**Page time** is a concept in theoretical physics related to the study of black holes and their thermodynamic properties. It is named after the physicist [[Don Page]], who introduced the idea in the context of black hole evaporation and information theory. Here’s a detailed explanation of what Page time is and its significance: ### Definition and Concept **Page time** is the time at which a black hole has radiated away half of its original entropy through Hawking radiation. This concept is particularly important in the study of the black hole information paradox, which addresses the question of whether information that falls into a black hole is lost forever or can be recovered. ### Key Points 1. **Hawking Radiation**: - Black holes emit radiation due to quantum effects near their event horizons, a phenomenon predicted by Stephen Hawking. This radiation causes the black hole to lose mass and energy over time, eventually leading to its evaporation. 2. **Entropy and Information**: - The entropy of a black hole is a measure of the amount of information hidden within it. According to the laws of black hole thermodynamics, the entropy is proportional to the area of the event horizon. - As the black hole emits Hawking radiation, it loses mass and its entropy decreases. Page time marks the point when the black hole has emitted half of its entropy. 3. **Significance in Information Paradox**: - The black hole information paradox arises from the question of whether the information that falls into a black hole is permanently lost when the black hole evaporates completely. - Page time is significant because it represents a milestone in the evaporation process. It is hypothesized that if information is preserved and can be recovered, it should start to become apparent around Page time. 4. **Page Curve**: - The [[Page curve]] is a graphical representation of the entropy of a black hole as a function of time. Initially, as the black hole radiates, the entropy increases, reaching a maximum at Page time. After Page time, the entropy decreases as the black hole continues to emit radiation and lose mass. - The shape of the Page curve is essential for understanding the dynamics of black hole evaporation and the fate of the information encoded in the black hole. ### Mathematical Representation The exact calculation of Page time depends on the specifics of the black hole, such as its mass and the nature of its Hawking radiation. However, a rough estimate can be given by: $t_{\text{Page}} \approx \frac{M^3}{m_p^4}$ where: - $M$ is the initial mass of the black hole. - $m_p$​ is the Planck mass. ### Implications for Quantum Gravity Page time and the associated Page curve have important implications for theories of quantum gravity. They provide a framework for testing ideas about how information is preserved in black hole evaporation. In particular, they challenge physicists to reconcile the principles of quantum mechanics, which forbid information loss, with the classical description of black holes. ### Recent Developments Recent advancements in holography and the `AdS/CFT` correspondence (a theoretical framework in which a gravitational system can be described by a conformal field theory) have provided insights into the black hole information paradox. These developments suggest that information may indeed be preserved and that the Page curve correctly describes the entropy evolution of a black hole. ### Conclusion Page time is a crucial concept in the study of black hole thermodynamics and the information paradox. It represents the halfway point in the black hole’s evaporation process and serves as a key milestone for understanding how information might be preserved in the presence of black holes. The study of Page time and the Page curve continues to provide valuable insights into the nature of black holes and the fundamental laws of physics. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Page time" ) sort title, subject, modified ```