam or the speed of the light soure. It seemed inoneivable to 19 th Century physiists that light and other eletromagneti waves, in ontrast to all other kinds of waves, ould propagate without a medium. It seemed to be a logial step to postulate suh a medium, alled the aether (or ether), even though it was neessary to assume unusual properties for it, suh as zero density and perfet transpareny, to aount for its undetetability. This aether was assumed to fill all spae and to be the medium with respet to whih eletromagneti waves propagate with the speed. It followed, using Newtonian relativity, that an observer moving through the aether with veloity v would measure a veloity for a light beam of ( + v). If the aether exists, an observer on Earth should be able to measure hanges in the veloity of light due to the Earth s motion through the aether. The Mihelson-Morley experiment attempted to do just this. When 19th Century physiists seleted the aether as the medium for the propagation of eletromagneti waves they were merely borrowing and adapting an existing onept. The fat that ertain physial events propagate themselves through astronomi spae led long ago to the hypothesis that spae is not empty but is filled with an extremely fine substane, the aether, whih is the arrier or medium of these phenomena. Indeed the aether was proposed as the arrier of light in Rene Desartes Dioptris, whih in 1638 beame the first published sientifi work on optis. In this work, Desartes proposed that the aether was all-pervasive and made objets visible by transmitting a pressure from the objet to the observer s eye. Robert Hooke in 1667 developed pressure wave theories that allowed for the propagation of light. In these theories, luminous objets set up vibrations that were transmitted through the aether like sound waves through air. The Duthman Christiaan Huygens published a full theory on the wave nature of light in Aording to Huygens, light was an irregular series of shok waves that proeeded with great veloity through a ontinuous medium the luminiferous aether. This aether onsisted of minute elasti partiles uniformly ompressed together. The movement of light through the aether was not an atual transfer of these partiles but rather a ompression wave moving through the partiles. It was thought that the aether partiles were not paked in rows but were irregular in their orientation so that a disturbane at one partile would radiate out from it in all diretions In 1817 the Frenh engineer Augustine Fresnel and the English sientist Thomas Young independently dedued that light was a transverse wave motion. This required a rethink of the nature of the aether, whih until this time had been onsidered by most sientists to be a thin fluid of some kind. Transverse waves an only travel through solid media (or along the surfae of fluids). Clearly, the aether had to be a solid. The solid also had to be very rigid to allow for the high veloity at whih light travelled. Clearly, this posed a problem, sine suh a solid would offer great resistane to the motion of the planets and yet no suh resistane had been noted by astronomers. In 1845 George Stokes attempted to solve the dilemma by proposing that the aether ated like pith or wax whih is rigid for rapidly hanging fores but is fluid under the ation of fores applied over long periods of time. The fores that our in light vibrations hange extremely quikly ( times per seond) ompared with the

7 relatively slow proesses that our in planetary motions. Thus, the aether may funtion for light as an elasti solid but give way ompletely to the motions of the planets. In 1865 the great Sottish physiist James Clerk Maxwell published his theory of eletromagnetism, whih summarised the basi properties of eletriity and magnetism in four equations. Maxwell also dedued that light waves are eletromagneti waves and that all eletromagneti waves travelled at m.s -1 relative to the aether. The aether was now alled the eletromagneti aether rather than the luminiferous aether and beame a kind of absolute referene frame for eletromagneti phenomena. Fig.. Aether proposed as the medium for the propagation of eletromagneti waves. Classial onept the speed of light depends on the relative motion of the Earth through the aether. AETHER proposed medium for the propagation of eletromagneti waves Property of aether Evidene Fills spae, permeates all matter light travels everywhere Stationary light travels in straight lines Transparent an t see it Extremely low density an t be deteted Great elastiity medium must be elasti otherwise energy dissipated MICHELSON MORLEY EXPERIMENT In 1887, Albert Mihelson & Edward Morley performed a very areful experiment to measure the motion of the Earth relative to the aether and thereby demonstrate that the aether existed. Their method involved using the phenomenon of the interferene of light to detet small hanges in the speed of light due to the Earth s motion through the aether. The apparatus was mounted on a solid stone blok for stability and was floated in a bath of merury so that it ould be rotated smoothly about a entral axis - the Earth, together with the apparatus is assumed to be travelling through the aether with a uniform veloity v of about 30 km.s -1, equivalent to the Earth at rest with the aether streaming past it at a veloity v.

