Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of physis as desribed by observers that are moving relative to eah other To quantitatively desribe motion, we need to speify positions and times, whih is done with the use of oordinate systems To desribe the motion of an objet, the observer uses a oordinate system relative to whih she is at rest A oordinate system that is used to quantitatively speify the positions and times of events is alled a frame of referene (or referene frame) 2 An event is an ourrene, whih is ompletely speified by its position and the time at whih it ourred Clearly, the motion of an objet is a suession of events (the arrival of the objet at a position along its trajetory at a ertain time) The value of a field (eg, a magneti field) at a given loation and time is also an event Thus, physis an be desribed in terms of events Relativity desribes the relationship between events that are observed from different referene frames We note here that when we speak of "observing," we mean "making quantitative measurements" Thus, observers of events make measurements of quantities that haraterize the event (position, time, magneti field, et) Inertial referene frames (IRF) are fundamentally important in relativity theory In Newtonian mehanis, an IRF is one in whih Newton s first law is valid (We shall see an improved definition later) Thus, in an IRF, an objet that is ated upon by zero net fore will remain at rest or ontinue moving with onstant veloity Any referene frame that moves with onstant veloity relative to an established IRF is itself an IRF An IRF must be unaelerated beause in an aelerated referene frame, Newton's first law is violated [Show diagrams of aelerated train ar] In the most general ase, one referene frame aelerates relative to another The omplete desription of events under these irumstanes is the domain of the general theory of relativity General relativity turns out to be a theory of gravitation beause motion in a gravitational field is indistinguishable from motion in an appropriate aelerated referene frame In this ourse, we shall fous upon the speial theory of relativity, whih desribes physis as observed from inertial referene frames Experimental Underpinnings of Relativity Speial relativity supersedes Newtonian mehanis in the sense that it is more general than Newtonian mehanis, whih fails when speeds approah the speed of light in vauum, See Resnik (972) for a good disussion of relativity See also Benson's University Physis 2 See Six Ideas Unit R Cubi lattie with loks that are synhronized Modern Physis Leture Notes, Chapter Shand (//2) Page of 7
However, for speeds muh less than, whih orrespond to our normal experiene, Newton s laws beome an exellent approximation to the dynamial equations of speial relativity Thus, it is justifiable to use Newton s laws to desribe motion for speeds v << All of this has been experimentally verified The Mihelson-Morley experiment (to be studied later) was among the first to learly show a deviation from Newtonian expetations Other experiments sine then have shown the departure from Newtonian mehanis more dramatially [Show figure of kineti energy of eletron vs speed] Let us begin our disussion of relativity with a look at relativity from a lassial, Newtonian perspetive Rotated Coordinate Systems and Relativity Consider a point P whose position is speified by x-y oordinates (x P, y P ) Now rotate the oordinate systems about the origin so we have a new system x'- y' Obviously, the oordinates of P (x P ', y P ') are different in this system than in the x-y system How does this affet our desription of the motion? The oordinates are different, but do Newton s laws still hold? Yes! The position vetor of the point P is the same in both systems (Its magnitude and diretion [relative to, for example, the "fixed" stars] are idential in both systems) Displaements of P will also be the same Thus, its veloity in both frames will be the same and the aelerations will also be idential Rotation of the axes only hanges the omponents; it does not hange the physial quantities themselves It also makes sense intuitively If you drop a ball from a ertain height, you should get the same answer for the time to hit the ground regardless of the orientation of the spatial oordinate system This is beause Newton s laws are valid regardless of the orientation of the oordinate system as we have taitly shown many times in doing inlined plane problems Choosing a oordinate system or referene frame suh that x is parallel to the surfae of the inline simplifies the problem But the same answer is obtained if we hoose the x'- y' oordinate system, where x' is horizontal and y' is vertial Newton s laws do not hange under rotation and/or translation (displaement) of the oordinate system Galilean (Newtonian) Relativity y' y y x P x P ' y' S' P y P y P ' r x' x m Consider two inertial referene frames S and S', with S' moving at a onstant veloity v relative to S If Newton's laws of motion are valid in S, are they valid in S'? If so, then the laws would be valid in all inertial frames and therefore all inertial frames would be equivalent S O O' R r x x' Modern Physis Leture Notes, Chapter Shand (//2) Page 2 of 7
