Abstract
The near-horizon geometry of evaporation black holes is determined according to the semi-classical Einstein equation. We consider spherically symmetric configurations in which the collapsing star has already collapsed below the Schwarzschild radius. The back-reaction of the vacuum energy-momentum, including Hawking radiation, is taken into account. The vacuum energy-momentum plays a crucial role in a small neighborhood of the apparent horizon, as it appears at the leading order in the semi-classical Einstein equation. Our study is focused on the time-dependent geometry in this region.
Highlights
The possibility of black-hole-like configurations with wormhole-like structures was proposed over 10 years ago [4, 5, 9,10,11]
In refs. [2, 3], this idea was realized in a static solution to the semi-classical Einstein equation, which adopted the vacuum energymomentum tensor derived from a 2D quantum field theory [12, 13]
We will first consider the approximation in which the exterior geometry of the black hole changes very slowly with time — as it is expected since the Hawking radiation is extremely weak — so that, at any instant of time, the static solution can be viewed as the 0-th order approximation
Summary
We define the semi-classical Einstein equation and review the model for the vacuum energy-momentum tensor proposed in refs. [12, 13]. We define the semi-classical Einstein equation and review the model for the vacuum energy-momentum tensor proposed in refs. The functions fu(u) and fv(v) comes from the integration constants for the conservation law They correspond to the outgoing and ingoing energy fluxes at r → ∞ if (u, v)-coordinates are chosen such that C → constant in r → ∞. Properly account for the back reaction of the vacuum energy-momentum tensor to the classical geometry of spacetime when the curvature is much smaller than the Planck scale. For this model, it is shown that static solutions have no horizon in the absence of outgoing or ingoing energy fluxes (fu(u) = fv(v) = 0) [2]. The Buchdahl theorem is circumvented as the quantum energy in vacuum can violate the weak energy condition
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