nearingzero

1. http://www.nearingzero.net

2. Time Dilation Length Contraction Relativity and Electricity & Magnetism “If you are out to describe the truth, leave elegance to the tailor.”—A. Einstein

3. Fall 2003 Digression: on the constancy of c. Recent research suggested that c may not be constant. Several researchers in Australia have been studying light absorbed by distant gas clouds about 12 billion years ago. The fine structure in spectral lines (the spacing of multiple lines close together) depends on the fine structure constant: In this formula, e is the charge of an electron, 0 is a constant you encountered in Physics 24, c is the speed of light, and h is another constant we will learn about soon. I’ll skip this digression. You can read the notes yourself, if you want.

4. Everything in the formula for  is a constant.  seems to have changed by 0.001% in 12000000000 years. That’s a change of 0.00000000000008% per year. The data obtained by the Australian group suggest that  has a larger value now than it did when the light they observed was emitted. If  has been increasing over time, then either the charge on an electron has been increasing, or 0, h, or c have been decreasing. According to a commentary* put out by the American Physical Society… *So it is an admittedly biased opinion of the commentary author.

5. “Since the effect on the laws of physics of increasing the electronic charge are too awful to contemplate, they figure light is going slower. That kills relativity, but my mail indicates nobody but physicists believe that stuff anyway.”—Bob Park The Australian researchers have reported on this work three times in recent years, including 2001 in Physical Review Letters* and August 2002 in Nature.** It seems like I am constantly hearing “new” reports of findings that c is decreasing, but it is all essentially this one group reporting their work as it progresses. *Arguably the most prestigious US physics journal. **The most prestigious science journal known to man.

6. To date no one else has reproduced this result. Some possibilities:  The Australians* could have made a mistake (unlikely?).  Their results could be a statistical fluke (unlikely).  A yet-undiscovered systematic error could have influenced their results.  The interpretation could be wrong.  They are correct.  ??? This is important enough that others will be investigating carefully. We should know the results within a few years. *Another Australian group disputes the necessity for c to have changed. See http://eprints.anu.edu.au/archive/00000797/00/Bicknell_Scott.pdf. Oh, and I’m not picking on Australians. They are as smart as we are.

7. What if they are right… Theories you will learn in this class have superseded theories you learned earlier (e.g., relativity will supersede Newtonian mechanics). You should not think of the earlier theories as being “wrong.” Rather, the new theories are better, and incorporate the old ones within them. You will learn that relativistic mechanics reduces to Newtonian mechanics in the limit of small relative velocities. So use the simpler Newtonian mechanics when the error introduced is small. Use relativity only if you must!

8. So if they are right…  There will be profound implications for cosmological theories.  Someone will have to re-think special relativity. Someone will have to come up with a new theory which incorporates all of special relativity but goes beyond it to include the slowly-changing value of c.  This may have profound implications for mankind (as did special relativity). It may not. We’ll see.  Newtonian mechanics will still work just fine as long as velocities are not too big.  Lots of physicists will have nice jobs for a long time to come. "If we knew what it was we were doing, it would not be called research, would it?“—A. Einstein

9. Fall 2004 Update: on the constancy of c. “New studies, conducted using the UVES spectrograph on Kueyen, one of the 8.2-m telescopes of ESO's Very Large Telescope array at Paranal (Chile), secured new data with unprecedented quality. These data, combined with a very careful analysis, have provided the strongest astronomical constraints to date on the possible variation of the fine structure constant. They show that, contrary to previous claims, no evidence exist for assuming a time variation of this fundamental constant.” (http://www.eso.org/outreach/press-rel/pr-2004/pr-05-04.html) This web page cites journal articles on which the above claim is based: http://www.sciencenews.org/articles/20040508/note10ref.asp

10. Getting back on topic…let’s consider another problem that time dilation helps us solve. Has anyone here ever felt a muon? Does anybody even know what a muon is? A muon is an elementary particle with a mass 207 times that of an electron, and a charge of either +e or –e. Muons are created in abundance at altitudes of 6 km* or more when cosmic rays collide with nuclei in the atmosphere. Fortunately, muons interact only very weakly with matter, which is why it is OK that many of them are passing through your body right now. *This is in the upper reaches of the troposphere, the part of the atmosphere in which we live.

