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Static Stellar Phase Transitions in General Relativity and a Generalized Buchdahl Limit

Moritz Reintjes, Ruochen Xia

TL;DR

This work constructs static, spherically symmetric GR stellar models with prescribed, potentially discontinuous density profiles by solving the TOV equation for the pressure. The authors introduce a general sufficient condition that guarantees a bounded p(r) and extend Buchdahl-type limits to nonuniform densities, providing a constructive ladder of staircase-density solutions that model phase transitions and connect to Schwarzschild exterior. They also prove a necessary Buchdahl-type condition and develop a convergence framework showing how general densities can be approximated by staircase profiles, yielding existence of solutions for broad classes of densities. The paper further demonstrates the practical implications via explicit mass–radius bounds for several density families (e.g., Tolman VII, power laws) and discusses the impact on phase-transition modeling and relativistic stellar structure. Overall, it advances the theory and numerical construction of static GR stars with phase transitions and refines mass–radius constraints beyond the classical constant-density Buchdahl limit.

Abstract

We give the first general construction of solutions of the static spherically symmetric Einstein-Euler equations, the Tolman-Oppenheimer-Volkoff (TOV-)equation, with prescribed density functions allowed to be discontinuous and non-uniform; these solutions describe stellar phase transitions in General Relativity. Boundedness of the resulting pressure functions solving the TOV-equations, from the boundary down to the stellar center, is obtained by identifying a novel condition on the prescribed density, in generalization of the classical Buchdahl limit. Moreover, we introduce a new necessary condition for the existence of such bounded pressure functions, which in the special case of a uniform density state reduces to the classical Buchdahl limit on the stellar mass-radius relationship. We present various examples to study the stellar mass-radius relationships resulting from our new conditions.

Static Stellar Phase Transitions in General Relativity and a Generalized Buchdahl Limit

TL;DR

This work constructs static, spherically symmetric GR stellar models with prescribed, potentially discontinuous density profiles by solving the TOV equation for the pressure. The authors introduce a general sufficient condition that guarantees a bounded p(r) and extend Buchdahl-type limits to nonuniform densities, providing a constructive ladder of staircase-density solutions that model phase transitions and connect to Schwarzschild exterior. They also prove a necessary Buchdahl-type condition and develop a convergence framework showing how general densities can be approximated by staircase profiles, yielding existence of solutions for broad classes of densities. The paper further demonstrates the practical implications via explicit mass–radius bounds for several density families (e.g., Tolman VII, power laws) and discusses the impact on phase-transition modeling and relativistic stellar structure. Overall, it advances the theory and numerical construction of static GR stars with phase transitions and refines mass–radius constraints beyond the classical constant-density Buchdahl limit.

Abstract

We give the first general construction of solutions of the static spherically symmetric Einstein-Euler equations, the Tolman-Oppenheimer-Volkoff (TOV-)equation, with prescribed density functions allowed to be discontinuous and non-uniform; these solutions describe stellar phase transitions in General Relativity. Boundedness of the resulting pressure functions solving the TOV-equations, from the boundary down to the stellar center, is obtained by identifying a novel condition on the prescribed density, in generalization of the classical Buchdahl limit. Moreover, we introduce a new necessary condition for the existence of such bounded pressure functions, which in the special case of a uniform density state reduces to the classical Buchdahl limit on the stellar mass-radius relationship. We present various examples to study the stellar mass-radius relationships resulting from our new conditions.

Paper Structure

This paper contains 7 sections, 7 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statement of Results
  4. Exact Solutions for Staircase Density Functions
  5. Sufficient Condition for a Bounded Pressure
  6. Necessary Condition
  7. Examples and Applications

Key Result

Theorem 3.1

Assume $\rho(r)$ is piece-wise continuous, non-increasing and bounded on $[0,R]$, $\rho(r)>0$ for all $r\in [0,R)$ and $\rho(r)=0$ for all $r>R$, and assume Outside_Schwarzschild holds. Assume further there exists a constant $\Delta\in(0,1)$ such that where the mass function $m(r)$ is given by m(r). Then there exists a unique solution $\{p(r),\nu(r)\} \in C^{0,1}([0,R])$ of the static spherically

Theorems & Definitions (12)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • ...and 2 more