Lorentz Invariant Vacuum Solutions in General Relativity
All Lorentz invariant solutions of vacuum Einstein's equations are found. It is proved that these solutions describe space-times of constant curvature.
physics.gen-ph↗
arXiv subjects
M. O. Katanaev
Publications and source records attributed to M. O. Katanaev.
Torsional Elastic Waves in Double Wall Tube
We describe the double wall tube with cylindrical dislocation in the framework of the geometric theory of defects. The induced metric is found. The dispersion relation is obtained for the propagation of torsional elastic waves in the double wall tube.
physics.class-ph↗
Torsional Elastic Waves in Cylindrical Waveguide with Wedge Dislocation
We describe propagation of torsional elastic waves in cylindrical waveguide with wedge dislocation in the framework of geometric theory of defects. The defect changes the dispersion relation. For positive deficit angles, it increases the phase velocity, and decreases the group velocity of the wave.
cond-mat.mtrl-sci↗
Passing the Einstein-Rosen bridge
We relax the requirement of geodesic completeness of a space-time. Instead, we require test particles trajectories to be smooth only in the physical sector. Test particles trajectories for Einstein--Rosen bridge are proved to be smooth in the physical sector, and particles can freely penetrate the bridge in both directions.
gr-qc↗
Point massive particle in General Relativity
It is well known that the Schwarzschild solution describes the gravitational field outside compact spherically symmetric mass distribution in General Relativity. In particular, it describes the gravitational field outside a point particle. Nevertheless, what is the exact solution of Einstein's equations with $δ$-type source corresponding to a point particle is not known. In the present paper, we prove that the Schwarzschild solution in isotropic coordinates is the asymptotically flat static spherically symmetric solution of Einstein's equations with $δ$-type energy-momentum tensor corresponding to a point particle. Solution of Einstein's equations is understood in the generalized sense after integration with a test function. Metric components are locally integrable functions for which nonlinear Einstein's equations are mathematically defined. The Schwarzschild solution in isotropic coordinates is locally isometric to the Schwarzschild solution in Schwarzschild coordinates but differs essentially globally. It is topologically trivial neglecting the world line of a point particle. Gravity attraction at large distances is replaced by repulsion at the particle neighbourhood.
gr-qc↗
On geometric interpretation of the Aharonov-Bohm effect
A geometric interpretation of the Aharonov--Bohm effect is given in terms of connections on principal fiber bundles. It is demonstrated that the principal fiber bundle can be trivial while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.
math-ph↗
On geometric interpretation of the Berry phase
A geometric interpretation of the Berry phase and its Wilczek--Zee non-Abelian generalization are given in terms of connections on principal fiber bundles. It is demonstrated that a principal fiber bundle can be trivial in all cases, while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.
math-ph↗
Adiabatic theorem for finite dimensional quantum mechanical systems
A new simple proof of the adiabatic theorem is given in the finite dimensional case for nondegenerate as well as degenerate states. The explicitly integrable two level system is considered as an example. It is demonstrated that the error estimate given by the adiabatic theorem can not be improved.
math-ph↗
Schrodinger equation in the space with cylindrical geometric defect and possible application to multi-wall nanotubes
The recently invented cylindrical geometric space defect is applied to the electron behaviour in the system which can be regarded as a simplified model of a double-wall nanotube. By solving the Schrodinger equation in the region of space with cylindrical geometric defect we explore the influence of such geometric defect on the energy gap and charge distribution. The effect is qualitatively similar to the one obtained earlier by means of traditional simulation methods. In general, the geometric approach can not compete with the known methods of theoretical study of the nanostructures, such as molecular dynamics. However it may be useful for better qualitative understanding of the electronic properties of the nanosystems.
cond-mat.mes-hall↗
Tube dislocations in gravity
We consider static massive thin cylindrical shells (tubes) as the sources in Einstein's equations. They correspond to $\dl$- and $\dl'$-function type energy-momentum tensors. The corresponding metric components are found explicitly. They are not continuous functions in general and lead to ambiguous curvature tensor components. Nevertheless all ambiguous terms in Einstein's equations safely cancel. The interplay between elasticity theory, geometric theory of defects, and General Relativity is analyzed. The elasticity theory provides a simple picture for defects creation and a new look on General Relativity.
