Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College

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1 Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College Propagation of waves through a medium As you ll reall from last semester, when the speed of sound is measured relative to the ground, it is faster for sound waves traveling downwind than upwind: The air is the medium whih arries the sound. The Mihaelson-Morley experiment In the late 1800 s, it was universally believed that light waves traveled through a medium as well; the medium was alled the ether. It was assumed that the ether was at rest with respet to the enter of the universe, or perhaps the enter of the galaxy. In 1887, A. A. Mihaelson and E. W. Morley set out to measure differenes in the speed of light aused by the motion of the earth relative to the ether: They made extremely preise measurements, and always found exatly the same value for the speed of light. The only logial onlusion was that light does not need a medium to travel through! It an travel through vauum! Another way of saying this is that the speed of light measured by any experimenter will always be the same, whether the experimenter is moving to the right, to the left, or is still. In fat, as we ve seen, Maxwell s equations show that the propagation of light is a basi form of eletromagnetism, whih propagates at a speed 1/ o o. Note that the speed of the observer doesn t appear in this equation, just as it doesn t appear in F = ma. So, in the same way that F = ma works in all onstant veloity (or inertial ) referene frames, 1/ o o works in all referene frames, i.e., light propagates with the same speed in all referene frames. (Again, this was shown by the Mihaelson-Morley experiment.)

2 The basi postulate of relativity: The laws of physis work equally well in all inertial referene frames. There is no preferred referene frame. This inludes the propagation of light, sine as we disussed above, light propagation is a onsequene of Maxwell s equations, and sine the Mihaelson-Morley experiment showed that light propagates at the same speed in a variety of referene frames. This postulate has immediate ounterintuitive onsequenes. For example, imagine two groups of observers. One group (S) is stationary, while the other group (S ) moves to the right at speed = 0.9 times the speed of light, i.e., = 0.9. One of the S observers turns on a flashlight, and the other S observers measure the speed at whih the wavefront propagates: The S observers, of ourse, measure a speed of for the wavefront. What speed do the S observers measure? In the way we re austomed to think, they would measure a speed of = 0.1. However, this is wrong. The light is propagating in the S frame as well as the S frame, so it must move with a speed of in S as well as in S! In fat, as we ll see, this strange way of adding veloities is not unique to light propagation. Similar effets our for any objet moving very lose to the speed of light. For example, we will show that if the observers in S throw a rok to the right with speed (measured in S) of 0.99, then the observers in S will measure a speed of 0.83 for the rok, instead of the speed of 0.09 that one might expet. (You an see that the effet is most extreme for the ase of something propagating at exatly, sine both sets of observers measure the same speed for it, despite their large relative veloities, while on the example we just did they measure similar but not idential veloities (0.99 in S and 0.83 in S ). The Four Basi Results There are four basi results from speial relativity. We ll derive them in the later setions of this doument, but first let s just look at them, and see how they fit together to form a self-onsistent piture: distane 1), so long as all three are measured in the same referene frame. You already know this one, time but it s reassuring to know that it still works, even when high speeds are involved. ) Time Dilation: tother tproper, where tproper is the time interval between two events as measured in the frame for whih both events our at the same plae, tother is the time interval between these two events as measured in some other referene frame, and (the Greek letter gamma ) is defined as 1. Note that when is written by hand, it looks like this:. Finally, is the veloity of the 1

