On the nature of the black hole information problem
Thiago T. Bergamaschi
TL;DR
This work surveys the classical and semiclassical framework of black holes to formulate the information problem: under semiclassical gravity, black hole formation and complete evaporation imply a transition from a pure quantum state to a mixed state due to entanglement between interior and exterior degrees of freedom, yielding information loss. It builds the argument from general relativity’s causal structure and horizon properties through quantum field theory in curved spacetime, invoking Hadamard states to ensure well-defined energy-momentum expectations and deriving Hawking radiation with spectrum $\mathrm{d}N_\omega \sim \frac{1}{e^{\frac{2\pi(\omega-m\Omega)}{\kappa}}-1}$ and temperature $T=\frac{\hbar\kappa}{2\pi c k_B}$. The paper also discusses possible resolutions—remnants, firewalls, fuzzballs, and holographic dualities like AdS/CFT—while emphasizing that a complete quantum gravity theory is needed to determine whether information is truly lost or preserved, and how the semiclassical picture might be reconciled with a unitary evolution. Overall, it highlights how the interplay of gravitation, quantum fields, and thermodynamics at horizons points to deep, foundational questions about determinism, entropy, and the microscopic origin of spacetime degrees of freedom.
Abstract
The aim of this work is to present the black hole information problem and discuss the assumptions and hypotheses necessary for its formulation. As the problem arises in the framework of semiclassical gravity, we first review the necessary notions to describe Lorentzian manifolds equipped with physical properties, as well as the physical concepts of the theory that describes the gravitational interaction as the curvature of spacetime, general relativity. From its classical perspective, we develop the formalism to study the dynamical aspects of black holes in spacetimes obeying suitable causality conditions. Equipped with conjectures that nature censors naked singularities and that black holes reach a stationary configuration after they form, the black hole uniqueness theorems allow us to review several relations for the geometrical quantities associated with them. Following considerations of the other fundamental interactions, which are described by quantum field theory, we review the arguments in the formalism of quantum field theory in curved spacetime that give rise to the effective particle creation effect, its approximately thermal character, and the concept of black hole evaporation. With a precise quantification of information in quantum mechanics and assuming that the condition for physically acceptable states is given by the Hadamard condition, we review the result that entanglement between causally complementary regions is an intrinsic feature of quantum field theory. As a consequence, we discuss how the formation and complete evaporation of black holes leads to information loss. Conscious that such a prediction follows if no deviations from the semiclassical picture occur at the Planck scale, we discuss alternatives to this nonunitary dynamical evolution and formulate the black hole information problem. Lastly, we analyze the assumptions and hypotheses that lead to the problem.