Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Verified constant-density static star and vacuum exterior

Physical scope

Exact constant-density family in the adopted static Lambda=0 model for 0<u<8/9, including regular center and explicit C1 piecewise-C2 vacuum matching; the bound is family-specific, with no realistic causal EOS, general Buchdahl conclusion, empirical claim, or stability result.

Statement

Assume the constant-density preparation Specified constant-density star and vacuum boundary model and the exact $Z F + {A C}_{ω}$ mathematical convention used by Exact regular constant-density solution with verified matching. Put $a = \sqrt{1 - u}$ and $B (r) = \sqrt{1 - u r^{2} / R^{2}}$ . For $0 \leq r \leq R$ , the adopted model has the explicit profiles $M (r) = M_{R} (r / R)^{3}, b (r) = B (r)^{2}, P (r) = ε_{0} \frac{B (r) - a}{3 a - B (r)}, e^{ϕ (r)} = \frac{3 a - B (r)}{2} .$ For $r \geq R$ , set $M (r) = M_{R}$ , $b (r) = 1 - 2 G M_{R} / (c^{2} r)$ , $P (r) = 0$ , $ε (r) = 0$ , and $e^{ϕ (r)} = \sqrt{b (r)}$ . Then the interior is a regular static Einstein-fluid solution through the center, the exterior is vacuum, and the join at $r = R$ is $C^{1}$ and piecewise $C^{2}$ with no delta-supported surface stress. The central pressure is finite and equals $P (0) = ε_{0} (1 - a) / (3 a - 1)$ . This conclusion is specific to the constant-density branch $0 < u < 8 / 9$ .

Facts & Assumptions

Given: The constant-density EOS, $ρ_{0} > 0$ , $R > 0$ , $u \in (0, 8 / 9)$ , the adopted static Einstein-fluid model, and the exact current mathematical supplier under $Z F + {A C}_{ω}$ .

[F1]

Specified constant-density star and vacuum boundary model supplies $ε_{0} = ρ_{0} c^{2}$ , $M_{R} = 4 π ρ_{0} R^{3} / 3$ , $u = 2 G M_{R} / (c^{2} R) \in (0, 8 / 9)$ , and the constant EOS on $I_{P} = (0, \infty)$ together with its smooth extension $\hat{E}$ to an open interval containing zero, so the interior boundary value $P (R) = 0$ is defined. It prescribes the vacuum exterior separately from the fluid EOS.

[F2]

Exact regular constant-density solution with verified matching: under those parameters and the geometry of Smooth static spherical geometry and state domain, the displayed $M, b, P, e^{ϕ}$ solve the interior Einstein/conservation equations, extend to a $C^{2}$ Cartesian center, and match to the exterior at $r = R$ as a $C^{1}$ piecewise- $C^{2}$ metric with no distributional surface stress; the bound is only for this family.

[A1]

The Einstein equation and constant-density boundary preparation in [F1] are adopted physical premises, not observations.

Proof

1.1givenF1algebra

Since $u < 8 / 9$ , $a = \sqrt{1 - u} > 1 / 3$ . For $0 \leq r \leq R$ , $a \leq B (r) \leq 1$ , so $b = B^{2} > 0$ and $3 a - B (r) \geq 3 a - 1 > 0$ . Thus the stated interior pressure and lapse are finite, $P (R) = 0$ , and $P (0) = ε_{0} (1 - a) / (3 a - 1)$ .

2.1A1F1F2step 1.1∎

The inputs in [F1] meet the hypotheses of [F2] exactly. That mathematical result verifies the mass, lapse, pressure, and vacuum equations and the regular center and interface; therefore these fields solve the adopted physical model. The discontinuous density at the boundary is part of the stated piecewise model, and the verified $C^{1}$ matching supplies no delta-supported surface stress.

Verified constant-density static star and vacuum exterior

Statement

Facts & Assumptions

Proof

Depends on

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