3 3 other frame relative to the proper frame. For example, say I m moving to your right at onstant speed, and I throw a piee of halk into the air and then ath it. For me, the two events of 1) throwing the halk and ) athing it our at the same plae (at my hand), whereas for you they our in different plaes. Therefore, I measure tproper between these two events, and you measure tother. As we ll see eventually, is always less than or equal to, so is always greater than 1. This effet only beome easily notieable when is greater than about 0.1, as you an see from the plot of shown here. erifiation Of Time Dilation [This paragraph is taken from a textbook.] A striking onfirmation of time dilation was ahieved in 1971 by an experiment arried out by J.C. Hafele and R.E. Keating. They transported very preise esium-beam atomi loks around the world on ommerial jets. Sine the speed of a jet plane is onsiderably less than, the time-dilation effet is extremely small. However, the atomi loks were aurate to about 10 9 s, so that the effet ould be measured. The loks were in the air for 45 hours, and their times were ompared to referene atomi loks kept on earth. The experimental results revealed that, within experimental error, the readings on the loks on board the planes were different from those on earth by an amount that agreed with the predition of relativity. An espeially important example of time dilation is for two subsequent tiks of my wath (again, assume I m moving to your right at onstant speed). For me, the two tiks our at the same plae (on my wrist), so I measure tproper = 1 seond between tiks, and you measure tother. If 5, then you would measure 5 seonds between my tiks. You would say that my wath is running slow by a fator of 5. In fat, everything about me is moving slowly, inluding my heartbeat, the rate of hemial reations in my body, the rate of my aging, et. So, the Time Dilation effet an also be stated as, The other person s lok runs slow by a fator. The really odd thing about this is that, of ourse, it works both ways you see my wath running slow, but I see your wath running slow! This should seem paradoxial to you. We an only resolve the paradox by understanding effets 3 and 4 below, and then seeing how everything fits together. 3) Fitzgerald ontration: Lother, where L r is the rest length of an objet, i.e. the length of the objet in the frame for whih the objet is at rest, and L other is the length measured in another frame. (The fator is as defined before, where now is the veloity of the objet.) In words, this says that the length of an objet along the diretion of travel shrinks by a fator. 4) Synhronization: The hasing lok leads by tsynh. Imagine that frame S is moving to your right at speed. The people in S have two loks, with a separation as measured by them of L r. They have arefully synhronized these loks. However, this equation says that, to you the loks are mis-synhronized, so that, at any instant, the one on the left (whih appears to hase the one on the right) reads a time tsynh greater than the one on the right. This is the effet whih most people find the most onfusing.

4 4 However, speial relativity is only internally onsistent when you inlude this effet. In partiular, you an only resolve the time dilation paradox mentioned above by thinking about this synhronization effet. How the four effets fit together to resolve the time dilation paradox Again, let s assume that people in frame S are stationary, while those in S move to the right at speed. (Of ourse, from the equally-valid point of view of the folks in S, they re the ones that are stationary, and the people in S are moving to the left at speed.) How an it be that people in S see the loks in S running slow, but people in S see the loks in S running slow? The best way to answer this is with a numerial example. Sine things will be moving fast, we need to have large distanes, so we ll measure distane in units of light seonds. One light seond is the distane that light travels in one seond, and is written 1 s. One of the nie things about expressing distanes in these units is the way the fators of anel out. For example, to find the time it takes light to travel one light distane 1s seond: time 1s, where in the last step we simply anel the in the numerator and veloity denominator. In frame S, whih travels to the right at speed, let observer A be 15 s to the left of O (as measured in S ). There is a single observer O in frame S. All three observers have digital loks with readouts in seonds, as shown by the square boxes in the drawings on the next page. There are two events of interest: 1) when O passes O and ) when A passes O. When two observers pass by eah other (e.g. when A passes O), they both agree on what their loks read, but their explanations for these readings are quite different as we ll see. We arrange things so that, when O passes O both their loks 3 read 0. The two loks in S are synhronized (as seen by observers in S ). We ll pik, whih gives You an pretty well see from the drawing how things fit together, but if you like I ll walk you through it in the following paragraphs. Let s start with the upper left drawing, whih shows the situation as seen in S at event 1 (when O passes O ). We ve arranged things so that the loks of O and O both read 0 for this event. Sine the loks in S are synhronized, the lok of A reads the same as that of O, i.e. it reads 0. Moving down to the lower left drawing, we wait for O to move from O to A. O must over a distane of 15 s, and is moving at a speed of 3/5, so this takes 5 s. Thus, for event (when O passes A ), the loks of A and O both read 5 s. The lok of O started out (at event 1) reading 0, and it runs slow by a fator. 5 s Thus, during the 5 s that have elapsed, it only tiks off 0 s, giving a final reading of s = 0 s. The upper right drawing shows the situation as seen in S for event 1. The loks of O and O both read 0, as we ve arranged. However, A has the hasing lok, so it leads the lok of O by tsynh = 9 s. The distane between A and O is Fitzgerald-ontrated to Lother = 1 s. Moving down to the lower right drawing, we wait for A to move to O. A must over a distane of 1 s, and is moving at a speed of 3/5, so this takes 0 s. Thus, for event (when A passes O), the lok of O reads 0s. The lok of A started out (at event 1) reading 9 s, and it runs slow by a fator. Thus, during the 0 s 0 s that have elapsed, it only tiks off an additional 16 s, giving a final reading of (9 s) + (16 s) = 5 s. (The lok of O also advanes by 16 s during the interval between events 1 and.) Note that, for event, everyone agrees what is shown on the loks of A and O, but their explanations are quite different. The observers in S agree that the lok of A shows a larger reading than that of O, but only

