Exceptional points and cusp singularities in relativistic stellar oscillations with causal bulk relaxation

Abstract

Abstract Causal relativistic hydrodynamics promotes dissipative stresses to independent dynamical variables obeying first-order relaxation equations. When such relaxation channels are coupled to the second-order Einstein--fluid system, the stellar oscillation problem is generically elevated from a quadratic self-adjoint eigenvalue problem to a cubic non-Hermitian spectral problem in the complex frequency.

We demonstrate that the resulting cubic operator possesses a structured non-Hermitian spectral geometry governed globally by its generalized discriminant. When the unperturbed stellar background is treated rigorously as an open system emitting gravitational radiation, the baseline quadratic operator acquires an intrinsic dissipation parameter $\gamma$. We solve the complete spectral problem and prove that the strict $A_3$ (cusp) organizing center ceases to be fixed. First, we demonstrate that a consistent causal weak formulation naturally anchors the entire exceptional-point skeleton, including the cusp, firmly within the physically admissible parameter domain $(\Lambda>0)$. Second, we show that open-system radiation damping continuously drives this cusp along an algebraic trajectory within this physical space. The physical curves of exceptional points (EP2) observed in the causal transport domain are shown to be the structural unfolding of this shifted open-system singularity.

The effective spectral surface exhibits generic non-analytic behaviour, with square-root splitting near generic exceptional points, whereas the cubic-root scaling characterises the strict $A_3$ organizing center now natively rooted in the physical domain. These features arise from the structural interplay between second-order gravitational--fluid dynamics and first-order causal relaxation, characterising the robust topological skeleton of the system. The results establish the global geometric organisation of the non-Hermitian spectrum of relativistic stellar oscillations with causal bulk relaxation.