Quantum Hair from Gravity

https://arxiv.org/abs/2110.09386

Xavier Calmet, Roberto Casadio, Stephen D. H. Hsu, and Folkert Kuipers

We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. A typical semiclassical matter source results in an entangled asymptotic graviton state. We exhibit a purely quantum gravitational effect which causes the subleading asymptotic behavior of the graviton state to depend on the internal structure of the source. These observations establish the existence of ubiquitous quantum hair due to gravitational effects.From the introduction:

Classical no-hair theorems limit the information that can be obtained about the internal state of a black hole by outside observers [1]. External features (``hair'') of black hole solutions in general relativity are determined by specific conserved quantities such as mass, angular momentum, and charge. In this letter we investigate how the situation changes when both the matter source (black hole interior state) and the gravitational field itself are quantized.

We begin by showing that the graviton state associated with an energy eigenstate source is determined, at leading order, by the energy eigenvalue of the source. These graviton states can be expressed as coherent states of non-propagating graviton modes, with explicit dependence on the source energy eigenvalue. Semiclassical matter sources (e.g., a star or black hole) are superpositions of energy eigenstates with support in some band of energies, and produce graviton states that are superpositions of the coherent states. ... We discuss implications for black hole information and holography in the conclusions.General relativity relates the spacetime metric to the energy-momentum distribution of matter, but only applies when both the metric (equivalently, the gravitational field) and matter sources are semiclassical. A theory of quantum gravity is necessary to relate the quantum state of the gravitational field to the quantum state of the matter source. However, as we show in section 2 one can deduce this relationship either from a simple gedanken experiment or from careful study of how the Gauss law affects quantization. It turns out the latter is common to both ordinary gauge theory (cf Coulomb potential) and gravity.