Two vessels of water at different temperatures are separated by a thin partition. As long as the partition is intact, each vessel lives its own life: in one, the water is hot; in the other, it's icy. But once the partition is removed, the temperatures begin to equalize, and eventually both vessels reach the same state. Something similar happens with two horizons of a black hole when quantum fields are placed between them.
In a two-dimensional model of gravity, where spacetime has only one spatial dimension, one can construct a black hole with two horizons — boundaries beyond which light cannot escape. These horizons have different temperatures: the outer one has a temperature proportional to the parameter a, while the inner one has zero temperature. This situation resembles the two vessels from the example, except that instead of water, quantum fields fill the space between the horizons.
When the field is in a thermal state, its energy density near the horizon with nonzero temperature turns out to be infinite. This is not just a mathematical difficulty: infinite energy means that quantum fields strongly affect the geometry, curving it even further. Such backreaction can completely change the initial picture.
To understand what actually happens, the researchers solved the equations of gravity with quantum fields taken into account self-consistently — not approximately, but exactly within the model. It turned out that the backreaction rearranges the geometry: the dilaton field becomes constant, and the metric function takes an exponential form. The new geometry is defined on the entire X-axis and has two horizons at infinities, both with zero temperature.
Thus, two vessels with different temperatures, connected by a quantum partition, come to a common state: the temperatures equalize. However, unlike water, here the equalization occurs not through heat transfer but through a change in the geometry of spacetime itself. Quantum fields, like an invisible architect, reshape space to eliminate the temperature difference between the horizons.
This result shows that quantum effects can stabilize spacetimes with multiple horizons. Instead of leading to infinities and the breakdown of the theory, they rearrange the geometry into a new, consistent state. Two horizons that initially had different temperatures become indistinguishable in their thermal properties after accounting for quantum fields.
In two-dimensional gravity, the horizon temperature is proportional to its surface gravity, and for the horizon at r = 0 it is zero, even though the horizon exists. This resembles an extreme black hole, but here the zero temperature arises from the degeneracy of the metric function.