Consider the IRFs S and S' as shown in the diagram At some instant in time t, the position vetor of the objet as observed from frame S is r and the position vetor as observed from S' is r The origin O' of the frame S' is at a position R = vt relative the origin O of S At time t = t' = 0, the origins oinided Vetor algebra tells us that the position vetors are related thus: r = r R () Sine R = vt, Eq () beomes r = r vt (2) Eq (2) (along with the absolute-time assumption disussed below) is alled the Galilean Transformation Usually, it is written in omponent form, as we shall see later Now let us take the derivative of Eq (2) with respet to time In lassial physis, time is absolute, ie, t' = t Thus, if two loks are synhronized in one inertial frame, then the loks will be synhronized in all inertial frames Therefore, we have dr = d ( r vt ), (3) d t dt ie, u = u v (4) Eq (4) is alled the Galilean Veloity Transformation This is equivalent to the vetor addition of veloities Let us now take the time derivative of Eq (4) We find du = d ( u v ) (5) d t dt Beause v is onstant, we obtain a = a Thus, the aelerations are equal in both frames and therefore in all inertial frames The aeleration is said to be invariant in inertial referene frames Let us now assume that Newton's first law is valid in inertial frame S, whih is onsistent with experiment Thus, as observed in S, when no net fore ats on the objet, it moves with onstant veloity (u ) To simplify the disussion, we assume that the objet is totally isolated, ie, all fores ating on it are negligible From Eq (4), we dedue that the veloity of the objet as observed (measured) in S' is u = u v, whih is onstant beause v is onstant Therefore, Newton's first law is also valid in frame S' beause the objet also moves with onstant veloity as measured in that frame It follows that Newton's first law is valid in all inertial referene frames Next, we assume that Newton's seond law is valid in inertial frame S, as dedued from experiment Thus, in frame S, Fnet = ma (7) Let us further assume that the net fore is due to a position-dependent fore This is not very restritive beause ommonly enountered fores suh as gravity and the eletri fore are position-dependent These fores really depend on the relative position of one objet with respet to another with whih it interats Thus, the fores are a funtion of the differene in position of the two interating objets: F = F( r2 r ) We an alulate the orresponding relative position in S' by using Eq (2): (6) Modern Physis Leture Notes, Chapter Shand (//2) Page 3 of 7
r 2 r = ( r 2 + vt) ( r + vt) = r 2 r (8) Hene, the relative positions are equal; therefore, the fores must also be equal in both frames: F net = Fnet (9) The same onlusion would be obtained if the fore were dependent on the relative veloity or both relative position and relative veloity A fore that is dependent on absolute position or absolute veloity would not be the same in both frames; however, experiments are onsistent with the invariane of the fore (Note: frition and air resistane are really dependent on the relative veloity of two objets) What about the masses? Classial physis simply delares that the masses are equal: m = m (0) Thus, mass is an invariant in lassial physis Measurements made under normal irumstanes with equipment available in an undergraduate laboratory would onfirm this Finally, we have previously seen [Eq (6)] that the aelerations as measured in both frames are equal, ie, a = a Sine the fores are equal, the masses are equal, and the aelerations are equal, it follows that Newton's seond law must also be valid in S': F net = m a () If we assume Newton's third law is valid in S, then its validity in S' follows diretly from the validity of the seond law in both frames We onlude that Newton's laws are valid in all inertial referene frames and they have exatly the same form Sine the transformation of oordinates from one inertial frame to another is the Galilean transformation, we say that Newton's laws are ovariant under a Galilean transformation This is the formal expression of Galilean (or Newtonian) relativity Sine Newton's laws are exatly the same in all inertial frames, it follows that no mehanial experiment an distinguish one IRF from another If you were in a windowless soundproof rail ar, you ould not tell whether it was "at rest" or moving with a onstant non-zero veloity [Group disussion: pendulum in a uniformly moving or aelerating rail-ar] It should be noted that the lassial onservation laws for momentum and energy are also ovariant under a Galilean transformation, ie, the laws are valid in all inertial frames This follows from the fat that the onservation laws an be derived from Newton's laws, whih are ovariant as seen above Galilean Relativity and the Speed of Light The other great edifie of lassial physis is eletromagnetism One result of lassial eletromagnetism is that the speed of light in vauum () is the same in all diretions, independent of the S S' v Modern Physis Leture Notes, Chapter Shand (//2) Page 4 of 7
motion of the soure 3 One quikly sees that this is inonsistent with the Galilean veloity transformation u = u v sine the speed of light would depend on the relative motion of the referene frames (see figure) A related problem was the fat that light travels in a vauum Sine pre-20 th entury physiists believed that all waves needed a medium in whih to propagate, a medium alled the ether was invented for light to travel in The ether frame was therefore the unique frame in whih the speed of light was All frames moving relative to the ether would measure a speed of light different from in general The ether had distintly unphysial properties in that it had to have infinite elastiity and inertia, no mass, and was undetetable However, the ether onept persisted Experiments showed that the Earth had to travel through the ether (eg, experiments on aberration of starlight) [See Resnik; also, Anderson] This being the ase, one should then be able to detet the ether wind, ie, the motion of the Earth through the ether, by showing that light beams on Earth have different speeds depending upon the veloity of the Earth through the ether, as we saw above (see figure) Mihelson-Morley Experiment The most famous experiment to