109. 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. |
110. 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 |
111. 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. |
112. J.Gruszczak, M.Heller
Differential Structure of Space-Time and its Prolongations to Singular Boundary
Int J Theor Phys, vol. 32, pp. 625-648 (1993).
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Abstract:
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113. Andrzej Woszczyna
A dynamical systems approach to the cosmological structure formation - Newtonian universe
Mon. Not. R.A.S., vol. 225, p. 701 (1992).
[journal] |
Abstract:
Abstract |
114. Andrzej Woszczyna
Gauge invariant cosmic structures : A dynamic systems approach
Phys. Rev. D, vol. 45, pp. 1982-1988 (1992).
[abstract] [journal] |
Abstract:
Gravitational instability is expressed in terms of the dynamic systems theory. The gauge-invariant Ellis-Bruni equation and Bardeen's equation are discussed in detail. It is shown that in an open universe filled with matter of constant sound velocity the Jeans criterion does not adequately define the length scale of the gravitational structure.
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