[Paper Review] Static Solutions of the Einstein Equations for Spherically Symmetric Elastic Bodies

Jiseong Park|arXiv (Cornell University)|Oct 4, 1998

Elasticity and Material Modeling11 references18 citations

TL;DR

This paper establishes the existence and uniqueness of regular, spherically symmetric static solutions to the Einstein equations for elastic bodies with a general constitutive relation. By reducing the system to a Fuchsian ODE for strain invariants, it proves local existence near the center and, for quadratic constitutive relations, global regularity up to the boundary, ensuring a smooth match to Schwarzschild vacuum with mass-radius ratio bounded by 1/2.

ABSTRACT

The paper is concerned with the Einstein equations for a spherically symmetric static distribution of anisotropic matter. The equations are cast into a system of Fuchsian type ODE for certain scalar invariants of the strain. And then the existence and regularity of this ODE is studied under general constitutive relation. In the case the constitutive relation is given by a quadratic form of strain, it is also shown that the solutions stay regular up to the boundary of the material ball.

Motivation & Objective

• To resolve the lack of general existence theorems for spherically symmetric elastic bodies in general relativity.
• To establish the existence of regular solutions near the center for arbitrary constitutive relations satisfying compatibility conditions.
• To show that solutions remain regular up to the boundary when the constitutive relation is quadratic in strain.
• To prove that the mass-radius ratio is bounded above by 1/2, ensuring asymptotic flatness and smooth vacuum matching.
• To extend the analysis to non-flat material manifolds, showing the results hold under mild geometric assumptions.

Proposed method

• Reducing the spherically symmetric Einstein equations to a system of Fuchsian ODEs for scalar invariants of the strain tensor.
• Using the Rendall-Schmidt theorem on Fuchsian ODEs to prove local existence and uniqueness of solutions near the center.
• Defining strain via the logarithmic map of the pullback metric relative to the spacetime metric, ensuring well-defined, symmetric, and spatial strain tensors.
• Employing a constitutive relation that expresses energy density and stress as functions of strain, enabling reduction to a closed ODE system.
• Deriving a bound on the mass-radius ratio via a method analogous to Rein (2001) for the Vlasov case, ensuring global regularity and vacuum matching.
• Extending the results to non-flat material manifolds by modifying the auxiliary equation for the radial derivative of the strain invariant.

Experimental results

Research questions

• RQ1 Under what conditions does a unique regular solution exist for a spherically symmetric elastic body in general relativity with a given constitutive relation and central pressure?
• RQ2 Can solutions remain regular up to the boundary of the material ball when the constitutive relation is quadratic in strain?
• RQ3 What is the upper bound on the mass-radius ratio for elastic bodies under spherical symmetry, and does it ensure a smooth match to Schwarzschild vacuum?
• RQ4 How does the theory extend to non-flat material manifolds, and does the local existence result still hold?
• RQ5 Is the tangential stress regular a priori, or is its regularity a consequence of the mass-radius ratio bound?

Key findings

• For any given constitutive relation and central pressure satisfying compatibility conditions, a unique regular solution exists in a neighborhood of the center.
• When the constitutive relation is quadratic in strain, the solution remains regular up to the boundary where the radial stress vanishes.
• The mass-radius ratio is bounded above by a constant less than 1/2, ensuring the spacetime is asymptotically flat.
• The tangential stress remains bounded and regular throughout the body, even though its regularity is not assumed a priori.
• The results extend to non-flat material manifolds with non-negative, bounded, and increasing material metric functions.
• The system remains well-posed under the same Fuchsian framework when the material metric is non-flat, provided the function f(y) satisfies g(0)=1 and f'(y) ≥ 0.