Back in the "Uncomfortable Questions" thread, Thony C suggested that I should do running updates on the course I'm teaching now. I meant to get to this sooner, but last weekend's bout with norovirus kind of got in the way...
I like the idea, though, so below the fold are a bunch of comments on the classes I've had thus far this term:
Class 1: Introduction to Relativity. I do a quick recap of the two classical physics classes that are pre-requisites for my class, showing the various conservation laws, and Maxwell's equations. I then set up a version of the problem that led to relativity, showing that Maxwell's equations predict a single value for the speed of light, independent of the speed of the source. Then I talk about the Michelson-Morley experiment, showing that there is no observable change in the speed of light from a moving object.
This treatment is slightly ahistorical-- it's not clear whether Einstein knew about Michelson-Morley before he developed special relativity-- but it provides a narrative that's easy to follow, and gets the course off to a good start. Homework is a couple of math problems relating to the binomial approximation, which gets used over and over.
Class 2: I start off by comparing the Galileian transformation to a proof that 1=2-- it looks reasonable at first glance, but on closer inspection, it turns out to be flawed because of an error that's subtle enough to be missed. In the case of a proof that 1=2, it's usually a divide-by-zero; in the case of the Galileian transformation, it's the assumption that time is the same for all observers.
Then I go through the "light clock" thought experiment to show that moving observers will disagree about the timing of events, and derive the formula for time dilation. This is a special case of the Lorentz transformation, which is the next thing introduced, and I work through one example relating to muon creation and decay to demonstrate both time dilation and length contraction.
As usual, I made a total hash of the explanation of length contraction. There's a subtle definitional point about who measures what length that is really easy to screw up, and I was rushing to get things done before the end of class, and screwed it up completely. Based on the homework I just graded, they got the basic ideas, but more in spite of my lecture than because of it.
Class 3: On the schedule, this is Paradox Day: I go through the "barn and pole" paradox to show the importance of keeping your frames straight and getting the timing of events right. Then it's the "twin paradox," talking about the distinction between inertial and non-inertial frames. This often confuses students into thinking that acceleration causes the time difference, so the homework includes reading an American Journal of Physics article on a variation of the "twin paradox" in which both twins accelerate.
I say "on the schedule," because this was the day that the norovirus hit. I had to go to Albany to get Kate from her work, and hastily arranged for a colleague to cover my class.
Class 4: I wrapped up the twin paradox (which my colleague hadn't gotten to), and went through the transformation of velocities in special relativity. This is one of a half-dozen classes in this course that drive math majors crazy, as the quasi-derivations used to come up with the velocity addition formulae use a bunch of swashbuckling physicist tricks ("We have a dx here, and a dt there, so we divide through by dt, and hey, there's the velocity, dx/dt..."). There's nothing actually wrong with it, but I don't go through the formal steps needed to make that leap, and it always makes mathematicians squirm in their seats.
Class 5: Relativistic momentum. The bulk of this class is taken up with a very long demonstration that the formula for relativistic momentum does, in fact, leave momentum as a conserved quantity in a two-dimensional elastic collision. Lots and lots of gory algebra, and it's easy to get lost in the details. I'm not sure how well they got the idea, but it's got to be done.
Class 6: Relativistic energy. I go through the derivation of kinetic energy via the work-energy theorem (another wonderful swashbuckling physicist moment, where a "du/dt" multipled by "dt" becomes just a "du," magically changing the integration variable). I talk about the implications of E=mc2: pair creation and annihilation, binding energy, conservation of mass-energy. I end with one example of a particle physics problem, using the final energy and momentum of two decay products to find the mass and momentum of the original particle.
Again, this was rushed a bit toward the end, but I think they got the idea.
That's pretty much it for relativity in this class. Today's lecture is the Special Bonus Topic of showing how you can see a magnetic field as an electric field in a moving frame. It doesn't lend itself to homework problems or exam questions, but it does bring everything together nicely.
The first mid-term exam is on Thursday. Then it's on to quantum mechanics.
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No 4-vectors?
No 4-vectors in the sophomore "modern physics" course. The book doesn't talk about them at all, and I don't have enough time to do it. It's also not guaranteed that the students will have the math for it-- it depends on where they started in the calculus sequence.
This is a recurring problem with the course. Wednesday's lecture, for example, will be the swashbuckling physicist's introduction to complex exponentials, because the math department doesn't teach Euler's theorem in the calculus sequence.
"Today's lecture is the Special Bonus Topic of showing how you can see a magnetic field as an electric field in a moving frame. It doesn't lend itself to homework problems or exam questions, but it does bring everything together nicely."
Indeed. This was not mentioned in the modern physics class I took as a physics undergrad, probably for that reason, that it doesn't really fit on an exam. I saw it first when I started studying general relativity (as a philosophy grad student--go figure!), and I almost literally had one of those light bulbs pop up over my head, like in a cartoon. "Oh! *Now* I get it!"