try to detet the motion of the Earth through the ether is the Mihelson-Morley experiment Albert Mihelson designed an interferometer that ould detet extremely small differenes in the path lengths traveled by interfering oherent light beams (The Mihelson interferometer will be used in the Modern Physis Laboratory ourse to measure extremely small differenes in wavelength) In the Mihelson interferometer, the half-silvered mirror (beam-splitter) partially transmits and partially reflets the beam from the soure along paths and 2 (See figure below) The split beams are refleted by mirrors M and M 2 and brought together at the beam-splitter where they are partially transmitted and partially refleted The overlapping beams may then be observed using a telesope or another imaging devie If 3 In general, Maxwell s equations are not ovariant under a Galilean transformation [See Tipler and Llewellyn (simple example); Griffiths; or Reitz, Milford, and Christy] Modern Physis Leture Notes, Chapter Shand (//2) Page 5 of 7
mirrors M and M 2 are nearly perpendiular, a set of nearly parallel interferene fringes will be observed in the telesope At the position of a bright fringe, the interfering beams are in phase At the position of a dark fringe, the beams are out of phase Changes in the path length (along the horizontal or vertial paths) result in the fringes moving aross the field of vision In the figure, v is the veloity of the interferometer (and Earth) relative to the ether To understand the Mihelson-Morley experiment and its ramifiations, we will alulate the phase differene between beams and 2 To do this, we need to alulate the total time beam spends along path and the total time beam 2 spends along path 2 The phase differene is related to the differene between these two times We need to alulate times beause the speed of light along path will be different from that along path 2 aording to the Galilean veloity transformation [The fringe that is seen at a partiular position depends on what phases (rest or trough or something in between) of the two wave trains arrive simultaneously at that position] Sine the path lengths shown are in the frame of the interferometer, we need to determine the veloity of the light relative to the interferometer We let the S frame be the ether frame and S' be the interferometer Then, aording to Eq (4), u = u v Sine the speed of light relative to the ether is, u = For path, when the beam is moving away from M, u = v and when the beam is toward M, u = + v Thus, the total time to traverse path (bak and forth) is l l 2 2l t = + = l 2 2 = 2 2 (2) v + v v v The total time to traverse path 2 an be easily alulated by reognizing that this situation is idential to that of a boat moving aross a river, perpendiular to the urrent The path of the light shown in the diagram above is the path traveled in the interferometer (toward mirror M 2 ) We need the speed of light along this path Reall that u = u v The vetor diagram to the right shows how the veloities add Using the Pythagorean theorem, the speed of the light relative to the 2 interferometer is u = v 2 By onstruting a veloity triangle similar to that shown, one finds that the speed of the light relative to the interferometer while traveling away from M 2 is the same as that while moving toward M 2 Thus, the total time taken to and from M 2 is given by l2 2l2 t2 = 2 = 2 2 2 2 v v (3) The differene in traversal times is 2 l2 l Δ t = t2 t = (4) 2 2 2 2 v v If this time differene at a partiular position equals a whole number of periods of the light wave, the wave trains will be in phase and a bright fringe will be seen at that position Now, let us rotate the entire apparatus by 90º so that paths and 2 essentially swith plaes Then the transit time differene in this ase is given by v u' u = Modern Physis Leture Notes, Chapter Shand (//2) Page 6 of 7
Δt r = t 2r t r = 2 l 2 v 2 l 2 v 2 2 (5) Thus, the rotation hanges the time differenes by Δt r Δt = Δt r = t 2r t r = 2 l 2 + l v 2 l 2 + l (6) 2 v 2 2 It seems reasonable to assume that the speed v of the Earth relative to the ether should be approximately equal to the orbital speed of the Earth around the Sun The orbital speed of the Earth is approximately 3 0 4 m/s and so v 2 2 0 8, whih is muh less than We therefore use the binomial expansion up to the first order term to approximate Eq (6): Δt r Δt 2(l 2 + l ) + v2 2 2 This orresponds to a shift in the number of fringes given by ΔN = Δt Δt r T = Δt r Δt λ / ( ) v 2 2 = (v2 2 ) λ = l + l 2 v2 (7) 2 ( l + l 2 ) (8) For a typial set up, l = l 2 = 0 m and λ = 500 nm, whih gives ΔN 04 This fringe shift is easily observable The experiment was repeated many times at different times of year (to aount for the differene in the Earth s orbital veloity), and no fringe shift has ever been observed (The preision was less than one-hundredth of a fringe) Thus, one is fored to onlude that the ether frame does not exist, sine the null result proves that the speed of light is the same in all diretions in a frame that supposedly moves relative to the ether Many more experiments support this onlusion Thus, Maxwell s eletromagneti theory is valid in all inertial frames, but is inonsistent with Galilean relativity How ould this ugly disrepany be retified? Einstein believed deeply that all physial laws should be invariant in all inertial referene frames This belief is the foundation of the speial theory of relativity, whih shows that Galilean relativity must be abandoned in order to obtain a theory that desribes mehanis and eletromagnetism onsistently Disuss modern Lorentz invariane experiments (See inside front binder over inserts, Physis Today, July 2004, p 40) Modern Physis Leture Notes, Chapter Shand (//2) Page 7 of 7