Soft Hair on Black Holes

Abstract: It has recently been shown that BMS supertranslation symmetries imply an infinite number of conservation laws for all gravitational theories in asymptotically Minkowskian spacetimes. These laws require black holes to carry a large amount of soft ($i.e.$ zero-energy) supertranslation hair. The presence of a Maxwell field similarly implies soft electric hair. This paper gives an explicit description of soft hair in terms of soft gravitons or photons on the black hole horizon, and shows that complete information about their quantum state is stored on a holographic plate at the future boundary of the horizon. Charge conservation is used to give an infinite number of exact relations between the evaporation products of black holes which have different soft hair but are otherwise identical. It is further argued that soft hair which is spatially localized to much less than a Planck length cannot be excited in a physically realizable process, giving an effective number of soft degrees of freedom proportional to the horizon area in Planck units.

• The paper demonstrates that soft hair, manifesting as zero-energy excitations from supertranslation symmetries, offers a novel mechanism for encoding quantum information.
• The paper reveals that soft electric hair, correlated with soft photon emissions, establishes new electromagnetic conservation laws near black holes.
• The paper proposes that treating the black hole horizon as a holographic plate localizes quantum states, providing an innovative approach to resolving the information paradox.

On Soft Hair and Black Hole Information Paradox

The paper "Soft Hair on Black Holes" by Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger extends the exploration of black hole physics by addressing the longstanding information paradox within the framework of recent discoveries surrounding the infrared structure of quantum gravity. This research fundamentally examines how soft hair, comprised of supertranslations and soft photons, can be ascribed to black holes, thus presenting a pathway for possibly resolving key aspects of the information paradox.

Summary of the Core Concepts

The investigation begins with the recognition that BMS (Bondi-van der Burg-Metzner-Sachs) supertranslation symmetries, inherently present in asymptotically flat spacetimes, guarantee an infinite number of conservation laws across gravitational theories. BMS symmetries enrich the conventional view of gravitational dynamics by suggesting that black holes are not characterized merely by mass, charge, and angular momentum. Instead, they propose that black holes possess an extended set of 'soft hair', which manifests as zero-energy excitations, or soft particles, on the horizon.

Key Theoretical Insights:

1. Supertranslation Hair: Supertranslations are diffeomorphisms that alter the physical state of a black hole. The paper provides a characterization of this 'soft hair', which is viewed in terms of soft gravitons or photons—conceptualized as Goldstone bosons—on the horizon's future boundary. The study delineates an infinite set of exact relations between the quantum states of the evaporation products of black holes, connected by these conservation laws.
2. Soft Electric Hair: Similarly, when a Maxwell field is present, the black hole is implied to have soft electric hair. The detailed analysis connects these dynamics with the soft photon theorem, suggesting novel electromagnetic conservation laws that parallel those in gravitational theories.
3. Quantum Information Localization: The authors suggest treating the black hole horizon as a holographic plate, wherein the information about the quantum state of the black hole is localized. This contributes to the discourse on how information might be retained rather than lost, hinting at potential solutions to the information paradox by proposing conservation of quantum purity through transition among degenerate vacua.
4. Effective Soft Degrees of Freedom: Considerations of the physical process required to excite soft degrees of freedom reveal practical restrictions. These excitations are spatially limited to scales larger than the Planck length, imposing a quantifiable limit proportional to the horizon area as measured in Planck units.

Implications and Outlook

The implications of this work are multifaceted, touching both theoretical and practical aspects of black hole physics and quantum gravity. The study provides a mechanism by which information that falls into a black hole could be encoded in its 'soft hair', preserving information in a manner consistent with quantum mechanical principles. Furthermore, this paper sets a coherent framework for incorporating such soft excitations as integral components of the gravitational and electromagnetic field dynamics near black holes.

Future Directions and Speculations:

The notion of soft hair as a solution to the information paradox opens up new research avenues in gravitational theory and quantum mechanics. Verification and further exploration of these ideas might involve:

• Deepening our understanding of superrotation symmetries, which might result in "thicker" kinds of hair.
• Elucidating the role of soft symmetries in a broader set of interaction scenarios, including those in higher-dimensional spacetimes.
• Harnessing insights from string theory and other quantum gravity approaches to establish a fully fleshed-out hypothesis for the resolution of the paradox.
• Developing observational or computationally viable criteria to test these theoretical predictions through advanced gravitational wave detectors or numerical simulations.

In conclusion, the exploration of soft hair on black holes offers an intellectually stimulating view of the universe's fundamental laws, presenting concepts that might, when fully understood, resolve the infamous information loss problem in black hole thermodynamics. While this paper lays the groundwork for a crucial theoretical advancement, much work remains to consolidate these findings within the broader tapestry of physics.