Order-Unity Correction to Hawking Radiation

When a black hole first forms, the properties of the emitted radiation as measured by observers near future null infinity are very close to the 1974 prediction of Hawking. However, deviations grow with time and become of order unity after a time $t\sim M_i^{7/3}$, where $M_i$ is the initial mass in Planck units. After an evaporation time, the corrections are large: the angular distribution of the emitted radiation is no longer dominated by low multipoles, with an exponential falloff at high multipoles. Instead, the radiation is redistributed as a power-law spectrum over a broad range of angular scales, all the way down to the scale $\mathrm{\Delta }\theta \sim 1/M_i$, beyond which there is exponential falloff. This effect is a quantum gravitational effect, whose origin is the spreading of the wave function of the black hole’s center-of-mass location caused by the kicks of the individual outgoing quanta, discovered by Page in 1980. The modified angular distribution of the Hawking radiation has an important consequence: the number of soft hair modes that can effectively interact with outgoing Hawking quanta increases from the handful of modes at low multipoles $l$ to a large number of modes, of order $\sim M_i^2$. We argue that this change unlocks the Hawking-Perry-Strominger mechanism for purifying the Hawking radiation.

Figure 1

An illustration of the standard Unruh state of an evaporating nonspinning black hole, at a particular instant of retarded time at future null infinity, in a reference frame that is displaced from the black hole center of mass by several Schwarzschild radii. The quantity plotted is a typical realization of the Gaussian random process on the sphere whose two-point function is given by taking the two-point function of a scalar field in the Unruh state at future null infinity and subtracting the two-point function of the out vacuum. Fluctuations in individual wave packet modes give rise to fluctuations on the sphere that are confined to concentric thin strips, giving rise to a characteristic angular scale that is small compared to unity.