Post-Newtonian extension of the Newton-Cartan theory

The theory obtained as a singular limit of General Relativity, if the reciprocal velocity of light is assumed to tend to zero, is known to be not exactly the Newton-Cartan theory, but a slight extension of this theory. It involves not only a Coriolis force field, which is natural in this theory (although not original Newtonian), but also a scalar field which governs the relation between Newtons time and relativistic proper time. Both fields are or can be reduced to harmonic functions, and must therefore be constants, if suitable global conditions are imposed. We assume this reduction of Newton-Cartan to Newton`s original theory as starting point and ask for a consistent post-Newtonian extension and for possible differences to usual post-Minkowskian approximation methods, as developed, for example, by Chandrasekhar. It is shown, that both post-Newtonian frameworks are formally equivalent, as far as the field equations and the equations of motion for a hydrodynamical fluid are concerned.Comment: 13 pages, LaTex, to appear in Class. Quantum Gra

Exactly Soluble Sector of Quantum Gravity

Cartan's spacetime reformulation of the Newtonian theory of gravity is a generally-covariant Galilean-relativistic limit-form of Einstein's theory of gravity known as the Newton-Cartan theory. According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate `metric' structure of spacetime remains fixed with two mutually orthogonal non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric gravitational connection-field is dynamical in the sense that it is determined by matter distributions. Here, this generally-covariant but Galilean-relativistic theory of gravity with a possible non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schroedinger-field (bosonic or fermionic), is successfully cast -- for the first time -- into a manifestly covariant Lagrangian form. Then, exploiting the fact that Newton-Cartan spacetime is intrinsically globally-hyperbolic with a fixed causal structure, the theory is recast both into a constraint-free Hamiltonian form in 3+1-dimensions and into a manifestly covariant reduced phase-space form with non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schroedinger system is non-perturbatively quantized using the standard C*-algebraic technique combined with the geometric procedure of manifestly covariant phase-space quantization. The ensuing unitary quantum field theory of Newtonian gravity coupled to Galilean-relativistic matter is not only generally-covariant, but also exactly soluble.Comment: 83 pages (TeX). A note is added on the early work of a remarkable Soviet physicist called Bronstein, especially on his insightful contribution to "the cube of theories" (Fig. 1) -- see "Note Added to Proof" on pages 71 and 72, together with the new references [59] and [61

On Relating Theories: Proof-Theoretical Reduction

The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. Feferman and Sieg (Buchholz et al., Iterated inductive definitions and subsystems of analysis. Springer, Berlin, 1981, Chap. 1) and Feferman in (J Symbol Logic 53:364–384, 1988) made first steps to delineate it in more formal terms. The first goal of this paper is to corroborate their view that this notion has the greatest explanatory reach and is superior to others, especially in the context of foundational theories, i.e., theories devised for the purpose of formalizing and presenting various chunks of mathematics. A second goal is to address a certain puzzlement that was expressed in Feferman’s title of his Clermont-Ferrand lectures at the Logic Colloquium 1994: “How is it that finitary proof theory became infinitary?” Hilbert’s aim was to use proof theory as a tool in his finitary consistency program to eliminate the actual infinite in mathematics from proofs of real statements. Beginning in the 1950s, however, proof theory began to employ infinitary methods. Infinitary rules and concepts, such as ordinals, entered the stage. In general, the more that such infinitary methods were employed, the farther did proof theory depart from its initial aims and methods, and the closer did it come instead to ongoing developments in recursion theory, particularly as generalized to admissible sets; in both one makes use of analogues of regular cardinals, as well as “large” cardinals (inaccessible, Mahlo, etc.). (Feferman 1994). The current paper aims to explain how these infinitary tools, despite appearances to the contrary, can be formalized in an intuitionistic theory that is finitistically reducible to (actually Π02 -conservative over) intuitionistic first order arithmetic, also known as Heyting arithmetic. Thus we have a beautiful example of Hilbert’s program at work, exemplifying the Hilbertian goal of moving from the ideal to the real by eliminating ideal elements

Tourism in protected and conserved areas amid the COVID-19 pandemic

The COVID-19 pandemic has had a global impact on the tourism sector. With tourism numbers dramatically reduced, millions of jobs could be lost, and progress made in equality and sustainable economic growth could be rolled back. Widespread reports of dramatic changes to protected and conserved1 area visitation have negative consequences for conservation finances, tourism businesses and the livelihoods of people who supply labour, goods and services to tourists and tourism businesses. This paper aims to share experiences from around the world on the impacts of the COVID-19 pandemic on protected area tourism; and considers how to build resilience within protected area tourism as a regenerative conservation tool

Strukturieren eines algebraischen Ausdrucks als Herstellen von Bezügen

Das Strukturieren eines algebraischen Ausdrucks ist ein individueller Prozess, bei dem eine Person Teile des Ausdrucks aufeinander bezieht. Durch die Beschreibung der dabei hergestellten Bezüge wird erstens explizit gemacht, in welche Bündel ein Proband die algebraische Zeichenreihe aufteilt und als was er sie interpretiert, und zweitens, wie die Person die Bezüge gebraucht. Das erlaubt die Rekonstruktion der internen Bedeutung, welche die Person dem Ausdruck zuschreibt. Dieses Konzept wird in diesem Artikel zur Analyse von Interviews verwendet. Es konnten empirisch vier Ebenen des Herstellens von Bezügen identifiziert werden: In einem Ausdruck Bezüge herstellen kann heißen, ihn optisch einfacher machen, ihn ändern, Teile umdeuten oder den Ausdruck klassifizieren. Dies ist bedeutsam im Algebraunterricht für das Aushandeln der Bedeutung von individuellen Strukturen