11. Muons travel with speeds of about 0.998 c (fast!) and have an average lifetime of 2.2 s (2.2x10-6 s). How far can an average muon travel during its lifetime? d = v t d = 0.998 · 3·108· 2.2·10-6 = 0.66 km. How can muons get through the 6 or more kilometers of atmosphere between their birthplace and us if they only live long enough to travel 0.66 km? OK, some will go more than 0.66 km, and some less, but mostly not by very much. So the question stands.

12. Time dilation! I say the muon’s clock ticks slow. I say that while the muon thinks* its clock ticks 2.2 s, I observe that it actually ticks During this time the muon travels a distance d = 0.998 · 3·108 · 34.8·10-6 = 10.4 km, so the average muon will reach me before decaying. *Of course, a muon doesn’t “think” anything, but we use words like that to help us form a mental image of the process. If you prefer, imagine a nano-human riding on the muon and reporting what he/she sees.

13. Double-check: what is the event, who is the observer, and who measures the proper time. The event is the muon “living.” The event does not take place at a single location in my reference frame, so I measure the dilated time, and the calculation was correct.

14. One important aspect of relativity is that there is only one reality. If I see the muon arrive at the surface of the earth, the muon must agree that it actually did arrive at the surface of the earth. Our average muon “says” there is no doubt whatsoever that its lifetime is 2.2 s, and during that time it travels 0.66 km.I say the muon reaches the surface of the earth. The muon says it doesn’t?? “I thought you said time dilation would help us solve the muon problem.” We seem to have created a new problem. Either we have encountered two different realities, or else there is… “Relativity teaches us the connection between the different descriptions of one and the same reality.”—A. Einstein

15. Length Contraction If two observers in relative motion measure different times for an identical event, what makes us think they should measure the same lengths for an identical object? The formula for length contraction is not terribly difficult to derive. I’ll lend you a book if you are curious. Here is the formula. The Proper Length, L0, of an object is its length as measured in its own rest frame. “The faster you go, the shorter you are.”—A. Einstein

16. An observer measuring the length of an object moving relative to him will measure a length L less than the length L0 he would measure if he were not moving relative to the object. Let me demonstrate length contraction using a meter stick… The length contraction occurs only along the direction of relative motion. A spacecraft moving past an observer at nearly the speed of light will seem to be very short in length and normal diameter.

17. A muon created at an altitude of 10.4 km would say that during its lifetime it saw an atmosphere of length I say the muon gets to earth because its lifetime is longer. The muon says it gets to earth because the atmosphere is shorter. Different descriptions of the same reality. Be careful when you talk about the lifetime of a particle moving with v close to c. You need to specify the reference frame in which the lifetime is measured!

18. A B A 20 light years The Twin Paradox A and B are 20 year old twins. A travels on a spaceship at v = 0.8c to a star 20 light years* away and returns. *A light year, y, is the distance light travels in one year. Thus, y = (1 year)·(c). If D is a distance expressed in light years, then the number of years it takes to travel that distance at a speed of v is found from time = (distance) / velocity. Thus: time in years = (distance in light years) / (velocity expressed as a fraction of c).

19. A B A 20 light years B, left behind on earth, says the trip takes 2·20/0.8 = 50 years. B is 70 years old when A returns. B also observes that A’s clock (which is identical to B’s) ticks slowly, and records less time. If the event in question is the ticking of A’s clock, then the 50 years calculated above is the dilated time t (why?).

