[Paper Review] A class of perfect-fluid cosmologies with polarised Gowdy symmetry and a Kasner-like singularity

K. Anguige|arXiv (Cornell University)|May 19, 2000

Black Holes and Theoretical Physics1 references6 citations

TL;DR

This paper constructs a new class of spatially inhomogeneous perfect-fluid cosmologies with polarized Gowdy symmetry and Kasner-like singularities using the Fuchsian algorithm. By perturbing Bianchi I solutions and solving a system of Fuchsian equations with analytic initial data, the authors prove the existence of unique solutions depending on four free analytic functions, extending previous results to include fluid dynamics under relaxed symmetry constraints.

ABSTRACT

We prove the existence of a class of perfect-fluid cosmologies with polarised Gowdy symmetry and a Kasner-like singularity. These solutions of the Einstein equations depend on four free functions of one space coordinate and are constructed by solving a system of Fuchsian equations.

Motivation & Objective

• To extend existing results on Gowdy-symmetric vacuum cosmologies to include perfect fluids with a $γ$-law equation of state.
• To establish the existence of a one-parameter family of inhomogeneous perfect-fluid solutions with polarized Gowdy symmetry and Kasner-like asymptotics at early times.
• To demonstrate that these solutions depend on the maximum number of free analytic functions (four) allowed within the symmetry class.
• To rigorously prove the existence and uniqueness of such solutions using the Fuchsian algorithm for singular initial value problems.
• To verify that the constraints of the Einstein-fluid system are preserved under the evolution equations, ensuring physical consistency.

Proposed method

• Perturb exact Bianchi I solutions in one spatial direction to construct inhomogeneous solutions with polarized Gowdy symmetry.
• Express the metric in conformal coordinates and introduce a time variable $ t $ defined via an integral transformation from the original time $ \tau $.
• Introduce perturbation variables $ \tilde{A}, \tilde{R}, \tilde{W}, \phi, \tilde{\psi} $ to describe deviations from the model Bianchi I solution.
• Rewrite the Einstein-perfect fluid equations as a first-order system of the form $ t\partial_t u + N(x)u = t^\delta H(t,x,u,u_x) $, where $ u $ includes all dynamical variables and $ H $ is analytic.
• Apply the Fuchsian algorithm to prove existence and uniqueness of analytic solutions with $ u(0) = 0 $, leveraging the positive eigenvalues of the matrix $ N(x) $.
• Verify that the constraints $ C_0 $ and $ C_1 $ vanish identically by showing their rescaled versions $ \tilde{C}_0 $ and $ \tilde{C}_1 $ satisfy a system with zero initial data and decay to zero.

Experimental results

Research questions

• RQ1 Can a class of perfect-fluid cosmologies with polarized Gowdy symmetry and Kasner-like singularities be constructed that depend on the maximum number of free functions within the symmetry class?
• RQ2 Does the Fuchsian algorithm yield unique analytic solutions for the Einstein-perfect fluid system under these symmetry and asymptotic conditions?
• RQ3 How do the constraints of the Einstein-fluid system evolve under the proposed perturbative framework, and are they preserved?
• RQ4 What conditions on the initial data (e.g., $ p_1(x) $, $ \alpha(x) $, $ m(x) $) ensure the existence of such solutions?
• RQ5 Can the perturbation framework be extended to include fluid dynamics while maintaining the analytic structure and singularity behavior?

Key findings

• The paper proves the existence of a unique solution to the Einstein-perfect fluid equations with polarized Gowdy symmetry and Kasner-like singularity, depending on four free analytic functions.
• The solution is constructed via the Fuchsian algorithm, ensuring existence and uniqueness for analytic initial data satisfying $ -\frac{1}{3} \leq p_1(x) < \gamma - 1 - k $ for small $ k > 0 $.
• The asymptotic behavior at $ t \to 0 $ matches that of a Bianchi I fluid model, with $ \rho \sim \mu $, $ v^1 \sim t^{(\gamma-1-p_1)/(1-p_1)} $, and logarithmic scaling in $ t $ for $ A $, $ W $, and $ R $.
• The constraints $ C_0 $ and $ C_1 $ are identically zero for the constructed solutions, confirming consistency of the evolution system.
• The time variable $ t $ is defined implicitly via $ t = \int_0^\tau B^{p_1 + \gamma - 5/3}(s) s^{-p_1} ds $, ensuring the correct asymptotic behavior.
• The solution framework allows for inhomogeneous $ \alpha(x), m(x), c(x), p_1(x) $, all required to be analytic and strictly positive where needed.