5 5 beause it was mis-synhronized so that it started with a reading of 9 s instead of 0 s they say that the lok of A runs slow, sine it showed only 16 s elapsing, when the atual time interval between the two events was 0 s. On the other hand, the people in S say that the atual time interval was 5 s, and that the lok of O only shows a reading of 0 s beause it runs slow.

6 6 Derivations of the four fundamental effets distane 1). This is really just the definition of veloity. time ) Time Dilation The light lok Part of the reason that veloities lose to don t add in the way we expet is that time is pereived differently in S and S. To investigate this, we ll use an unusual lok, the light lok : A devie sends out a flash of light whih travels upward, bounes off a small mirror, and then returns to a detetor, whih is at the same position as the flash unit. (For graphial larity, the detetor is shown just to the right of the flash unit.) As soon as this detetor sees the refleted light flash, it triggers another flash. Eah of these yles is one tik of the light lok. Let s put one of these light loks in the moving frame S. We ll show that the rate at whih this lok tiks, as pereived by the observers in S, depends on, the relative veloity between S and S. Light loks vs. ordinary loks It s important to realize that the results we ll get are not limited to light loks; any other lok in S would display exatly the same variation. To see this, assume that the person in S is initially at rest relative to S. She has a onventional lok, whih she adjusts so that it has the same tik rate as the light lok. Now she starts moving. Sine there is no preferred referene frame (by the basi postulate of relativity), there should be no way for her to tell that she was previously stationary and is now moving, rather than the other way around. For example, sine the regular lok and the light lok were synhronized when she was stationary, they should remain synhronized now that she is moving. We ould make a similar argument using her heartbeat. If there are a ertain number of tiks of the lightlok per heartbeat when she is stationary, there must be the same number when she is moving. It is still possible that the rate at whih all these loks tik (the light lok, the regular lok, and her heart) might vary in unison, as seen by the people in S. All that the person in S an tell is that the loks remain synhronized. Sine we ould make these arguments using a hemial reation or any other time-dependent phenomenon instead of her heartbeat, we see the results we will derive for the variation in the tik rate of the light lok in S (as seen by the people in S) are not limited to the behavior of the light lok itself, but are atually statements about the way time itself is passing in S (as seen by the people in S).

7 Derivation of time dilation First, let s think about the path followed by the light, as seen in S and then as seen in S. The situation is very similar to a person walking at onstant speed who throws a ball straight up (as seen by the walking person) into the air, and then athes it. To the person who is walking, the ball goes straight up and then straight down. However, to a stationary observer, the ball follows 7

