Spacetimes with multiple horizons, such as the Schwarzschild–de Sitter black hole, raise the question of consistency of quantum field theory. If the horizon temperatures do not match, the thermal state of the field leads to a divergence of the stress-energy tensor at one of them. There are already works on this topic: some seek regular states, others solve the backreaction equations perturbatively. This work follows the self-consistent approach but applies it for the first time to a two-dimensional model with two horizons at different temperatures.
We consider two-dimensional dilaton gravity with an action depending on the parameter ω. We choose the case ω = -4/3, which gives a metric with two horizons: at r = 0 and r = 1/a, with temperatures 0 and a/(2π), respectively. A massless scalar field is added, quantized canonically in Eddington–Finkelstein coordinates. The field is taken in a thermal state with inverse temperature β. The regularized stress-energy tensor is computed using a known formula that includes the conformal anomaly contribution. Then, analogues of the semiclassical Einstein equations for dilaton gravity are written, where the right-hand side contains the quantum expectation value of the stress-energy tensor, and solved numerically with the boundary condition that the horizon at r = 0 remains unchanged.
For the original geometry, the stress-energy tensor of the thermal field diverges at one of the horizons if the field temperature does not match the canonical horizon temperature. Choosing β = 0 (Fock vacuum) makes the tensor regular at the horizon r = 0 but divergent at r = 1/a. Numerical solution of the self-consistent equations shows that the backreaction changes the geometry: the dilaton field becomes a constant, and the metric function $e^{2ν(x)}$ takes an exponential form. The new geometry is defined on the entire X-axis and has two horizons at X = ±∞ with zero temperature. Thus, after accounting for backreaction, the horizon temperatures coincide.
This work differs from neighboring ones by solving the backreaction equations self-consistently rather than perturbatively, and by showing that a divergence in the stress-energy tensor does not necessarily imply an incorrect state but leads to a restructuring of the geometry. This supports the idea that quantum fields can stabilize spacetimes with multiple horizons by equalizing their temperatures. In the context of neighboring works that studied laboratory black holes and horizon temperatures, this paper provides a clear example of a temperature equalization mechanism.
The method can be extended to higher dimensions, where the exact form of the regularized stress-energy tensor is unknown, but approximations can be used. It is also interesting to consider self-interacting fields and to test the stability of the obtained solutions. Another path is to study the dynamics of the transition from the original geometry to the new one, which would require solving the equations in real time.
The results affect the understanding of quantum effects in spacetimes with multiple horizons, including cosmological models like de Sitter and black holes in the presence of a cosmological constant. This may also be useful for analog gravity models in condensed matter systems.
Check the stability of the found solutions under small perturbations of the metric and the field state. Generalize the approach to self-interacting fields and higher dimensions.
This work is related to the cosmological constant problem and the stability of de Sitter space: if quantum fields equalize horizon temperatures, this may indicate a mechanism that prevents instabilities. It also touches on the finiteness of horizon entropy and the thermodynamics of multi-horizon spacetimes.
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What the work does
This work examines two-dimensional dilaton gravity with a black hole that has two horizons at different temperatures. A quantum scalar field in a thermal state creates a divergence in the stress-energy tensor at one of the horizons. The self-consistent semiclassical equations are solved numerically, showing that backreaction leads to a geometry with two horizons at zero temperature.
Where its strength is
The strength of the work lies in demonstrating that the divergence of the stress-energy tensor does not imply the state is ill-defined, but rather leads to a restructuring of the geometry. The self-consistent approach, rather than a perturbative one, reveals the mechanism of spacetime stabilization.
Where it could go next
Worth trying
Why this may matter
Understanding how quantum fields reshape geometries with multiple horizons could be important for the cosmological horizon problem and for interpreting quantum states in spacetimes with black holes.
Nearby works in our archive
The closest work in the archive sits at 0.68 by meaning — the usual level for our corpus.
as written by the authors arXiv:2401.07645
We consider $$(1+1)$$-dimensional dilatonic black hole with two horizons, canonical temperatures of which do not coincide. We show that the presence of quantum fields in such a background leads to a substantial backreaction on the metric: 2D dilatonic analog of the semiclassical Einstein equations are solved self-consistently, and we demonstrate that taking into account of backreaction leads to a geometry with two horizons with coinciding temperatures.
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.