[Paper Review] Instanton representation of Plebanski gravity: IX. Hamiltonian minisuperspace dynamics in undensitized momentum space variables

Eyo Eyo Ita|arXiv (Cornell University)|Jul 3, 2007

Cosmology and Gravitation Theories11 references3 citations

TL;DR

This paper investigates Hamiltonian minisuperspace dynamics in Plebanski gravity's instanton representation using undensitized momentum space variables, deriving new exact solutions in both degenerate and nondegenerate sectors. It demonstrates that the degenerate and nondegenerate sectors are preserved under time evolution and introduces a fixed-point iteration algorithm for constructing general solutions via expansion about isotropic initial conditions, avoiding the big bang singularity classically.

ABSTRACT

In this paper we illustrate the dynamics of the instanton representation in the description of vacuum GR in minisuperspace for undensitized variables. We uncover a new class of general solutions in both the degenerate and the nondegenerate sectors of the theory. Additionally, the individual sectors are preserved under Hamiltonian evolution. Finally, we present an algorithm for constructing general solutions by expansion about the isotropic sector.

Motivation & Objective

• To explore classical dynamics of the instanton representation of Plebanski gravity in minisuperspace using undensitized momentum variables.
• To identify and classify new exact solutions in both degenerate and nondegenerate sectors of the theory.
• To demonstrate that the degenerate and nondegenerate sectors remain invariant under Hamiltonian evolution.
• To develop a systematic algorithm for constructing general solutions through asymptotic expansion about the isotropic sector.
• To show that the big bang singularity can be avoided classically in this formulation without quantum corrections.

Proposed method

• Formulates the Hamiltonian action for the instanton representation in minisuperspace, reducing spatial gradients to zero for spatially homogeneous configurations.
• Derives Hamilton's equations of motion for the CDJ matrix $\Psi_{ae}$ and its conjugate velocities $\dot{X}^{ae}$, using undensitized variables.
• Reduces the constraints—Hamiltonian, diffeomorphism, and Gauss’ law—to minisuperspace, showing they depend linearly on $\psi_d$ and are thus redundant.
• Introduces a fixed-point iteration scheme using $\varphi_n(t)$ to generate asymptotic expansions about the isotropic solution, with $\varphi_0$ as the initial value.
• Applies the method to both nondegenerate and degenerate cases, solving the resulting equations via integration and trigonometric inversion.
• Uses the identity $\Psi^{-1}_{ae} = (\sigma^{-1})^a_i B^i_e$ under the CDJ ansatz to recover Ashtekar variables in the nondegenerate limit.

Experimental results

Research questions

• RQ1 Can new classical solutions be derived in the instanton representation of Plebanski gravity using undensitized momentum variables?
• RQ2 Are the degenerate and nondegenerate sectors of the theory preserved under Hamiltonian evolution?
• RQ3 Can a general solution be systematically constructed via expansion about the isotropic sector?
• RQ4 Does the big bang singularity emerge or can it be avoided in this classical formulation?
• RQ5 What is the role of the CDJ matrix $\Psi_{ae}$ as a fundamental momentum variable in minisuperspace dynamics?

Key findings

• The paper uncovers a new class of general solutions in both degenerate and nondegenerate sectors of the instanton representation, preserving sector structure under time evolution.
• The Hamiltonian dynamics in the nondegenerate sector leads to a solution involving inverse trigonometric functions of $\varphi(t)$, with $\varphi(t)$ evolving via $\sqrt{\varphi(t;\epsilon)} = \Bigl{(}\sqrt{\varphi_0} + \frac{3}{\Lambda\sqrt{\varphi_0}}\Bigr{)}\Bigl{[}1 - \sqrt{\frac{\Lambda\varphi_0}{3}}\tan\bigl{(}\frac{\sqrt{3\Lambda}}{2}\int_0^t E(\varphi,\epsilon)dt'\bigr{)}\Bigr{]}^{-1} - \frac{3}{\Lambda\sqrt{\varphi_0}}$.
• In the degenerate sector, the theory admits solutions labeled by eight arbitrary complex constants $\epsilon_{ae}$, with the initial value $\varphi_0$ determining whether the universe remains degenerate or transitions to nondegenerate.
• The fixed-point iteration scheme defined by $\varphi_{n+1}(t) = \sqrt{\frac{3}{\Lambda}}\tan\Bigl{[}\tan^{-1}\bigl{(}\sqrt{\frac{\Lambda\varphi_0}{3}}\bigr{)} + \frac{\sqrt{3\Lambda}}{2}\int_0^t E(\varphi_n(t'),\epsilon)dt'\Bigr{]}$ converges to a general solution under suitable conditions.
• The big bang singularity is avoided classically in this formulation, as the equations of motion remain well-defined even at arbitrarily small times, independent of quantum corrections.
• The configuration space variables $X^{ae}$ are not globally integrable, but their velocity combinations $\dot{X}^{ae}$ still allow for a consistent time evolution and solution construction.