20. The proper time, which in this case is amount of time recorded by a clock in the spacecraft, is found by solving our time OSE for t0: According to B (who was left back on earth), A’s clock only ticked 30 years, so that A is 20+ 30 = 50 years old on return to earth. At the end of the trip, B, left behind, is 70 years old. A, who made the trip, is 50 years old. Can this be possible? Yes! Absolutely! and it was verified experimentally in the jets-around-the-world experiment mentioned earlier.

21. Now here’s the paradox. A moving clock ticks slower. This applies to all observers. A, on the spacecraft, sees B move away and then come back.* A says B’s clock ticks slower. A does the calculation presented on the last slide and concludes that at the end of the trip, B is 50 and A is 70. That’s the famous twin paradox. It would appear that each twin rightfully claims the other aged less. Have we discovered an example of the existence of two different, mutually exclusive realities? *Remember, there is no absolute reference frame for specifying motion. Motion is relative! An observer is free to say “I am at rest; you are the one moving!”

22. When you encounter a paradox like this you can be sure that someone has pulled a fast one on you. In this case, an unwarranted calculation was made. Special relativity applies only to observers in inertial (non-accelerated) reference frames. A had to accelerate (very rapidly) to leave earth and get up to speed, and again when turning around to head home, and a third time when landing on earth. A is not allowed to use the equations of special relativity! B is, and B’s calculation is correct: A comes back 20 years younger.

23. If you examine the problem carefully, it’s only the turning around part that causes A trouble. What’s poor A to do? Doesn’t a moving clock tick slower? Yes, so evidently during A’s period of extreme acceleration, B’s clock (as observed by A) would tick incredibly fast. Isn’t A allowed to use the laws of physics? Yes, but it would have to be general relativity. We won’t have completely eliminated the paradox unless we can find a description for A’s reality that agrees with B’s reality.

24. A B 12 light years A 20 light years A, in the spacecraft, needs to reconsider the distance traveled. During the “out” portion of the trip, A will say that the actual distance traveled was and that the back portion was also 12 light years. 24 light years at a speed of 0.8 c takes 30 years so A ages 30 years during the trip, and comes back at age 50.

25. B tells A “you are younger because your clock ticked slower.” A says “I am younger because the trip covered less distance than you thought.” Same reality, two different descriptions. There are a number of famous paradoxes* based on relativistic calculations. Typically, someone makes an invalid calculation (usually on purpose, to see if they can trick you). *In another famous problem, where a very fast runner tries to put a 10 meter pole in a 5 meter barn, a paradox arises because…

26. Electricity and Magnetism The material we’ve been studying is fascinating and thought-provoking, but it is not how Einstein’s theory of relativity came into being. “What led me more or less directly to the special theory of relativity was the conviction that the electromagnetic force acting on a body in motion in a magnetic field was nothing else but an electric field.”—A. Einstein. In other words, Einstein believed that what you and I might call a magnetic force is really just an electric force in another inertial reference frame.

27. - + + + + - - - - + + + - - - - - + + + + - Consider a conducting wire and a positive test charge. + What force does the test charge “feel” due to the charges in the wire?

28. - + + + + - - - - + + + - - - - - + + + + - Attraction, because there is a – closest to the test +? No net charge inside the conductor. No electric field outside the conductor. + No force!

29. - - - - + + + + - - - - - - - - + + + - - - -` - - - - + + + + - + What does the test charge see when an electric field is applied and current flows? E The test charge “observes” that the space between the moving electrons is contracted. There are more electrons in the part of the conductor nearest the test charge!

30. The test charge “observes” that the moving electrons are closer together than the stationary protons, and therefore "feels" a Coulomb attraction. A human observer is unable to see the electrons, and attributes the attraction to a “magnetic force” generated by the moving charges. Same reality, two different descriptions! And both descriptions are mildly troublesome, as we will see shortly… Beiser’s presentation of this material is different, but equivalent.

