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H. -J. Schmidt

Publications and source records attributed to H. -J. Schmidt.

At least 55 records · Page 3Linked to original sources
arXiv2001

The metric in the superspace of Riemannian metrics and its relation to gravity

The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion. The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula. Our approach is a local one: the essence is a metric in the space of all symmetric rank-two tensors, and then the space becomes a warped product of the real line with an Einstein space.

gr-qc↗
arXiv2001

The massive scalar field in a closed Friedmann universe model - new rigorous results

For the minimally coupled scalar field in Einstein's theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove that D = 1 or D > 1, where D is the (fractal) dimension of the set of solutions which can be integrated up to t to infinity. (D > 0 was conjectured by PAGE (1984)). We discuss concepts like ``the probability of the appearance of a sufficiently long inflationary phase" and argue that it is primarily a probability measure q in the space V of solutions (and not in the space of initial conditions) which has to be applied. q is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of q can be given. We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented.

gr-qc↗
arXiv2001

Independent magnon states on magnetic polytopes

For many spin systems with constant isotropic antiferromagnetic next-neighbour Heisenberg coupling the minimal energies E_{min}(S) form a rotational band, i.e. depend approximately quadratically on the total spin quantum number S, a property which is also known as Lande interval rule. However, we find that for certain coupling topologies, including recently synthesised icosidodecahedral structures this rule is violated for high total spins. Instead the minimal energies are a linear function of total spin. This anomaly results in a corresponding jump of the magnetisation curve which otherwise would be a regular staircase.

cond-mat.mtrl-sci↗
arXiv2001

Spherically Symmetric solutions in Multidimensional Gravity with the SU(2) Gauge Group as the Extra Dimensions

The multidimensional gravity on the principal bundle with the SU(2) gauge group is considered. The numerical investigation of the spherically symmetric metrics with the center of symmetry is made. The solution of the gravitational equations depends on the boundary conditions of the ``SU(2) gauge potential'' (off-diagonal metric components) at the symmetry center and on the type of symmetry (symmetrical or antisymmetrical) of these potentials. In the chosen range of the boundary conditions it is shown that there are two types of solutions: wormhole-like and flux tube. The physical application of such kind of solutions as quantum handles in a spacetime foam is discussed.

gr-qc↗
arXiv2001

Heisenberg exchange parameters of molecular magnets from the high-temperature susceptibility expansion

We provide exact analytical expressions for the magnetic susceptibility function in the high temperature expansion for finite Heisenberg spin systems with an arbitrary coupling matrix, arbitrary single-spin quantum number, and arbitrary number of spins. The results can be used to determine unknown exchange parameters from zero-field magnetic susceptibility measurements without diagonalizing the system Hamiltonian. We demonstrate the possibility of reconstructing the exchange parameters from simulated data for two specific model systems. We examine the accuracy and stability of the proposed method.

cond-mat.stat-mech↗
arXiv2001

Cosmic strings and strings in gravitational waves

We consider strings with the Nambu action as extremal surfaces in a given space-time, thus, we ignore their back reaction. Especially, we look for strings sharing one symmetry with the underlying space-time. If this is a non-null symmetry, the problem of determining the motion of the string can be dimensionally reduced. We get exact solutions for the following cases: straight and circle-like strings in a Friedmann background, straight strings in an anisotropic Kasner background, different types of strings in the metric of a gravitational wave. The solutions will be discussed.

gr-qc↗
arXiv2001

On Ellis' programme within fourth order gravity

For the non-tachyonic curvature squared action we show that the expanding Bianchi-type I models tend to the dust-filled Einstein-de Sitter model for t tending to infinity if the metric is averaged over the typical oscillation period. Applying a conformal equivalence between curvature squared action and a minimally coupled scalar field (which holds for all dimensions > 2) the problem is solved by discussing a massive scalar field in an anisotropic cosmological model.

