Hawking radiation inside a Schwarzschild black hole

Abstract

The boundary of any observer's spacetime is the boundary that divides what the observer can see from what they cannot see. The boundary of an observer's spacetime in the presence of a black hole is not the true (future event) horizon of the black hole, but rather the illusory horizon, the dimming, redshifting surface of the star that collapsed to the black hole long ago. The illusory horizon is the source of Hawking radiation seen by observers both outside and inside the true horizon. The perceived acceleration (gravity) on the illusory horizon sets the characteristic frequency scale of Hawking radiation, even if that acceleration varies dynamically, as it must do from the perspective of an infalling observer. The acceleration seen by a non-rotating free-faller both on the illusory horizon below and in the sky above is calculated for a Schwarzschild black hole. Remarkably, as an infaller approaches the singularity, the acceleration becomes isotropic, and diverging as a power law. The isotropic, power-law character of the Hawking radiation, coupled with conservation of energy-momentum, the trace anomaly, and the familiar behavior of Hawking radiation far from the black hole, leads to a complete description of the quantum energy-momentum inside a Schwarzschild black hole. The quantum energy-momentum near the singularity diverges as $r^{-6}$, and consists of relativistic Hawking radiation and negative energy vacuum in the ratio $3:-2$. The classical back reaction of the quantum energy-momentum on the geometry, calculated using the Einstein equations, serves merely to exacerbate the singularity. All the results are consistent with traditional calculations of the quantum energy-momentum in 1+1 spacetime dimensions.

v1: 21 pages, 7 figures. v2: more references. v3: additional material on Heun functions. v4: 45 pages, 10 figures; typos fixed, new sections on the illusory horizon, and on 1+1 dimensions

References in corpus (7)

• A Primer for Black Hole Quantum Physics
• Static, stationary and inertial Unruh-DeWitt detectors on the BTZ black hole
• Static detectors and circular-geodesic detectors on the Schwarzschild black hole
• A quantum peek inside the black hole event horizon
• Stereoscopic visualization in curved spacetime: seeing deep inside a black hole
• Exact Solutions of Regge-Wheeler Equation
• Hawking-like radiation and the density matrix for an infalling observer during gravitational collapse

Cited by in corpus (6)

• Semiclassical instability of inner-extremal regular black holes
• Hawking radiation inside a charged black hole
• The Vaidya metric: expected and unexpected traits of evaporating black holes
• Hawking radiation inside a rotating black hole
• A Note on the Entropy Force in Kinetic Theory and Black Holes
• Pancakification and negative Hawking temperatures