Hi all, I'd like to elaborate slightly on something robphy said:
robphy said:
One outstanding problem in pedagogy is that the accelerated observer in SR has not been satisfactorally presented in textbooks yet.
It matters whether one is discussing the education of
future physicists or a
general audience in a broad sense (e.g. electrical engineers may not require exposure to conceptual subtleties in relativistic physics). My comments here concern the problem of educating the
former group.
There is a sizeable and (amazingly enough) still-growing crank literature by dissidents who don't accept the Lorentz transformation; in particular, who disagree with what standard textbooks say about how they work. I will pass over this literature without further comment, on the grounds that
any physics student who fails to grasp the mathematics of the Lorentz group has no chance of succeeding in his studies (unless, just possibly, at a stretch, he somehow contrives to stay far away from relativistic physics throughout his career in physics).
For the benefit of the nonscientists here, let me stress that implicitly suggesting that failing to learn how to compute correctly with the Lorentz group should be grounds for summary dismissal from any graduate program in physics is
not at all equivalent to telling students that they must accept the (ludicrous) proposition that "str is 'true' once and for all". Science simply doesn't deal in 'eternal truth' in any mystical or dogmatic sense, nor does it deal in models which can never be overturned under any circumstances, and
no one is suggesting the latter procedure! Rather, I am suggesting that physics apprentices need to be comfortable with the mathematics of relativistic physics in order to function as professional physicists in the 21st century. We simply do not need to consider here the unanswerable question of whether or not this will still be true in the 23rd century.
Passing on: I feel that the phenomenon which should really intrigue contemporary educators of potential future physicists is the existence of a considerable literature founded upon more subtle conceptual (and sometimes mathematical) errors. The authors of these papers may no longer be educable, but their lamentable history might suggest that better education of physics majors might result in fewer careers being compromised in this way in the future. Since
educators must strive to prepare students to function, not to fail to function, it follows that they should be very interested in improving the education of future potential physicists in ways specifically aimed at helping them to avoid falling into these conceptual traps.
The existence of misinformed eprints dealing with various versions of Bell's paradox and Ehrenfest's paradox (among others) is both evidence that some positive fraction of prospective physicists are not learning in school some stuff they really need to know in order to function effectively as physics researchers, and a fertile source of "missing topics" which are overdue for consideration in any curriculum reform.
Those who have studied such papers and know what went wrong will, I warrant, recognize that some particular
fundamental topics, which can be considered to fall within the scope of relativistic physics in flat spacetimes, have not yet been adequately covered in the textbooks. ("Adequately" in the sense which I have just explained.) Off the top of my head, I'd list the following as missing topics:
1. thinking about and computing with vector fields as first order linear partial differential operators, or equivalently as flows (local actions by the group of reals), or as uniparameter subgroups of diffeomorphisms, etc.,
2. computing and interpreting the kinematical quantites defined for any congruence (acceleration vector, expansion tensor, vorticity tensor) which are invaluable even in flat spacetime,
3. computing and interpreting frame fields and computations of tensors with respect to a frame field (this is how one models the physical experience of a family of observers; in order to study one observer it is often technically and conceptually advantageous to consider him one of a family of observers),
4. spatial hyperplanes and "space at a time" decompositions of tensor fields valid at the level of jet spaces ("higher order" infinitesimal approximations at an event; a tangent space is a first order jet space),
5. the existence of multiple competing notions of distance/speed "in the large" even in the case of (accelerating observers in) flat spacetime,
6. various algebraic/geometric phenomena regarding the Lorentz group and its analogues, including the Steenrod twist algebra, Penrose-Terrell and other optical phenomena, etc.,
7. familiarity with important alternatives to the usual Cartesian and polar spherical charts on Minkowski spacetime, such as the Coll canonical chart for flat spacetime and its relationship with the Bondi k-calculus,
8. distinguishing between visual experience and length and time measurements in the sense used in elementary discussions of str.
That's just off the top of my head; no doubt I have made some silly omissions in this list.
Note: various books like Eric Poisson, A Relativist's Toolkit, do discuss many of these topics, but their appearance in a book whose subtitle mentions black holes appears to fool casual readers into concluding (quite incorrectly) that the above mentioned techniques "belong to gtr".
robphy said:
It's not clear to me if topics like relativistic thermodynamics and relativistic (Hamiltonian and Lagrangian) mechanics have been fully worked out for SR (let alone GR)... not to mention presented in textbooks.
Some graduate level textbooks do discuss relativistic thermodynamics (e.g. MTW). There is a significant literature on all these topics but I'd agree that much of this hasn't made it into the textbooks, possibly because it is not generally considered sufficiently mature/useful by most would-be textbook authors.
