RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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97. Guido Magnano, Leszek M. Soko這wski
Can the local stress-energy conservation laws be derived solely from field equations?
Gen. Rel. Grav., vol. 30, pp. 1281-1288 (1998).
[abstract] [preprint]

Abstract:
According to a recent suggestion [1], the energy--momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that the property observed by Accioly et al. in [1] is the consequence of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. However, we also show that only in particular cases can this identity be used to obtain the actual form of the stress-energy tensor, while in general the method leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique.

98. Zdzislaw Pogoda, Leszek M. Soko這wski
Does Mathematics Distinguish Certain Dimensions of Spaces? Part II.
The American Mathematical Monthly, vol. 105, pp. 456-463 (1998).
[journal]

Abstract:
Abstract

99. Michael Heller, Wies豉w Sasin, and Dominique Lambert
Groupoid approach to noncommutative quantization of gravity
J. Math. Phys., vol. 38, p. 5840 (1997).
[abstract] [preprint] [journal]

Abstract:
We propose a new scheme for quantizing gravity based on a noncommutative geometry. Our geometry corresponds to a noncommutative algebra A = Gc∞(G,C) of smooth compactly supported complex functions (with convolution as multiplication) on the groupoid G = E◁Γ being the semidirect product of a structured space E of constant dimension (or a smooth manifold) and a group Γ. In the classical case E is the total space of the frame bundle and Γ is the Lorentz group. The differential geometry is developed in terms of a Z(A)-submodule V of derivations of A and a noncommutative counterpart of Einstein’s equation is defined. A pair (A,math), where math is a subset of derivations of A satisfying the noncommutative Einstein’s equation, is called an Einstein pair. We introduce the representation of A in a suitable Hilbert space, by completing A with respect to the corresponding norm change it into a C∗-algebra, and perform quantization with the help of the standard C∗-algebraic method. Hermitian elements of this algebra are interpreted as quantum gravity observables. We introduce dynamical equation of quantum gravity which, together with the noncommutative counterpart of Einstein’s equation, forms a noncommutative dynamical system. For a weak gravitational field this dynamical system splits into ordinary Einstein’s equation of general relativity and Schr鐰inger’s equation (in Heisenberg’s picture) of quantum mechanics. Some interpretative questions are considered.

100. Zdzislaw Pogoda, Leszek M. Soko這wski
Does Mathematics Distinguish Certain Dimensions of Spaces? Part I.
The American Mathematical Monthly, vol. 104, pp. 860-869 (1997).
[journal]

Abstract:
Abstract

101. Marco Litterio, Leszek M. Soko這wski, Zdzis豉w A. Golda, Luca Amendola, Andrzej Dyrek
Anisotropic inflation from extra dimensions
Phys. Lett., vol. B382, pp. 45-52 (1996).
[abstract] [preprint] [journal]

Abstract:
Vacuum multidimensional cosmological models with internal spaces being compact $n$-dimensional Lie group manifolds are considered. Products of 3-spheres and $SU(3)$ manifold (a novelty in cosmology) are studied. It turns out that the dynamical evolution of the internal space drives (power-law) inflation in the external world. This inflationary solution brings two extra bonuses: 1) it is an attractor in phase space (no fine-tuning required); 2) it is determined by the Lie group space solely and not by any arbitrary inflaton potentials. Only scalar fields representing the anisotropic scale factors of the internal space appears in the four dimensions. The size of the volume of the internal space at the end of inflation is compatible with observational constraints. This simple and natural model can be completed by some extended-inflation-like mechanism that ends the inflationary evolution.

102. Marek Szyd這wski, Michael Heller, and Wies豉w Sasin
Geometry of spaces with the Jacobi metric
J. Math. Phys., vol. 37, p. 346 (1996).
[abstract] [journal]

Abstract:
The generalized Maupertuis principle is formulated for systems with the natural Lagrangian and an indefinite form of the kinetic energy. The generalization is applied to the theory of gravity and cosmology. For such systems, the metric determined by the kinetic energy form has a Lorentz signature. The theorem is proved concerning the behavior of trajectories in a neighborhood of the boundary of the region admissible for motion. This region is not a smooth manifold but turns out to be a differential space of constant differential dimension. This fact allows us to use geometric methods analogous to those elaborated for smooth manifolds. It is shown that singularities of the Jacobi metric are not dangerous for the motion; its trajectories are smooth in the sense of the theory of differential spaces.

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