Renormalized energy-momentum tensor of quantized scalar perturbations for compact objects

In this thesis, we study the Hadamard renormalization of the energy-momentum tensor of quantized scalar perturbations on static spherically symmetric spacetimes. We reduce the problem to an auxiliary scalar field theory in the two-dimensional conformally flat time-radial sector and approximate the Regge-Wheeler-type effective potential as a finite potential well. The field is quantized in terms of normal modes characterized by definite parity, a discrete low-energy spectrum below the potential height, and a continuous high-energy spectrum above it. Owing to the step discontinuity of the potential, the renormalized expectation value of the energy-momentum tensor operator (REVEMTO) is explicitly constructed piecewise in the interior and exterior regions following Hadamard's renormalization and by imposing matching conditions on the field and its first derivative at the well's surfaces. Inside the well, the REVEMTO has the structure of a Zeldovich stiff fluid with homogeneous proper energy density receiving low- and high-energy contributions. Without the Hadamard parametrix, the low-energy contribution remains finite and regular, but the high-energy contribution becomes ultraviolet divergent. Outside, the low-energy contribution has a dust structure with proper energy density decaying exponentially away from the well. In addition, the exterior REVEMTO contains a state-independent contribution with a vacuum-energy structure induced by the potential. It remains to be established whether the exterior high-energy contribution ensures the REVEMTO's regularity across the well's surfaces without spurious divergences, and whether these effective perfect-fluid structures persist for more realistic effective-potential models.