34 research outputs found
A rigorous formulation of the cosmological Newtonian limit without averaging
We prove the existence of a large class of one-parameter families of
cosmological solutions to the Einstein-Euler equations that have a Newtonian
limit. This class includes solutions that represent a finite, but otherwise
arbitrary, number of compact fluid bodies. These solutions provide exact
cosmological models that admit Newtonian limits but, are not, either implicitly
or explicitly, averaged
How do professional development programs on comparing solution methods and classroom discourse affect students' achievement in mathematics? The mediating role of studentsâ subject matter justifications
Comparing solution methods fosters strategy flexibility in equation solving. Productive classroom discourse such as Accountable Talk (AT) orchestrated by teachers can improve students' justifications during classroom discussions and achievement. Do students' subject matter justifications during classroom discourse mediate the effect of teachers' professional development (PD) programs focused on comparing and AT on studentsâ mathematics achievement? We investigated whether two PD programs (comparing or comparing+AT) compared to a control group increased the number of students justifications, and whether this affected mathematics achievement (strategy flexibility, procedural knowledge, and conceptual knowledge). The study (739 9th and 10th grade students in 39 classes) had an experimental pre-post control group design. Both PD programs significantly increased students justifications compared to the control group. The results of our multilevel path models showed significant small mediation effects in the comparing+AT group on procedural and conceptual knowledge. No mediation effects were found in the comparing group
A Dynamical Study of the Friedmann Equations
Cosmology is an attracting subject for students but usually difficult to deal
with if general relativity is not known. In this article, we first recall the
Newtonian derivation of the Friedmann equations which govern the dynamics of
our universe and discuss the validity of such a derivation. We then study the
equations of evolution of the universe in terms of a dynamical system. This
sums up the different behaviors of our universe and enables to address some
cosmological problems.Comment: Needs IOP LaTeX class; 17 pages, 9 figure
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
Cosmological post-Newtonian expansions to arbitrary order
We prove the existence of a large class of one parameter families of
solutions to the Einstein-Euler equations that depend on the singular parameter
\ep=v_T/c (0<\ep < \ep_0), where is the speed of light, and is a
typical speed of the gravitating fluid. These solutions are shown to exist on a
common spacetime slab M\cong [0,T)\times \Tbb^3, and converge as \ep
\searrow 0 to a solution of the cosmological Poisson-Euler equations of
Newtonian gravity. Moreover, we establish that these solutions can be expanded
in the parameter \ep to any specified order with expansion coefficients that
satisfy \ep-independent (nonlocal) symmetric hyperbolic equations
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
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 53:364â384, 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 Fefermanâs title of his Clermont-Ferrand lectures at the Logic Colloquium 1994: âHow is it that finitary proof theory became infinitary?â Hilbertâs 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 âlargeâ 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 Î 02
-conservative over) intuitionistic first order arithmetic, also known as Heyting arithmetic. Thus we have a beautiful example of Hilbertâs program at work, exemplifying the Hilbertian goal of moving from the ideal to the real by eliminating ideal elements
Bruchterme -- Handeln wie Experten
Fachliche Expertise beim Umgang mit Bruchtermen und Bruchtermgleichungen zeichnet sich durch ein angemessenes Handeln in jenen Situationen aus, die ungewohnt sind. Um die
SchuÌler und SchuÌlerinnen zu einer solchen FlexibilitĂ€t hinzufuÌhren, wird hier vorgeschlagen, das, was oftmals unausgesprochen â aber handlungsleitend â ist, sichtbar zu machen. Am Beispiel des Umformens von Bruchtermen und des Lösens von Bruchtermgleichungen wird dargestellt, wie mit den SchuÌlerdokumenten gearbeitet werden kann, um dieses
Unausgesprochene der Klasse zugÀnglich zu machen