Parity nonconservation and stellar collapse
ACCORDING to the current theory, the evolution of sufficiently massive stars leads eventually to gravitational instability and collapse. After thermonuclear burning has stopped, dense stellar cores become unstable and collapse to form neutron stars (pulsars) or black holes. In both cases energy of the order of 0.1 Mc2 is released, mostly in the form of neutrinos and gravitational waves. As the radius of the core decreases, its magnetic field grows (magnetic flux is conserved) and can reach huge values of order 1012–1013 G. Even greater azimuthal fields (∼3 × 1015 G) can be generated by differential rotation of the core1. It will be shown here that neutrino emission in the collapse is asymmetric, as a result of parity nonconservation in weak interactions. Neutrinos and antineutrinos are emitted preferentially in the direction parallel to the local magnetic field of the collapsing core.
The neutrino radiation for a hot neutron star formation and the envelope outburst problem
With the equations of neutrino heat conductivity being used, the neutrino light curve is calculated for the spherically symmetrical collapse of an iron-oxygen 2M⊙ star (Figure 1) up to the formation of a hot hydrostatically equilibrium neutron star. The total energy, radiated in the form of muon and electron neutrinos, is 5.8×1053 erg (0.16Mc2). The mean neutrino particle energy is ∼12 MeV for all the time the collapse proceeds. The maximum neutrino luminosity value is equal to 3×1053 erg s−1. For a 10M⊙ star collapse, the luminosity maximum 3×1054 erg s−1 takes place just at the moment of the formation of a black hole inside the collapsing star. The total radiated energy in this case is about 0.08Mc2. The set of calculations, allowing for the deposition of momentum by means of neutrino-nuclear coherent scattering, brings us to a conclusion that the envelope outburst is only possible if the scattering cross-section is 50 times larger than the value experimentally accepted (inequality 20)).
Decidability of some intuitionistic predicate theories
Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T. Question. Is T decidable? One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows: Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.