Dynamics of a self-gravitating shell of matter

Dynamics of a self-gravitating shell of matter is derived from the Hilbert variational principle and then described as an (infinite dimensional, constrained) Hamiltonian system. A method used here enables us to define singular Riemann tensor of a non-continuous connection {\em via} standard formulae of differential geometry, with derivatives understood in the sense of distributions. Bianchi identities for the singular curvature are proved. They match the conservation laws for the singular energy-momentum tensor of matter. Rosenfed-Belinfante and Noether theorems are proved to be still valid in case of these singular objects. Assumption about continuity of the four-dimensional spacetime metric is widely discussed.Comment: publishe

Energy-minimizing two black holes initial data

An attempt to construct the ``ground state'' vacuum initial data for the gravitational field surrounding two black holes is presented. The ground state is defined as the gravitational initial data minimizing the ADM mass within the class of data for which the masses of the holes and their distance are fixed. To parameterize different geometric arrangements of the two holes (and, therefore, their distance) we use an appropriately chosen scale factor. A method for analyzing the variations of the ADM mass and the masses (areas) of the horizons in terms of gravitational degrees of freedom is proposed. The Misner initial data are analyzed in this context: it is shown that they do not minimize the ADM mass.Comment: Minor corrections, 2 references adde

Hesed: Redeemed Brokenness in a Multimedia Retelling of the Biblical Story of Ruth

Through the powerful interaction between the visual arts and music, an ancient story of brokenness and redemption is retold. This thesis seeks to give greater insight into this multimedia retelling of the biblical book of Ruth. Scholarly sources were reviewed to deepen understanding, and works from professional visual artists and musicians were examined for this project to come together. The end product is this thesis as well as a body of art and a five-movement piece of music. This combination of visual art and music allows the relevance of the biblical book of Ruth to be seen in the transforming journey of a grieving woman and her daughter

Hesed: Discovering Redeeming Brokenness in a Retelling of the Biblical Story of Ruth

Through the powerful interaction between the visual arts and music, an ancient story of brokenness and redemption is told. This thesis seeks to give greater insight into this multimedia retelling of the biblical book of Ruth. Scholarly sources were reviewed to deepen understanding, and works from professional visual artists and musicians were examined for this project to come together. The end product is this thesis paper as well as a body of art and a five-movement piece of music. This combination of visual art and music allows the relevance of the biblical book of Ruth to be seen in the transforming journey of a grieving woman and her daughter

Unconstrained Hamiltonian formulation of General Relativity with thermo-elastic sources

A new formulation of the Hamiltonian dynamics of the gravitational field interacting with(non-dissipative) thermo-elastic matter is discussed. It is based on a gauge condition which allows us to encode the six degrees of freedom of the ``gravity + matter''-system (two gravitational and four thermo-mechanical ones), together with their conjugate momenta, in the Riemannian metric q_{ij} and its conjugate ADM momentum P^{ij}. These variables are not subject to constraints. We prove that the Hamiltonian of this system is equal to the total matter entropy. It generates uniquely the dynamics once expressed as a function of the canonical variables. Any function U obtained in this way must fulfil a system of three, first order, partial differential equations of the Hamilton-Jacobi type in the variables (q_{ij},P^{ij}). These equations are universal and do not depend upon the properties of the material: its equation of state enters only as a boundary condition. The well posedness of this problem is proved. Finally, we prove that for vanishing matter density, the value of U goes to infinity almost everywhere and remains bounded only on the vacuum constraints. Therefore the constrained, vacuum Hamiltonian (zero on constraints and infinity elsewhere) can be obtained as the limit of a ``deep potential well'' corresponding to non-vanishing matter. This unconstrained description of Hamiltonian General Relativity can be useful in numerical calculations as well as in the canonical approach to Quantum Gravity.Comment: 29 pages, TeX forma

Rigid spheres in Riemannian spaces

Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct such foliations. For this purpose we define a special family of topological two-spheres, which we call "rigid spheres". We prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold (in case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere). The rigid spheres can be used as building blocks for various ("spherical", "bispherical" etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. \cite{LHH}) and the foliations used by us (cf. \cite{JKL}) in the analysis of the two-body problem.Comment: 23 page

Podstawowa bariera procesowa uzyskania statusu uchodźcy w Rzeczypospolitej Polskiej

Zdigitalizowano i udostępniono w ramach projektu pn. Rozbudowa otwartych zasobów naukowych Repozytorium Uniwersytetu w Białymstoku, dofinansowanego z programu „Społeczna odpowiedzialność nauki” Ministra Edukacji i Nauki na podstawie umowy SONB/SP/512497/202123925

Ochrona gruntów rolnych - prawda czy fikcja?

Zdigitalizowano i udostępniono w ramach projektu pn. Rozbudowa otwartych zasobów naukowych Repozytorium Uniwersytetu w Białymstoku, dofinansowanego z programu „Społeczna odpowiedzialność nauki” Ministra Edukacji i Nauki na podstawie umowy SONB/SP/512497/202111613

Energy and angular momentum of the weak gravitational waves on the Schwarzschild background -- quasilocal gauge-invariant formulation

It is shown that the axial and polar perturbations of the spherically symmetric black hole can be described in a gauge-invariant way. The reduced phase space describing gravitational waves outside of the horizon is described by the gauge-invariant quantities. Both degrees of freedom fulfill generalized scalar wave equation. For the axial degree of freedom the radial part of the equation corresponds to the Regge-Wheeler result (Phys. Rev. 108, 1063-1069 (1957)) and for the polar one we get Zerilli result (Phys. Rev. D2, 2141-2160 (1970)), see also Chandrasekhar (The Mathematical Theory of Black Holes,(Clarendon Press Oxford, 1983)), Moncrief (Annals of Physics 88, 323-342 (1974)) for both. An important ingredient of the analysis is the concept of quasilocality which does duty for the separation of the angular variables in the usual approach. Moreover, there is no need to represent perturbations by normal modes (with time dependence $\exp(-ikt)$), we have fields in spacetime and the Cauchy problem for them is well defined outside of the horizon. The reduced symplectic structure explains the origin of the axial and polar invariants. It allows to introduce an energy and angular momentum for the gravitational waves which is invariant with respect to the gauge transformations. Both generators represent quadratic approximation of the ADM nonlinear formulae in terms of the perturbations of the Schwarzschild metric. We also discuss the boundary-initial value problem for the linearized Einstein equations on a Schwarzschild background outside of the horizon.Comment: 23 page