What happens to a black hole when you include the energy of quantum fields? This paper presents a new analytic solution to the semiclassical Einstein equations that describes an evaporating Schwarzschild black hole in 3+1 dimensions. “Semiclassical” here means the matter fields are treated with quantum theory while spacetime itself remains classical. The authors obtain an explicit backreacted geometry that reproduces the outgoing Hawking radiation far away and the corresponding ingoing negative-energy flux at the horizon.

The key technical advance is a reformulation of the Hadamard renormalization prescription. Hadamard renormalization is a standard way to remove short-distance infinities in the quantum field’s energy and make the stress-energy tensor finite. The new formulation makes the part of the stress tensor that depends on the quantum state explicit and compact. Instead of reconstructing a full two-point function by summing modes, the computation is reduced to specifying a single scalar function ω(x) and solving a universal first-order covariant equation for a traceless tensor. In symmetric spacetimes this equation can be integrated in closed form.

Using this approach the authors build explicit, renormalized stress-energy tensors for scalar fields on a Schwarzschild background. They construct Unruh-like quantum states. The Unruh-like states are stationary, regular on the future horizon, and match the Hawking flux at future null infinity. By imposing the asymptotic Hawking flux, the free integration constants that appear in the solution are fixed. The family of solutions then reproduces the expected outgoing flux at large radius and the expected ingoing negative-energy flux at the horizon.

Putting the stress tensor back into Einstein’s equations, the authors obtain analytic, quasi-stationary metrics that describe black-hole evaporation with backreaction. One notable feature of these metrics is a timelike apparent horizon. In plain terms, the surface that traps light is not exactly lightlike in these solutions, which gives a new analytical setting to study how evaporation changes a black hole’s causal structure. The paper therefore opens a new route to study semiclassical backreaction in four dimensions, extending analytical control previously available mainly in two-dimensional toy models.