8 A beam of light from the soure S is split into two beams by a half-silvered mirror K, half of the beam travels from K to M1 and is then refleted bak to K - the other half refleted from K to M and then refleted from M bak to K. At K part of the beam from M1 is refleted to the observer O and part of the beam from M is transmitted to O. Fig. 3. Mihelson Morley Interferometer. The light from refleted by the two mirrors produes an interferene pattern at the loation of the observer. Although the mirrors M1 and M are the same distane from K, it is virtually impossible to have the distanes travelled by eah beam exatly equal, sine the wavelength of light is so small ompared with the dimensions of the apparatus. Thus, the two beams would arrive at O slightly out of phase and would produe an interferene pattern at O. There is a differene in the time taken by eah beam to traverse the apparatus and arrive at O, sine one beam travels aross the aether stream diretion while the other travels parallel and then anti-parallel to the aether stream diretion (see the note below). This differene in time taken for eah beam to arrive at O would also introdue a phase differene and would thus influene the interferene pattern. Now if the apparatus were to be rotated through 90 o, the phase differene due to the path differene of eah beam would not hange. However, as the diretion of the light beams varied with the diretion of flow of the aether, their relative veloities would alter and thus the differene in time required for eah beam to reah O would alter. This would result in a hange in the interferene pattern as the apparatus was rotated (hanges in the patterns of bright and dark fringes). NULL result: no shift in fringe pattern Expeted result: slight shift in position of fringes Fig. 4. Interferene patterns reorded at loation O. The pattern on the right was expeted where a fringe shift ours but a null result was obtained as shown by the interferene pattern on the left.

9 The Mihelson-Morley apparatus was apable of deteting a phase hange of as little as 1/100 of a fringe. The expeted phase hange was 4/10 of a fringe. However, no suh hange was observed - the result of the Mihelson-Morley experiment was that no motion of the Earth relative to the aether was deteted. Sine the experiment failed in its objetive, the result is alled a null result. After the original beam is split at K the half transmitted to M1 travels with veloity ( + v) relative to the stationary Earth, as it is travelling in the diretion of flow of the aether. When it is refleted from M1 it travels towards K with a veloity relative to the Earth of ( v) against the motion of the aether stream. Thus, the time taken for the total journey of this beam from K to M1 and bak again is: d d t1 v v However, the other beam travels with veloity towards M and then with the same speed in the opposite diretion away from M after refletion - the time for the total journey of the beam from K to M and bak again is v t vt t = v t +d d t = t t d u t = d / ( v ) t 1 and t learly different The Mihelson-Morley experiment is an exellent example of a ritial experiment in siene - the fat that no motion of the Earth relative to the aether was deteted suggested quite strongly that the aether hypothesis was inorret and that no aether (absolute) referene frame existed for eletromagneti phenomena this opened the way for a whole new way of thinking that was to be proposed by Albert Einstein in his Theory of Speial Relativity. The null result of the Mihelson-Morley experiment was suh a blow to the aether hypothesis in partiular and to theoretial physis in general that the experiment was repeated by many sientists over more than 50 years. A null result has always been obtained. QUESTIONS AND PROBLEMS VIEW ANSWERS How to answer a question: problem solving View periodi table (ited Aug 01) Numerial values for onstants and useful physial quantities P5950 P5886 P5950 Desribe an experiment that you ould perform in a referene frame to determine whether or not the frame was non-inertial. P5886 The idea of a universal aether was first proposed to explain the transmission of light through spae. Mihelson and Morley attempted to measure the speed of Earth through the aether. Evaluate the impat of the result of the Mihelson and Morley experiment on sientifi thinking.