8 8 As we ll see eventually, is always less than or equal to, so is always greater than 1. This equation says that the time between tiks as measured in S is greater than the time between tiks as measured in S by a fator of! The faster S is moving, the greater the size of this effet. Let s assume that when everyone is at rest, their hearts beat at the same rate. Sine the dilation of time derived above applies to all loks in S, inluding the heartbeat of the person in S, this means that, as measured by the people in S. there is a longer time between the heartbeats of the person in S than between their own heartbeats, and the faster she moves the longer this time beomes. Thus (aording to the people in S), the person is S is aging more slowly than they are! Proper time We just showed that t = t However, this seems to ontradit the fundamental postulate of relativity, sine the equation is not symmetrial between the two referene frames; the time interval as measured in S is longer than that measured in S, and the faster S goes, the more dramati this effet beomes. However, there is something about the experiment with the light lok itself whih makes a fundamental distintion between the two referene frames. The time interval t represents the time interval between two events: the first event is the flash, and the seond is the reeption of the flash. The fundamental distintion between the referene frames is that in S these two events our at the same plae (beause the flash unit and the detetor are in the same plae), while in S they our at different plaes. Thus, if we instead did the experiment with the light lok in S (instead of S ) then the roles of the two referene frames would be reversed, and we would find t = t, i.e. that the time as measured in S between the two events is longer than the time as measured in S. So, the two referene frames really are equally good, it just depends on how we do the experiment. We note that the shorter time is always measured in the referene frame in whih the two events our at the same plae: When the lok was in S, the shorter time was measured in S, whereas when the lok was in S, the shorter time was measured in S. We define the proper time between two events to be the time as measured in the referene frame in whih the two events our at the same plae. With this definition, we an summarize all suh experiments with a single equation: tother t proper

9 9 where tproper is the proper time (from now on, we ll simply write it as t p ), and tother is the time as measured in some other referene frame. For example, if we do the experiment with the light lok in S, then both events (the flash and the reeption) our at the same plae in S, so and tp t and tother t. However, if we do the experiment with the light lok in S then both events (the flash and the reeption) our at the same plae in S, so t t and t t p other.

10 10 4) Synhronization Let s return to the setup from page 4, but this time we ll keep all the variables in symbol form, rather than using partiular numeri values. As before, we set things up so that the spaing between A and O is L r and the loks of A and O are synhronized, as seen in frame S. Also, we arrange things so that the loks of O and O both read 0 at the moment that O passes O. Finally, now A has a green flashbulb. As shown in the top right piture on the next page, as seen in S, the bulb goes off at the instant that O L r passes O, i.e. at t = 0. Also, as seen in S, the spaing between A and O is Fitzgerald ontrated to. Therefore, the distane along the x-axis from the point of the flash to O is equal to -- we use this again in the bottom piture of the right olumn. In S, of ourse, the loks of both O and O read 0 at this instant, but observers in S see that the lok of A reads a value greater than zero. We ll all this reading tf -- this is the time displayed on the lok of A when she sets off her flashbulb, and this is our desired unknown. For a lassial situation, we would have t f = 0, so that the loks of A and O would appear synhronized even in S, but we ll show that t f 0. As shown in the top left piture, as seen in S the loks of A and O are synhronized. Thus, when O passes O, whih happens when the lok of O reads 0, the lok of A also reads zero. Assuming for the moment that I m right in asserting that t f 0, this means that A has not yet set off the flash, and that in S the flash is set off after O passes O, as shown in the seond piture down on the left. Continuing down the left olumn of pitures, the flash spreads out at speed from A, and eventually it passes O at a time (as measured in S ) that we ll all t. Beause the light had to travel a distane L r to reah O, she an infer that A must have set off the flash at L t t r f (1) Now let s go bak to the right olumn. After the flash is set off, the light must travel away from the flash point at speed. Thus, it spreads out in irles that are entered on the point of origin of the flash, as shown in the middle piture on the right, rather than being entered on A, who is moving to the right. As shown in the bottom piture on the right, eventually the flash athes up with O. This happens at time t, where the subsript stands for ath up. During the time from t = 0 to t = t, the light travels a distane t, while O travels a distane t. As shown in the figure, t t t () Now, during the time interval between t = 0 and t = t, the observers in S have seen that the lok of O runs slow by a fator. Therefore, the time it now shows is t t. (3) Substituting this into (1) gives L L t t 1 r r f Applying this result to the top piture in the right olumn, we see that indeed, as seen from S, the hasing lok (i.e. the lok of A ) leads by L r /.