Re "There's nothing actually wrong with it, but I don't go through the formal steps needed to make that leap, and it always makes mathematicians squirm in their seats."
In my first (pre measure theory based) mathematical stat course as an undergrad, our professor would, whenever he interchanged differentiation and integration/summation, or otherwise played fast and loose with manipulations of limits, convergence, etc., his standard comment was "let's just pretend we're physicists and do this: if things work in the end our steps are justified."
You are correct - that was always a bit disconcerting.
I took a course similar (almost exactly the same) to this last semester and I think it was wonderful. It was a good contrast to the other physics course I was taking (Fluids and Thermal Physics; Bernoulli's Eqn, Continuity Eqn, First and Second Laws of Thermodynamics, Heat engines and etc) and also a interesting change from CM and EM in first year.
If the mathematicians give you trouble, it's always handy to pass the buck to Professor Maxwell, one of whose earlier claims to fame are all these "Maxwell relation" derivatives for what is essentially a change of variable. Or you can trot out the fact that if the relevant sets being acted on by the derivative and integral operations aren't convergent (i.e., you don't get equivalent values from the two versions of the operation), then the operations are unsuited to representing physical phenomenon because the operation will blow up anyways. (This has the perk of making them squirm with math they don't know yet instead of physics they don't know yet.)
Maybe I'm wrong, but I thought the bazillion biographers had established pretty conclusively that Einstein had not heard about Michelson-Morley.
No mention of general relativity? Now that i think about it... I'm not sure my undergrad Modern course mentioned GR either, which kind of explains why I don't really know anything about GR except for what I picked up on the street.
Anyway, I guess the difference between physics majors and liberal arts students is... the concepts you just outlined in 6 class periods takes me until AFTER midterm to cover in my "Space, Time, and Einstein" course!
My view is that if you can do 3 vectors, you can do 4. It is the metric that is odd, not the idea.
Glad to see that you did the particle physics application. The one thing my introduction to relativity did not convey is just how powerful the invariants can be, particularly the total energy (more precisely, the square of the center of mass energy) manifest as the Mandelstam variable s.
http://en.wikipedia.org/wiki/Mandelstam_variables
This approach makes it trivial (fairly easy) to figure out lots of things of interest in particle physics or other scattering problems at high energy.
"Today's lecture is the Special Bonus Topic of showing how you can see a magnetic field as an electric field in a moving frame. It doesn't lend itself to homework problems or exam questions, but it does bring everything together nicely."
Actually, that was what gave me the most trouble in my basic physics courses when we hit electromagnetism - the magnetism stuff made no sense as presented, it all evaluated to "here are a bunch of tricks that take the place of what's really happening, which we're not going to cover". So I got a worse than usual grade because I went off trying to figure it out.
The book didn't cover it (how it can be viewed as a relativistic effect of electricity), and I haven't (yet) found any good sources on it - so, care to make a post on the subject? :-)
hehe - I just had a cool idea for a science course: don't teach anything - you give the students experiments and their results (they don't necessarily have to do the experiment, they just have to understand the experimental setup), and make them figure out explanations and how to test them.
For a simple example, instead of:
(a) teaching students about momentum and then have them crash objects into each other to observe it;
You would:
(b) have students crash objects into each other and figure out how to predict the results.
Impractical for "get it over with" or main-course classes, perhaps, but it could be very useful for those who do best with intuition and/or want to become experimental scientists. Perhaps an optional lab-type class (it would have to schedule the experiments *before* the background was taught, of course), or as an elective that covers a selection of interesting (and practically solvable) concepts/experiments (wouldn't you love to see the electron-mass oil experiment?). With a 'bonus' lab when a solar eclipse happens during the semester (hint: telescope and starchart)..
Dreikin,
You can always try Lorrain and Corson's Electromagnetic Fields and Waves book. It's been a while, but as I recall Chapters 5 and 6 covered relativity, with chapter 6 showing how the magnetic field can be found by considering charges in an inertial frame moving with respect to some observer.
I imagine that there are newer books out there, but I know this one covered it. The derivation was simply working through the relativistic force transformation of the electrostatic force.
