Evaporating Black Holes Coupled to Baths

An evaporation model becomes calculable by coupling a gravitating region to an external bath with explicit interface conditions, state preparation, and a real-time contour. A transparent nongravitating bath defines a clean radiation subsystem and energy sink; making the interface reflecting or the bath gravitating changes both the flux and the entropy algebra.

Required background. Black-Hole Evaporation and Mean Backreaction supplies the energy-balance approximation. Schwinger–Keldysh Contours and Real-Time Bulk Geometries fixes state and causal ordering.

Helpful background. Initial Density Matrices and Contour Boundary Conditions supplies the preparation data. Greybody Scattering and Flux Accounting and Evaporating Backgrounds and Adiabatic Backreaction supply the controlled approximations.

A transparent gravitating-region–bath model

Take a two-dimensional gravitating region $G$ with null coordinates $x^\pm=t\pm x$ and metric

$$ds^2=e^{2\rho(x^+,x^-)}dx^+dx^-,$$

which has signature $(+-)$. Glue it at a timelike interface to a nongravitating flat bath $B$ carrying the same CFT. Transparent matching identifies the induced metric and transmits the CFT stress tensor:

$$T_{nt}^{G}\big|_{\mathrm{int}} =T_{nt}^{B}\big|_{\mathrm{int}},$$

with no reflected component in the ideal limit. The gravitational fields obey their own boundary condition, such as fixed dilaton at the cutoff; they are not continued as dynamical fields into $B$.

The initial density matrix is prepared on a Schwinger–Keldysh contour. The outgoing Unruh-like state, switching protocol for the interface, and any incoming bath flux must be stated. A Euclidean state-preparation cap alone does not specify the evaporating Lorentzian contour.

Application: energy balance and the radiation region

Let $M(u)$ be the quasistatic black-hole energy and $F_{\mathrm{out}}(u)$ the renormalized bath flux measured at a fixed bath cut. Conservation gives

$$\frac{dM}{du} =-F_{\mathrm{out}}(u)+F_{\mathrm{in}}(u) +P_{\mathrm{interface}}(u).$$

For an adiabatic, transparent interface with no incoming quanta or external work,

$$\frac{dM}{du}=-F_{\mathrm{out}}(u).$$

In a two-dimensional CFT with slowly varying Hawking temperature, the ideal outgoing chiral flux is $F_{\mathrm{out}}\simeq \pi cT_H^2/12$, up to anomaly, greybody, and switching corrections. Integrating gives

$$M(u)=M(0)-\int_0^u du'\,F_{\mathrm{out}}(u').$$

The radiation region $R(u)$ must be a specified interval or half-line in the bath on a chosen Cauchy slice. Because the bath is nongravitating with a regulator, one can first define a tensor-factor entropy $S_{\mathrm{bath}}(R)$. The island prescription, if invoked later, computes a different semiclassical expression involving a candidate gravitating region; it does not alter the basic definition of the bath observable.

Models of this type underlie explicit Page-curve calculations, including the AdS black hole coupled to a bath in Almheiri and collaborators Almheiri et al. 2019.

Interface and algebra stress tests

Replace perfect transmission by a reflection coefficient $\mathcal R(\omega)$. Then

$$F_{\mathrm{out}}= \int_0^\infty\frac{d\omega}{2\pi}\, \omega\,[1-\mathcal R(\omega)]\,n_\omega$$

in the simplest stationary scattering picture. Both the evaporation rate and radiation entropy production change. A reflecting interface can put the system in equilibrium rather than produce an evaporating Page problem.

If the bath itself gravitates, its subregion is not automatically an independent tensor factor: gravitational constraints and dressing reach the interface, the generalized-entropy variational problem changes, and an asymptotic bath observer may no longer exist. Results derived with a nongravitating bath cannot be transferred unchanged.

Controlled limits and handoff

The quasistatic calculation requires $|\dot T_H|/T_H^2\ll1$, small curvature in cutoff units, controlled stress-tensor fluctuations, and negligible interface transients at the times used. It breaks near a quantum-gravity endpoint. Replica boundary conditions for $R(u)$ are developed on Evaporating Replicas, Radiation Entropy, and Entanglement-Wedge Transitions.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Almheiri, A., N. Engelhardt, D. Marolf, and H. Maxfield. “The Entropy of Bulk Quantum Fields and the Entanglement Wedge of an Evaporating Black Hole.” Journal of High Energy Physics 2019, 12 (2019): 063. DOI.
• Almheiri, A., R. Mahajan, and J. Maldacena. “Islands Outside the Horizon.” arXiv:1910.11077.