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12 1 Spaetime Coordinate Transformations (There s a good Star Trek phrase for you!) It an be hallenging to orretly take all four effets of relativity into effet for a given problem. It would be muh easier if we ould have a simple set of equations, whih, given the spaetime oordinates (i.e. given x, y, z, and t) for an event (suh as a flashbulb going off) in one referene frame would allow us to easily find the oordinates in any other referene frame. This set of equations is alled the Lorentz transformation / The Galilean Transformation For now, let s return to the intuitive arena where things move at speeds muh less than, and examine how we an express oordinates of an event as measured in S in terms of the oordinates as measures in S. By event I mean simply something with well-defined x, y and z oordinates and a well-defined time when it happens. Good examples inlude an explosion, a ollision, and a partiular tik on a partiular lok. Let s say an event ours at oordinates x, y, z, and t in the S referene frame. (Again, S moves to the right with speed relative to S.) What are the oordinates x, y, z and t of the event as measured by the observers in x? We assume as we will for the remainder of our treatment of relativity that the origins (x = y = z = 0) of the two referene frames oinide at t = t = 0 (as measured by loks at the origins). We lose no generality by doing this, sine we an always hoose where t = 0. Beause of the synhronization, we immediately have that t = t. Let s start with an easy ase: x = y = z = 0, i.e., the event ours at the origin of S at time t. Sine S moves to the right with speed, at time t = t, the S origin is at x = t = t, y = 0, z = 0. Now onsider an event that ours someplae else in S, at oordinates x, y, z, t. This is really just like the ase we just onsidered, exept now there is an offset relative to the S origin of x, y, z : These relations between the event oordinates x, y, z, t as measured on S and those as measured on S is alled the Galilean transformation. It s really nothing new, but just a formal way of writing what you already understand about things moving at relatively small speeds. The Galilean eloity Transformation A simple onsequene of the Galilean transformation is the veloity addition rule whih you re used to, as we ll show here. We onsider not just a single event whih ours at x, y, z, t, but rather an objet whih is moving, i.e., its oordinates x, y, and z depend on time. The omponents of its veloity, as measured in the

13 13 dx dy dz S frame, are found as usual by taking the derivatives with respet to time: u x uy uz. By dt dt dt taking the derivative of the Galilean transformation with respet to t, we an find the relationship between these veloities (measured in S ) and those measured in S: dx dx ux x x t d dt dt dt dy dy y y uy dt dt z z dz dz uz dt dt dx dx Sine t = t, we have that dt = dt, so u x, et. Substituting this into the above gives the Galilean dt dt veloity transformation: u u x x uy uy uz uz This should make intuitive sense to you, but as we ve just disussed, it doesn t work when the speeds involved are lose to. The Lorentz Transformation On the next assignment, you will use the four basi effets of relativity to show that the fully orret version of the Galilean transformation (the one that is orret both at low speeds and at speeds lose to ) is This is alled the Lorentz Transformation. x x' t' y y' z z' t t' x' / The Inverse Lorentz Transformation It is easy to find the inverse transform: S and S are ompletely symmetrial, exept that, in S the frame S is traveling with veloity all we do is interhange primed and unprimed variable, and substitute for : x' ( x t) y' y z' z t' ( t x / )

14 14 Reovery of Galilean Transformation at Low Speeds We know that the Galilean transformation works quite well when things are moving at reasonable speeds, so we need to show that the Lorentz transformation is onsistent with this. Reall that transform redues to: 1 1 /. For, this beomes 1 1. So, for, the Lorentz 1 0 x ( x' t') y y' z z' t ( t x' ) 1 x x t ' y y' z = z' t t' x' 0 sine 0 So, we indeed reover the Galilean transformation. In other words, the Galilean transformation, as in all of Newtonian physis, is a speial ase of the Lorentz transformation.