[Paper Review] Weakly regular T2 symmetric spacetimes. The global geometry of future developments

Philippe G. LeFloch, Jacques Smulevici|arXiv (Cornell University)|Dec 29, 2010

Black Holes and Theoretical Physics7 references3 citations

TL;DR

This paper establishes the global well-posedness of the vacuum Einstein equations for $T^2$-symmetric spacetimes under weak regularity assumptions, introducing a geometric framework that ensures existence, uniqueness, and stability of future developments. It proves that the area function $R$ grows to infinity and admits a global foliation by $R =$ const hypersurfaces, with metric coefficients satisfying optimal $L^p$ and Sobolev-type regularity bounds despite minimal differentiability requirements.

ABSTRACT

Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity assumptions are the minimal ones allowing to give a meaning to the Einstein equations under the assumed symmetry and to solve the initial value problem. First of all, we introduce a frame adapted to the symmetry in which each Christoffel symbol can be checked to belong to some Lp space. We identify certain cancellation properties taking place in the expression of the Riemann and Ricci curvatures, and this leads us to a reformulation of the initial value problem for the Einstein field equations when the initial data set has weak regularity. Second, we investigate the future development of a weakly regular initial data set. We check that the area R of the orbits of symmetry must grow to infinity in the future timelike directions, and we establish the existence of a global foliation by the level sets of R. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the orbits of symmetry are in the Sobolev space H1 and the remaining coefficients have even weaker regularity. We develop here the compactness arguments required to cover the natural level of regularity associated with the energy of the system of partial differential equations determined from Einstein's field equations.

Motivation & Objective

• To develop a fully geometric, weakly regular well-posedness theory for $T^2$-symmetric vacuum Einstein spacetimes.
• To identify the minimal regularity conditions under which the Einstein equations remain meaningful and solvable.
• To establish the global causal and geometric structure of the future development of weakly regular initial data sets.
• To prove the existence of a global foliation by level sets of the area function $R$, which tends to infinity in future timelike directions.
• To derive compactness and a priori estimates that ensure stability and uniqueness under weak regularity.

Proposed method

• Adopt a geometric, coordinate-independent formulation of the Einstein equations under $T^2$-symmetry, using a frame adapted to the symmetry group.
• Identify cancellation structures in the Riemann and Ricci curvatures to reformulate the initial value problem in weak regularity.
• Use areal coordinates to parametrize spacetime and derive evolution equations for metric components $U, A, u, a, G, H$, with $R$ as the time variable.
• Establish $L^p$ and Sobolev-type bounds on all metric components, including $a$ and its derivatives, via energy estimates and integration by parts.
• Prove that the area function $R$ is Lipschitz and grows to infinity in future timelike directions, enabling a global foliation.
• Develop novel compactness arguments based on weak convergence and uniform $L^1$ estimates to handle the low regularity of the initial data.

Experimental results

Research questions

• RQ1 What is the minimal regularity required for the initial data to give a meaningful formulation of the Einstein equations under $T^2$-symmetry?
• RQ2 Does the future development of a weakly regular $T^2$-symmetric initial data set admit a global foliation by hypersurfaces of constant area $R$?
• RQ3 Can one establish global existence, uniqueness, and stability of solutions to the Einstein equations under weak regularity, with only $H^1$-regularity on initial metric coefficients?
• RQ4 How does the area function $R$ behave asymptotically in the future, and does it grow to infinity?
• RQ5 What are the optimal $L^p$ and Sobolev-type estimates for the metric components and their derivatives under weak regularity assumptions?

Key findings

• The area function $R$ is Lipschitz continuous and grows to infinity along all future-directed timelike curves, ensuring global hyperbolicity.
• A global foliation by hypersurfaces $R = \text{const}$ exists on the entire future development.
• The metric coefficients $U, A$ belong to $C^0_R(H^1_\theta(S^1)) \cap C^1_R(L^2_\theta(S^1))$, with $\eta$ in $C^0_R(W^{1,1}_\theta(S^1)) \cap C^1_R(L^1_\theta(S^1))$, indicating optimal regularity.
• The coefficient $a$ and its inverse satisfy $a, a^{-1} \in C^0_R(W^{2,1}_\theta(S^1)) \cap C^1_R(W^{1,1}_\theta(S^1))$, with uniform $L^1$ bounds on second derivatives.
• The mixed derivative $a_{R\theta}$ is controlled pointwise by the energy density $E$, with $|a_\theta| \leq C^*$ uniformly.
• Higher-order derivatives of $a$, including $a_{RR}, a_{R\theta}, a_{\theta\theta}$, are uniformly bounded in $L^1$ norm, confirming the robustness of the solution structure.