Maximum compactness of static stars when the lapse is tied to the isotropic conformal factor, N = 2/ψ − 1

Abstract

In isotropic coordinates the Schwarzschild metric satisfies N = 2/ψ − 1 for every mass, so if the lapse is a function of the isotropic conformal factor alone, vacuum general relativity fixes that function (in areal coordinates the same argument gives a different closure, Jacobson 2007: the choice of the isotropic variable is an assumption). Imposing this closure inside matter, while keeping the Hamiltonian constraint and energy–momentum conservation of a perfect fluid, gives a phenomenological closure of the static equations in which the fluid pressure does not enter the field equations. Read with Einstein's equations, the stars are anisotropic. For uniform density, in closed form: maximum compactness 2^(1/3) − 2^(−1/3) = 0.4662 against Buchdahl's 4/9, surface redshift 2.847 against 2, and 0.4043 against 3/8 if p_c ≤ ρ is imposed. For five density profiles the maximum compactness is 3–5% above GR. At observed compactness the required central pressure is already lower than in GR (by 7% at C = 0.15), so realistic mass–radius curves differ. With the SLy and APR4 equations of state the closure raises the maximum mass by about 5% and the radius of a 1.4 solar-mass star by 0.1 km, about ten times less than present NICER errors: the closure is compatible with PSR J0437−4715 and PSR J0740+6620 but not distinguishable from general relativity with current data. The author is not a professional physicist; his motivation for the closure is explained in a separate optional note (WHERE_THIS_COMES_FROM.md), on which no result depends. Computations were done with AI assistants; all except the NICER comparison were reproduced by a second AI assistant, not by a human referee. All code is included.