New Horizons in Effective Field Theory?

TL;DR

This work analyzes a parity-symmetric scalar-tensor EFT in four dimensions with up to four-derivative terms and shows that horizons defined with respect to the fastest propagation cone align with metric Killing horizons for stationary black holes. Using generalized Gaussian null coordinates and analytic perturbation in the EFT scale $\ell$, it proves two theorems (A and B) ensuring that the horizon is a Killing horizon with constant surface gravity and that all propagation cones touch the horizon. The results argue that Killing-horizon thermodynamics remains robust in EFT settings and provide a censorship mechanism that avoids thermodynamic paradoxes tied to multiple propagation cones. Overall, the paper links the causal structure of high-derivative EFTs to familiar black-hole horizon concepts, with rotating cases encompassed under the assumptions.

Abstract

We consider the most general parity symmetric effective scalar tensor theory in four dimensions containing terms up to fourth derivative order in the Lagrangian. It has been shown [H.S. Reall, Phys. Rev. D 103 (2021), 084027] that this theory has three polarizations generically goverened by different (nested) propagation cones, neither of which in general coincides with the lightcone as defined by the metric. Consequently, the notion of black hole horizon must be defined relative to the widest propagation cone, and not with respect to the metric. We provide two theorems stating that, nevertheless, the horizon of a \emph{stationary} black hole is null with respect to the metric, and that, in fact, all three propagation cones touch on the horizon. The conditions in these theorems allow for rotating black holes. Our theorems thereby suggest that the notion of Killing horizon, central in most discussions of black hole thermodynamics, retains its fundamental status, and that certain thermodynamic paradoxes associated with multiple propagation cones are evaded.

Figures (6)

• Figure 1: A hypothetical spacetime containing two nested horizons $\mathscr H_A$ and $\mathscr H_B$ defined w.r.t. $g_{Aab}$ and $g_{Bab}$, respectively. Since $\chi^a$ is a "boostlike" Killing field, it does not vanish on $\mathscr H_A$, meaning that this set must be either a future or a past horizon w.r.t. $g_{Aab}$. In particular, this metric cannot have a "time reflection" symmetry, indicating a fundamental time-asymmetry of the setup.
• Figure 2: The collision between two particles with 4-momenta $p_{Aa}$ and $p_{Ba}$.
• Figure 3: The red, orange and blue cones indicate different propagation cones of the theory \ref{['lagrangian']} in the tangent space (quartic and quadratic cones). At the horizon $\mathscr{H}^+$, these touch at the horizon Killing vector field $\chi^a$ pointing along $\mathscr{H}^+$. At these points, $\chi^a$ is null w.r.t. $g_{ab}$ and satisfies $\chi^a \nabla_a \chi^b = \kappa \chi^b$ with constant $\kappa > 0$. $\chi^a$ is also co-linear with the tangent $\dot x^a$ of the null-geodesics ruling $\mathscr{H}^+$. This illustration is exaggerated because all cones should be close to each other in a weakly coupled (small $|\ell|$) theory.
• Figure 4: The figure shows the causal pasts of a point $p$ w.r.t. various cones terminating on some Cauchy surface $\Sigma$. Obviously, the domains of the dependence will be different.
• Figure 5: Generic behavior of the characteristic cone for a non-vanishing Weyl-like tensor \ref{['Wdef']}Reall:2021voz. The red and and the orange cone illustrate the two sheets of the quartic cone. The blue cone is the quadratic part of the characteristic cone, which smoothly touches the quartic cones along two principal null directions of the Weyl-like tensor.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Theorem 3.1
• Corollary 3.2
• Theorem 4.1
• Lemma D.1
• Lemma D.2