RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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103. Michael Heller and Wiesław Sasin
Noncommutative structure of singularities in general relativity
J. Math. Phys., vol. 37, p. 5665 (1996).
[abstract] [journal]

Abstract:
Initial and final singularities in the closed Friedman world model are typical examples of malicious singularities. They form the single point of Schmidt’s b‐boundary of this model and are not Hausdorff separated from the rest of space–time. The method of noncommutative geometry, developed by A. Connes and his co‐workers, is applied to this case. We rephrase Schmidt’s construction in terms of the groupoid math of orthonormal frames over space–time and carry out the ‘‘desingularization’’ process. We define the line bundle τ:Ω1/2→math over math and change the space of its cross sections into an involutive algebra. This algebra is represented in the space of operators on a Hilbert space and, with the norm inherited from these operators, it becomes a C∗‐algebra. The initial and final singularities of the closed Friedman model are given by two distinct representations of this C∗‐algebra in the space of operators acting on the Hilbert space L2(O(3,1)).

104. Zdzisław A. Golda, Marco Litterio, Leszek M. Sokołowski, Luca Amendola, Andrzej Dyrek
Pure Geometrical Evolution of the Multidimensional Universe
Ann. Phys. , vol. 248, pp. 246-285 (1996).
[abstract] [journal]

Abstract:
An exhaustive qualitative analysis of cosmological evolution for some multidimensional universes is given. The internal space is taken to be a compact Lie group Riemannian manifold. The space is generically anisotropic; i.e., its cosmological evolution is described by its (time-dependent) volume, the dilaton, and by relative anisotropic deformation factors representing the shear of the internal dimensions during the evolution. Neither the internal space nor its subspaces need to be Einstein spaces. The total spacetime is empty, and the cosmic evolution of the external, four-dimensional world is driven by the geometric ``matter'' consisting of the dilaton and of the deformation factors. Since little is known about any form of matter in the extra dimensions, we do not introduce any {\it ad hoc\/} matter content of the Universe. We derive the four-dimensional Einstein field equations (with a cosmological term) for these geometric sources in full generality, i.e., for any compact Lie group. A detailed analysis is done for some specific internal geometries: products of 3-spheres, and $SU$(3) space. Asymptotic solutions exhibit power law inflation along with a process of full or partial isotropization. For the $SU$(3) space all the deformation factors tend to a common value, whereas in the case of $S^3$'s each sphere isotropizes separately.

105. David H Lyth, Andrzej Woszczyna
Large scale perturbations in the open universe
Phys. Rev. D, vol. 52, pp. 3338-3357 (1995).
[abstract] [preprint] [journal]

Abstract:
When considering perturbations in an open (Omega<1) universe, cosmologists retain only sub-curvature modes (defined as eigenfunctions of the Laplacian whose eigenvalue is less than -1 in units of the curvature scale, in contrast with the super-curvature modes whose eigenvalue is between -1 and 0). Mathematicians have known for almost half a century that all modes must be included to generate the most general HOMOGENEOUS GAUSSIAN RANDOM FIELD, despite the fact that any square integrable FUNCTION can be generated using only the sub-curvature modes. The former mathematical object, not the latter, is the relevant one for physical applications. The mathematics is here explained in a language accessible to physicists. Then it is pointed out that if the perturbations originate as a vacuum fluctuation of a scalar field there will be no super-curvature modes in nature. Finally the effect on the cmb of any super-curvature contribution is considered, which generalizes to Omega<1 the analysis given by Grishchuk and Zeldovich in 1978. A formula is given, which is used to estimate the effect. In contrast with the case Omega=1, the effect contributes to all multipoles, not just to the quadrupole. It is important to find out whether it has the same l dependence as the data, by evaluating the formula numerically.

106. Michael Heller and Wieslaw Sasin
Structured spaces and their application to relativistic physics
J. Math. Phys., vol. 36, p. 3644 (1995).
[abstract] [journal]

Abstract:
A sheaf of functions on a topological space is called a differential structure if it satisfies an axiom of a closure with respect to composition with the Euclidean functions. A differential structure on a nonempty set is called a structured space. It is a generalization of the smooth manifold concept and of an earlier concept of differential space. Differential geometry on structured spaces is developed (tangent space, vector fields, differential forms, exterior algebra, linear connection, curvature, and torsion). Some of its techniques are applied to the classical singularity problem in general relativity. It turns out that Einstein’s equations can be defined on space–times with singularities. This can have important consequences for the search of the quantum theory of gravity.

107. Leszek M. Sokołowski
Universality of Einstein's General Relativity
GR14 Conference (Florence, Italy, Aug 1995) (1995).
[abstract] [preprint] [journal]

Abstract:
Among relativistic theories of gravitation the closest ones to general relativity are the scalar-tensor ones and these with Lagrangians being any function f(R) of the curvature scalar. A complete chart of relationships between these theories and general relativity can be delineated. These theories are mathematically (locally) equivalent to general relativity plus a minimally coupled self-interacting scalar field. Physically they describe a massless spin-2 field (graviton) and a spin-0 component of gravity. It is shown that these theories are either physically equivalent to general relativity plus the scalar or flat space is classically unstable (or at least suspected of being unstable). In this sense general relativity is universal: it is an isolated point in the space of gravity theories since small deviations from it either carry the same physical content as it or give rise to physically untenable theories

108. Guido Magnano, Leszek M. Sokołowski
On Physical Equivalence between Nonlinear Gravity Theories
Phys. Rev. D, vol. 50, pp. 5039-5059 (1994).
[abstract] [preprint] [journal]

Abstract:
We argue that in a nonlinear gravity theory, which according to well-known results is dynamically equivalent to a self-gravitating scalar field in General Relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical. We explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and backwards. We study energetics for asymptotically flat solutions. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space. The proof of this Positive Energy Theorem relies on the existence of the Einstein frame, since in the (Helmholtz--)Jordan frame the Dominant Energy Condition does not hold and the field variables are unrelated to the total energy of the system.

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