Einstein's Nemesis: DI Her Eclipsing Binary Stars Solution
The problem that the 100,000 PHD Physicists could not solve
This is the solution to the "Quarter of a century" Smithsonian-NASA Posted motion puzzle that Einstein and the 100,000 space-time physicists including 109 years of Nobel prize winner physics and physicists and 400 years of astronomy and Astrophysicists could not solve and solved here and dedicated to Drs Edward Guinan and Frank Maloney
Of Villanova University Pennsylvania who posted this motion puzzle and started the search collections of stars with motion that can not be explained by any published physics
For 350 years Physicists Astrophysicists and Mathematicians and all others including Newton and Kepler themselves missed the time-dependent Newton's equation and time dependent Kepler's equation that accounts for Quantum - relativistic effects and it explains these effects as visual effects. Here it is
Universal- Mechanics
All there is in the Universe is objects of mass m moving in space (x, y, z) at a location
r = r (x, y, z). The state of any object in the Universe can be expressed as the product
S = m r; State = mass x location
P = d S/d t = m (d r/dt) + (dm/dt) r = Total moment
= change of location + change of mass
= m v + m' r; v = velocity = d r/d t; m' = mass change rate
F = d P/d t = d²S/dt² = Force = m (d²r/dt²) +2(dm/d t) (d r/d t) + (d²m/dt²) r
= m γ + 2m'v +m"r; γ = acceleration; m'' = mass acceleration rate
In polar coordinates system
r = r r(1) ;v = r' r(1) + r θ' θ(1) ; γ = (r" - rθ'²)r(1) + (2r'θ' + rθ")θ(1)
F = m[(r"-rθ'²)r(1) + (2r'θ' + rθ")θ(1)] + 2m'[r'r(1) + rθ'θ(1)] + (m"r) r(1)
F = [d²(m r)/dt² - (m r)θ'²]r(1) + (1/mr)[d(m²r²θ')/d t]θ(1) = [-GmM/r²]r(1)
d² (m r)/dt² - (m r) θ'² = -GmM/r²; d (m²r²θ')/d t = 0
Let m =constant: M=constant
d²r/dt² - r θ'²=-GM/r² ------ I
d(r²θ')/d t = 0 -----------------II
r²θ'=h = constant -------------- II
r = 1/u; r' = -u'/u² = - r²u' = - r²θ'(d u/d θ) = -h (d u/d θ)
d (r²θ')/d t = 2rr'θ' + r²θ" = 0 r" = - h d/d t (du/d θ) = - h θ'(d²u/d θ²) = - (h²/r²)(d²u/dθ²)
[- (h²/r²) (d²u/dθ²)] - r [(h/r²)²] = -GM/r²
2(r'/r) = - (θ"/θ') = 2[λ + á» Ï (t)] - h²u² (d²u/dθ²) - h²u³ = -GMu²
d²u/dθ² + u = GM/h²
r(θ, t) = r (θ, 0) Exp [λ + á» Ï (t)] u(θ,0) = GM/h² + Acosθ; r (θ, 0) = 1/(GM/h² + Acosθ)
r ( θ, 0) = h²/GM/[1 + (Ah²/Gm)cosθ]
r(θ,0) = a(1-ε²)/(1+εcosθ) ; h²/GM = a(1-ε²); ε = Ah²/GM
r(0,t)= Exp[λ(r) + á» Ï (r)]t; Exp = Exponential
r = r(θ , t)=r(θ,0)r(0,t)=[a(1-ε²)/(1+εcosθ)]{Exp[λ(r) + ì Ï(r)]t} Nahhas' Solution
If λ(r) â 0; then:
r (θ, t) = [(1-ε²)/(1+εcosθ)]{Exp[á» Ï(r)t]
θ'(r, t) = θ'[r(θ,0), 0] Exp{-2á»[Ï(r)t]}
h = 2Ï a b/T; b=aâ (1-ε²); a = mean distance value; ε = eccentricity
h = 2Ïa²â (1-ε²); r (0, 0) = a (1-ε)
θ' (0,0) = h/r²(0,0) = 2Ï[â(1-ε²)]/T(1-ε)²
θ' (0,t) = θ'(0,0)Exp(-2á»wt)={2Ï[â(1-ε²)]/T(1-ε)²} Exp (-2iwt)
θ'(0,t) = θ'(0,0) [cosine 2(wt) - Ỡsine 2(wt)] = θ'(0,0) [1- 2sine² (wt) - Ỡsin 2(wt)]
θ'(0,t) = θ'(0,t)(x) + θ'(0,t)(y); θ'(0,t)(x) = θ'(0,0)[ 1- 2sine² (wt)]
θ'(0,t)(x) â θ'(0,0) = - 2θ'(0,0)sine²(wt) = - 2θ'(0,0)(v/c)² v/c=sine wt; c=light speed
Πθ' = [θ'(0, t) - θ'(0, 0)] = -4Ï {[â (1-ε) ²]/T (1-ε) ²} (v/c) ²} radians/second
{(180/Ï=degrees) x (36526=century)
Πθ' = [-720x36526/ T (days)] {[â (1-ε) ²]/ (1-ε) ²}(v/c) = 1.04°/century
This is the T-Rex equation that is going to demolished Einstein's space-jail of time
The circumference of an ellipse: 2Ïa (1 - ε²/4 + 3/16(ε²)²---) â 2Ïa (1-ε²/4); R =a (1-ε²/4)
v (m) = â [GM²/ (m + M) a (1-ε²/4)] â â [GM/a (1-ε²/4)]; m<joenahhas1958@yahoo.com