The radial wavefunctions of the hydrogen atom have an interesting mathematical behavior in the limit of infinite n, which is the infinite Rydberg limit. Finding the limiting wavefunctions corresponds to solving the radial Schroedinger equation for E = 0. Frobenius expansion gives two solutions, each of which is a convergent expansion, but neither is square-integrable. For bound states, the second Frobenius solution is discarded because a relation between the solutions at r = 0 is violated and the square-integrable solution survives. We can't rule out either one of the E = 0 solutions solutions on the same grounds, which highlights the fact that half the mathematical solutions for the energies E < 0 have been artificially wiped away. We have lived without a probability interpretation for all the unbound-state wave functions, so I conjecture here that there may be physical significance to the non-integrable wave functions at E < 0 that are ignored in textbooks.
It is generally thought that, when direct current flows in a stationary wire, no external electric field is produced. However, we show that if the Lorentz contraction of the assemblage of moving electrons is taken into account, special relativity theory predicts a nonzero electric field. Other theory also predicts nonzero electric fields through other mechanisms, and experimental works report confirmation of these mechanisms.
For a 'bola' with identical masses rotating freely in space, a Lorentz-boosted observer sees a time-varying momentum and kinetic energy, in violation of special-relativistic momentum and energy conservation laws. In the moving frame, the bola string is curved, not straight, and thrums with a period that is half the Lorentz-time-dilated period of the bola in the CM frame.
A one-dimensional box is posited with identical point masses synchronously and elastically hitting each other and opposite ends of the box. From a moving frame, Special-Relativistic kinematics says the center of mass CM of the box (exclusive of the point masses) moves uniformly, but momentum conservation says CM moves jerkily. Hence Special Relativity does not support momentum conservation.
A structure that can be interpreted as either affine or Euclidean is identified. That structure was invented to describe the manifold of colors, which has an undisputed affine symmetry (based on color matches) but a debated line element (based on color discrimination). The affine/Euclidean structure is reviewed here as a way to tune our notions of "invariance" and "covariance" in space-time physics. In particular, the structure displays a kind of manifest covariance that is rare in space-time physics, despite common invocations of the Principle of General Covariance in that field.
A gedanken experiment is described that exposes an apparent conflict between the treatment of proper timekeeping on geodesics according to general relativity theory, as customarily understood, and empirical evidence such as that of the Global Positioning System. The paradox is resolved by noting that there may be many geodesics between two spacetime events, only one of which represents a global maximum of proper time. The cardinality of such nonuniqueness (which may be that of the continuum) at first seems to violate the property that a geodesic between two events always incurs a (local) extremum of proper time. However, to first order (hence to observationally significant order), all free-fall orbits that have the same period have the same proper time, so no first variations of the orbits within our solution set change the proper time?a consistency check on the geodesic (extremum) interpretation of such orbits.
The traditional definition of entropy employed in statistical mechanics and in Shannon's information theory, −npn ln (pn), may be viewed as a noninvariant special case (associated with an implicit uniform prior) of an invariant covering theory based on −npn ln (pn/p), where pn refers to a posterior probability distribution, as affected by the arrival of ?new data,? and p refers to a Bayesian prior probability distribution. This generalized or explicitly Bayesian form of ?entropy? thus quantifies the transition between two states of knowledge, prior and posterior, exactly as does Bayes' theorem, and may be considered to have the same scope and information content as that theorem. Constrained extremalization of this form of entropy is demonstrated to be useful in solving three types of classical probability problems: (1) those for which the availability or presumption of single-parameter information allows a Poisson distribution to serve as ?universal prior,? (2) those for which additional prior information justifies a known departure from the Poisson law, and (3) those for which statistical sampling provides arbitrary nonuniform prior information; that is, prior to additional data input. In all cases the ?new data? must be of the aggregated or summed type expressible as Lagrange constraints. By reference to an example taken from Denting and by extension of the proof of Shannon's ?composition law? (hitherto thought to be valid only for the traditional form of entropy), it is shown that use of Bayesian entropy can broaden the scope of information theory, with interpretation of ?information? as that which quantifies a transition between states of knowledge. Shannon's ?monotonicity law? becomes superfluous and can be eliminated. This generalized form of entropy also promises a more powerful means of treating nonequilibrium thermodynamics, by freeing statistical-mechanical entropy from implicit connection to the equilibrium (uniform prior) or thermostatic state.
This paper is a speculation about the tectonic origins of the Ninety-East Ridge, based on the cartographic fact that the ridge is straight (adhering to a great circle on the Earth) over its 3,000 miles of extent...
Although general relativity purports to deal with noninertial effects in gravity, the noninertialness of the earthbound observer has not been incorporated in the theory as it touches experiment via the three fundamental tests. General-relativistic notions of reference frame are reviewed in light of this fact.