gr-qc↗
arXiv2001

Comparing selfinteracting scalar fields and R + R^3 cosmological models

We generalize the well-known analogies between m^2 ϕ^2 and R + R^2 theories to include the selfinteraction λϕ^4-term for the scalar field. It turns out to be the R + R^3 Lagrangian which gives an appropriate model for it. Considering a spatially flat Friedman cosmological model, common and different properties of these models are discussed, e.g., by linearizing around a ground state the masses of the resp. spin 0-parts coincide. Finally, we prove a general conformal equivalence theorem between a Lagrangian L = L(R), L'L" \ne 0, and a minimally coupled scalar field in a general potential.

gr-qc↗
arXiv2001

On a relation between the Bach equation and the equation of geometrodynamics

The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation plays the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a more general nature. We show the following non-trivial result: there exists a conformally invariant Lagrangian, whose field equation generalizes the Bach equation and has as solutions those Ricci tensors which are solutions to the equation of geometrodynamics.

gr-qc↗
arXiv2001

On static spherically symmetric solutions of the Bach-Einstein gravitational field equations

For field equations of 4th order, following from a Lagrangian `Ricci scalar plus Weyl scalar', it is shown (using methods of non-standard analysis) that in a neighbourhood of Minkowski space there do not exist regular static spherically symmetric solutions. With that (besides the known local expansions about r = 0 and r = infinity resp.) for the first time a global statement on the existence of such solutions is given. Finally, this result will be discussed in connection with Einstein's particle programme.

gr-qc↗
arXiv2001

Inhomogeneous Cosmological Models Containing Homogeneous Inner Hypersurface Geometry. Changes of the Bianchi Type

There are investigated such cosmological models which instead of the usual spatial homogeneity property only fulfil the condition that in a certain synchronized system of reference all spacelike sections t = const. are homogeneous manifolds. This allows time-dependent changes of the BIANCHI type. Discussing differential geometrical theorems it is shown which of them are permitted. Besides the trivial case of changing into type I there exist some possible changes between other types. However, physical reasons like energy inequalities partially exclude them.

gr-qc↗
arXiv2001

Surface Layers in General Relativity and Their Relation to Surface Tensions

For a thin shell, the intrinsic 3-pressure will be shown to be analogous to -A, where A is the classical surface tension: First, interior and exterior Schwarzschild solutions will be matched together such that the surface layer generated at the common boundary has no gravitational mass; then its intrinsic 3-pressure represents a surface tension fulfilling Kelvin's relation between mean curvature and pressure difference in the Newtonian limit. Second, after a suitable definition of mean curvature, the general relativistic analogue to Kelvin's relation will be proven to be contained in the equation of motion of the surface layer.

gr-qc↗
arXiv2001

Partition functions and symmetric polynomials

We find a close correspondence between certain partition functions of ideal quantum gases and certain symmetric polynomials. Due to this correspondence it can be shown that a number of thermodynamic identities which have recently been considered are essentially of combinatorical origin and known for a long time as theorems on symmetric polynomials. For example, a recurrence relation for partition functions appearing in the textbook of P. Landsberg is nothing else but Newton's identity in disguised form. Conversely, a certain theorem on symmetric polynomials translates into a new and unexpected relation between fermionic and bosonic partition functions, which can be used to express the former by means of the latter and vice versa.

cond-mat.stat-mech↗
arXiv2001

Asymptotic Freedom in Curvature-Satured Gravity

For a spatially flat Friedmann model with line element $ds^2=a^2 [ da^2/B(a)-dx^2-dy^2-dz^2 ] $, the 00-component of the Einstein field equation reads $8πG T_{00}=3/a^2$ containing no derivative. For a nonlinear Lagrangian ${\cal L}(R)$, we obtain a second--order differential equation for $B$ instead of the expected fourth-order equation. We discuss this equation for the curvature-saturated model proposed by Kleinert and Schmidt. Finally, we argue that asymptotic freedom $G_{\rm eff}^{-1}\to 0$ is fulfilled in curvature-saturated gravity.

gr-qc↗