Dirac stars in Einstein-Euler-Heisenberg theory

Abstract

We study self-gravitating configurations of a charged classical spinor field coupled to gravity and to Euler-Heisenberg (EH) nonlinear electrodynamics, truncated at quartic order. The spinor is gauged under a U(1) field and may carry a quartic self-interaction. For static and spherically symmetric configurations we derive the field equations, reduce them to a dimensionless eigenvalue problem for the spinor frequency and solve it numerically. In the uncharged limit our solutions coincide with the known Dirac stars of general relativity. The electric charge increases the frequency, the mass and the size of the stars, and the maximal mass grows from $0.709$ to $1.027$ (in units of $M_{\rm Pl}^2/μ$) when the charge coupling rises from zero to $0.75$. The quartic term changes the relation between the electric field and the enclosed charge, so that the field inside the star is weaker than in the Maxwell case. Our central result concerns stars with the same number of fermions, and therefore the same total charge. For them the quartic term changes the mass by less than $0.2\%$, but it makes the stars more compact and more strongly bound and shifts the turning point of the family to a smaller particle number. This shift is what lowers the maximal mass, by $9\%$ (from $0.924$ to $0.840$) at charge coupling $0.6$ as the dimensionless EH coupling goes from zero to $200$. In the weak-field regime of quantum electrodynamics (QED) the relative mass correction is $-1.5\times10^{-5}(E_{\max}/E_c)^2$, so the larger couplings describe the quartic model as an effective nonlinear electrodynamics rather than a controlled QED correction. All solutions satisfy the null, weak, strong and dominant energy conditions, show a positive pressure anisotropy, and remain free of horizons. Their stability is discussed through the binding energy and the turning point of the mass curve.