gr-qc↗
Global solutions in gravity. Euclidean signature
We consider a wide class of two-dimensional metrics having one Killing vector. The method is proposed for the construction of maximally extended surfaces with the given Riemannian metric which is the analog of the conformal block method for two-dimensional Lorentzian signature metrics. The Schwarzschild solution is considered as an example.
gr-qc↗
Inside the BTZ black hole
We consider static circularly symmetric solution of three-dimensional Einstein's equations with negative cosmological constant (the BTZ black hole). The case of zero cosmological constant corresponding to the interior region of a black hole is analyzed in detail. We prove that the maximally extended BTZ solution with zero cosmological constant coincides with flat three-dimensional Minkowskian space-time without any singularity and horizons. The Euclidean version of this solution is shown to have physical interpretation in the geometric theory of defects in solids describing combined wedge and screw dislocations.
gr-qc↗
Polynomial Hamiltonian form of General Relativity
Phase space of General Relativity is extended to a Poisson manifold by inclusion of the determinant of the metric and conjugate momentum as additional independent variables. As a result, the action and the constraints take a polynomial form. New expression for the generating functional for the Green functions is proposed. We show that the Dirac bracket defines degenerate Poisson structure on a manifold, and a second class constraints are the Casimir functions with respect to this structure. As an application of the new variables, we consider the Friedmann universe.
gr-qc↗
Polynomial form of the Hilbert-Einstein action
Configuration space of general relativity is extended by inclusion of the determinant of the metric as a new independent variable. As the consequence the Hilbert-Einstein action takes a polynomial form.
gr-qc↗
Geometric Theory of Defects
A description of dislocations and disclinations defects in terms of Riemann--Cartan geometry is given, with the curvature and torsion tensors being interpreted as the surface densities of the Frank and Burgers vectors, respectively. A new free energy expression describing the static distribution of defects is presented, and equations of nonlinear elasticity theory are used to specify the coordinate system. Application of the Lorentz gauge leads to equations for the principal chiral SO(3)-field. In the defect-free case, the geometric model reduces to elasticity theory for the displacement vector field and to a principal chiral SO(3)-field model for the spin structure. As illustrated by the example of a wedge dislocation, elasticity theory reproduces only the linear approximation of the geometric theory of defects. It is shown that the equations of asymmetric elasticity theory for the Cosserat media can also be naturally incorporated into the geometric theory as the gauge conditions. As an application of the theory, phonon scattering on a wedge dislocation is considered. The energy spectrum of impurity in the field of a wedge dislocation is also discussed.
cond-mat.mtrl-sci↗
Introduction to the Geometric Theory of Defects
We describe defects - dislocations and disclinations - in the framework of Riemann-Cartan geometry. Curvature and torsion tensors are interpreted as surface densities of Frank and Burgers vectors, respectively. Equations of nonlinear elasticity theory are used to fix the coordinate system. The Lorentz gauge yields equations for the principal chiral SO(3)-field. In the absence of defects the geometric model reduces to the elasticity theory for the displacement vector field and to the principal chiral SO(3)-field model for the spin structure.
cond-mat.mtrl-sci↗
One-dimensional topologically nontrivial solutions in the Skyrme model
We consider the Skyrme model using the explicit parameterization of the rotation group SO(3) through elements of its algebra. Topologically nontrivial solutions already arise even in the one-dimensional case because the fundamental group of SO(3) is Z_2. We explicitly find and analyze one-dimensional static solutions. Among them, there are topologically nontrivial solutions with finite energy among them. We propose a new class of projective models whose target spaces are arbitrary real projective spaces RP^d.
hep-th↗
Wedge Dislocation in the Geometric Theory of Defects
We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory reproduces quantitatively all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.
cond-mat.mtrl-sci↗