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S. W. Hawking | Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields | (1976)
Abstract
The principle of equivalence, which says that gravity couples to the energy-momentum tensor of matter, and the quantum-mechanical requirement that energy should be positive imply that gravity is always attractive. This leads to singularities in any reasonable theory of gravitation. A singularity is a place where the classical concepts of space and time break down as do all the known laws of physics because they are all formulated on a classical space-time background. In this paper it is claimed that this breakdown is not merely a result of our ignorance of the correct theory but that it represents a fundamental limitation to our ability to predict the future, a limitation that is analogous but additional to the limitation imposed by the normal quantum-mechanical uncertainty principle. The new limitation arises because general relativity allows the causal structure of space-time to be very different from that of Minkowski space. The interaction region can be bounded not only by an initial surface on which data are given and a final surface on which measurements are made but also a "hidden surface" about which the observer has only limited information such as the mass, angular momentum, and charge. Concerning this hidden surface one has a "principle of ignorance": The surface emits with equal probability all configurations of particles compatible with the observers limited knowledge. It is shown that the ignorance principle holds for the quantum-mechanical evaporation of black holes: The black hole creates particles in pairs, with one particle always falling into the hole and the other possibly escaping to infinity. Because part of the information about the state of the system is lost down the hole, the final situation is represented by a density matrix rather than a pure quantum state. This means there is no $S$ matrix for the process of black-hole formation and evaporation. Instead one has to introduce a new operator, called the superscattering operator, which maps density matrices describing the initial situation to density matrices describing the final situation.
Tags
Sample Definition And Size
The paper is a theoretical analysis by a single author (S. W. Hawking) and does not involve empirical data or a sample size. It is not a meta-analysis or review of multiple studies.
Study Type
The work is a theoretical physics paper, presenting conceptual and mathematical arguments regarding gravitational collapse, singularities, and quantum-mechanical limitations on predictability.
Conflicts Of Interest
No conflicts of interest are declared in the paper. As a theoretical physics article published in 1976, standard conflict-of-interest disclosures were not customary, and none are indicated in the available metadata.
Results Summary
Hawking argues that gravitational collapse leads to singularities where classical space-time and physical laws break down, imposing a fundamental limitation on predictability beyond quantum uncertainty. He introduces the concept of a “hidden surface” and a “principle of ignorance,” whereby observers have limited information (mass, angular momentum, charge) and must assign equal probability to all compatible particle configurations. Applied to black hole evaporation, this leads to loss of information into the black hole, resulting in a final state described by a density matrix rather than a pure quantum state. Consequently, there is no conventional S‑matrix for black hole formation and evaporation; instead, Hawking proposes a “superscattering operator” that maps initial density matrices to final ones. (No numerical statistics such as p‑values or confidence intervals are applicable.)
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Created: Feb 10, 2026