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Central pressure is unbounded at the constant-density branch limit

Physical scope

A counterexample within the explicit constant-density branch as u approaches 8/9 from below; no general Buchdahl theorem or claim about realistic matter.

Statement refuted

For fixed energy density $ε_{0} > 0$ , there is a finite constant $K_{0}$ such that every radius $R > 0$ with $u = 8 π G ε_{0} R^{2} / (3 c^{4}) \in (0, 8 / 9)$ in the constant-density family satisfies $P (0) \leq K_{0} ε_{0}$ .

Facts & Assumptions

Given: The constant-density family from Verified constant-density static star and vacuum exterior, including its strict range $0 < u < 8 / 9$ and central-pressure formula.

[F1]

Verified constant-density static star and vacuum exterior states $P (0) / ε_{0} = (1 - a) / (3 a - 1)$ with $a = \sqrt{1 - u}$ for every $0 < u < 8 / 9$ ; this is a family-specific formula, not a general compactness theorem.

Counterexample

Given: Fixed $ε_{0} > 0$ , varying radii with $u = 8 π G ε_{0} R^{2} / (3 c^{4}) \in (0, 8 / 9)$ , the uniform-bound statement above, and the family-specific formula [F1].

1.1givenF1constructalgebra

For each integer $n \geq 2$ , set $u_{n} = 8 / 9 - 1 / n$ and $R_{n} = c^{2} \sqrt{3 u_{n} / (8 π G ε_{0})} > 0$ . Keep $ρ_{0} = ε_{0} / c^{2}$ fixed and set $M_{n} = 4 π ρ_{0} R_{n}^{3} / 3$ . Then $2 G M_{n} / (c^{2} R_{n}) = 8 π G ε_{0} R_{n}^{2} / (3 c^{4}) = u_{n} \in (0, 8 / 9)$ , so every $(ρ_{0}, R_{n})$ is an admissible member of [F1] at the fixed density. Moreover $a_{n} = \sqrt{1 - u_{n}} = \sqrt{1 / 9 + 1 / n} > 1 / 3$ and $a_{n} ↓ 1 / 3$ .

2.1F1step 1.1algebra∎

By [F1], $P_{n} (0) / ε_{0} = (1 - a_{n}) / (3 a_{n} - 1)$ . The numerator tends to $2 / 3 > 0$ and the denominator tends to $0$ through positive values, so the ratios tend to $+ \infty$ . Therefore no finite $K_{0}$ can bound all ratios, refuting the displayed statement. At the limiting value $u = 8 / 9$ , the denominator is zero and the endpoint is not an admissible regular finite-pressure member.

Depends on

Verified constant-density static star and vacuum exterior

Dependency tree · 2 levels

Sources

MIT 8.962, General Relativity, Spring 2020 Lecture Guide, Lecture 21, PDF pp. 8–10