The paper discusses the concept of geometric and renormalized entropy in conformal field theory, particularly in the context of relativistic quantum field theory. The authors address the challenges posed by the infinite number of degrees of freedom per unit volume, which can lead to divergences in the microscopic entropy. They introduce the idea of renormalized entropy, defined as the entropy relative to the ground state, and provide a quantitative analysis using a simple model: the states of a conformal quantum field theory excited by a moving mirror. The paper then applies these concepts to the moving mirror model, showing how geometric entropy arises naturally in a dynamical context and how ultraviolet divergences can be regulated. The authors also explore the connection between the moving mirror model and black hole physics, suggesting that the renormalized entropy can be interpreted as a measure of correlations in the observed energy-momentum tensor, which are in excess of those expected in the vacuum. This interpretation is further supported by the fact that the renormalized entropy is negative when the observed correlations are more than expected, reflecting the state being more ordered than the vacuum. Finally, the paper reviews the connection between the moving mirror model and black hole collapse, demonstrating that the geometry of spherical collapse can be modeled by a moving mirror problem. The authors highlight the causal structure of the mirror problem, explaining how the reflection of rays off a rapidly receding mirror mimics the propagation of ingoing and outgoing waves in a black hole, leading to the creation of Hawking radiation. The paper concludes by discussing the implications of these findings for understanding black hole entropy and information loss.The paper discusses the concept of geometric and renormalized entropy in conformal field theory, particularly in the context of relativistic quantum field theory. The authors address the challenges posed by the infinite number of degrees of freedom per unit volume, which can lead to divergences in the microscopic entropy. They introduce the idea of renormalized entropy, defined as the entropy relative to the ground state, and provide a quantitative analysis using a simple model: the states of a conformal quantum field theory excited by a moving mirror. The paper then applies these concepts to the moving mirror model, showing how geometric entropy arises naturally in a dynamical context and how ultraviolet divergences can be regulated. The authors also explore the connection between the moving mirror model and black hole physics, suggesting that the renormalized entropy can be interpreted as a measure of correlations in the observed energy-momentum tensor, which are in excess of those expected in the vacuum. This interpretation is further supported by the fact that the renormalized entropy is negative when the observed correlations are more than expected, reflecting the state being more ordered than the vacuum. Finally, the paper reviews the connection between the moving mirror model and black hole collapse, demonstrating that the geometry of spherical collapse can be modeled by a moving mirror problem. The authors highlight the causal structure of the mirror problem, explaining how the reflection of rays off a rapidly receding mirror mimics the propagation of ingoing and outgoing waves in a black hole, leading to the creation of Hawking radiation. The paper concludes by discussing the implications of these findings for understanding black hole entropy and information loss.