31. If you think about it, this presentation is bothersome. To illustrate, I need to talk about conservation and invariance. A quantity is relativistically invariant if it has the same value in all inertial frames of reference. The speed of light is relativistically invariant. Time is not relativistically invariant. Length is not relativistically invariant. Electric charge is relativistically invariant.* *All observers agree on the total amount of charge in a system. A quantity is conserved if it has the same value before and after some event. Don’t confuse conservation with invariance.

32. It is a fact that electric charge is both conserved and relativistically invariant. Our thought experiment with the conductor and test charge suggests that a conductor which is electrically neutral in one reference frame might not be electrically neutral in another. How can we reconcile this with charge invariance? Our modern physics textbook author claims there is no problem, because you have to consider the entire circuit. Current in one part of the circuit will be balanced by opposite current in another part. Although the explanation is correct, I don’t find it satisfying.* Maybe the pole-in-barn paradox will help us understand. *It seems logical that if moving electrons are closer together in one part of the circuit, they ought to be closer in other parts of the circuit too, so that the conductor is no longer neutral and charge is not conserved.

33. The Pole-Barn Paradox A speedy runner carrying a 10-meter pole approaches a barn that is 5 meters long (short barn!), with open doors at each end. A farmer stands nearby, where he can see both front and back door at the same time. a) How fast does the runner have to go for the farmer to observe that the pole fits entirely in the barn? b) What will the runner observe?

34. The answer to a) involves a simple length contraction* calculation. For the pole to fit in the barn, the farmer must measure a contracted length L = 5 m for the pole of proper length L0 = 10 m. The result is v = 0.866 c. If the runner is going that fast, or faster, the farmer observes the pole to fit inside the barn. *Length contraction is often called the Lorentz contraction, named after the scientist who discovered the mathematical transformations which lead to the equation for length contraction.

35. The answer to b) (what will the runner observe?) starts with another length contraction calculation. The runner is moving … …no, the runner isn’t moving. The runner sees the barn moving towards him at a speed of v = 0.866 c. The runner says the speeding barn has a length equal to The pole can’t possibly fit inside the barn.

36. How do we explain this paradox? Which observation is the physical reality? The answer: both observations are correct! A detailed calculation (I can lend you the book it is in, if you are interested) shows that the runner observes the rear end of the barn arriving* at the front end of the pole long before the front end of the barn arrives at the rear end of the pole. The pole doesn’t fit! Events which are simultaneous in the farmer’s frame of reference (front pole arriving at back barn and back pole arriving at front barn) are not simultaneous in the runner’s frame of reference. *Remember, the runner sees the barn moving past him.

37. Simultaneity is not a “universal physical reality.” Now I’m no longer worried about the test-charge-plus-conductor example. At a certain instant in time I may observe an excess of moving negative charge in the portion of the circuit nearest me, but that does not mean I can claim there is a net excess of moving negative charge in the entire circuit at that instant in time. Now where were we before this interruption started… “Because simultaneity is a relative concept and not an absolute one, physical theories that require simultaneity in events at different locations cannot be valid.”—Beiser, Modern Physics, page 45.

38. An observer who doesn’t know about relativity, or even one who knows about relativity but invokes charge invariance, will claim that the conductor has a neutral charge density and invents a “magnetic” force to explain the attraction between test charge and current-carrying wire. But the “magnetic” force is present only when current is flowing. It is not valid to talk about a separate “magnetic” force. You must talk about the “electromagnetic” force. What you call “magnetic” force is just a manifestation of the Lorentz contraction and Coulomb’s law, and is not a separate force of nature.

39. The mathematical transformations which lead to our relativistic equations for length and time were actually derived by Lorentz in order to make Maxwell’s equations invariant in inertial reference frames.* Because Maxwell’s equations are invariant in inertial reference frames, special relativity does not demand that we correct them. On the other hand, when it comes to Newton’s Laws… *Part of Einstein’s genius was realizing that Lorentz was on to something big!