arXiv subjects

Panagiotis Kordas

Publications and source records attributed to Panagiotis Kordas.

2 recordsLinked to original sources

arXiv2021

Observables in terms of connection and curvature variables for Einstein's equations with two commuting Killing vectors

Einstein's equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection $A(ς, η, γ)=Ψ_{,γ}Ψ^{-1}$, where $γ$ the variable spectral parameter are considered. A transition matrix ${\cal T}= A(ς, η, γ)A^{-1}(ξ, η, γ)$ for $A$ is defined relating $A$ at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable pde's theory. A transition matrix on $ς={\mbox constant}$ is defined in an analogous manner. These transition matrices allow us to obtain a hierarchy of integrals of motion with respect to time, purely in terms of the trace of a function of the connections $g_{,ς}g^{-1}$ and $g_{,η}g^{-1}$. Furthermore a hierarchy of integrals of motion in terms of the curvature variable $B=A_{,γ}A^{-1}$, involving the commutator $[A(1), A(-1)]$, is obtained. We interpret the inhomogeneous wave equation that governs $σ=ln N$, $N$ the lapse, as a Klein-Gordon equation, a dispersion relation relating energy and momentum density, based on the first connection observable and hence this first observable corresponds to mass. The corresponding quantum operators are $\frac{\partial}{\partial t}$, $\frac{\partial}{\partial z}$ and this means that the full Poincare group is at our disposal.

gr-qc↗

arXiv2011

Transition Matrix, Poisson Bracket for gravitational solitons in the dressing formalism

The Hamiltonian methods of the theory of solitons are applied to gravisolitons in the dressing formalism. The Poisson bracket for the Lie-algebra valued one-form $A(ς, η, γ)=Ψ_{,γ}Ψ^{-1}$, for gravisolitons in the dressing formalism, for a specific background solution, is defined and computed, agreeing with results previously obtained. A transition matrix ${\cal T}=A(ς, η, γ) A^{-1}(η, ξ, -γ)$ for $A$ is defined relating $A$ at ingoing and outgoing light cones. It is proved that it satisfies equations familiar from integrable pde's with the role of time played by the null coordinate $η$. This is a new result mathematically, since there has not been a transition matrix for $A$ in the litterature, while physically it presents the possibility of obtaining integrals of motion (for appropriate boundary conditions), from the trace of the derivative with respect to the null coordinate $η$, of ${\cal T}$, in terms of classical relativity connections, since $A(γ= \pm 1)$ can be expressed in a simple way in terms of the classical Christoffel symbols. This may prove of use upon quantization since connections are fundamental variables of quantum gravity. The roles of $η$ and $ς$ may be reversed to obtain integrals of motion for $ς$, thus $ς$ playing the role of time. This ties well with the two-time interpretation and approach already established before.

gr-qc↗