Linear Stability and a Turning Point Principle for Barotropic Liquid Stars with a Free Boundary in General Relativity
Abstract
We study the linear stability of spherically symmetric, asymptotically flat steady states of the Einstein-Euler system describing a liquid star surrounded by vacuum. The equation of state has positive surface density, positive causal sound speed, and an asymptotically linear high-density limit. We construct the equilibrium family and derive the linearized equation with a Robin boundary condition. The linearized operator has discrete real spectrum, and its negative Morse index equals the number of growing modes. We prove that the linearized operator around steady states with large central redshift admits a growing mode, while such growing modes do not exist for steady states with small central redshift. A parameter derivative of the equilibrium family provides a regular interior zero solution of the linearized equation, whose boundary defect is a nonzero multiple of the parameter derivative of the total mass. This gives a turning point principle: the kernel is one-dimensional precisely at mass critical points, and the sign of the crossing eigenvalue is determined by the bending of the mass-radius curve.
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BibTeX
@misc{hao2026linear,
title = {{Linear Stability and a Turning Point Principle for Barotropic Liquid Stars with a Free Boundary in General Relativity}},
author = {Zeming Hao and Jianchao Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32138},
eprint = {2610.32138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32138},
}
RIS
TY - UNPB
TI - Linear Stability and a Turning Point Principle for Barotropic Liquid Stars with a Free Boundary in General Relativity
AU - Zeming Hao
AU - Jianchao Xu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32138
UR - https://arxiv.org/abs/2610.32138
AB - We study the linear stability of spherically symmetric, asymptotically flat steady states of the Einstein-Euler system describing a liquid star surrounded by vacuum. The equation of state has positive surface density, positive causal sound speed, and an asymptotically linear high-density limit. We construct the equilibrium family and derive the linearized equation with a Robin boundary condition. The linearized operator has discrete real spectrum, and its negative Morse index equals the number of growing modes. We prove that the linearized operator around steady states with large central redshift admits a growing mode, while such growing modes do not exist for steady states with small central redshift. A parameter derivative of the equilibrium family provides a regular interior zero solution of the linearized equation, whose boundary defect is a nonzero multiple of the parameter derivative of the total mass. This gives a turning point principle: the kernel is one-dimensional precisely at mass critical points, and the sign of the crossing eigenvalue is determined by the bending of the mass-radius curve.
ER -