We propose a new approach for studying $\textit{d}$ + 1 dimensional Euclidean Schwarzschild black holes with Hawking temperature near the Hagedorn temperature and Horowitz-Polchinski solutions. The worldsheet theory that describes some of these back-grounds is strongly coupled. We use its underlying affine SU(2) × SU(2) symmetry to continue to weak coupling, by varying the level of the current algebra from the small value relevant for black holes and HP solutions to a large value. In this limit, one can describe the dynamics by a solvable effective field theory, and the non-geometric features of the original problem are geometrized. The resulting construction is closely related to previous work on the non-abelian Thirring model, and sheds light on both problems.

United States

Jinwei Chu

David Kutasov

From Horowitz-Polchinski to Thirring and back

It is known that linearized perturbations of extremal black holes result in growing curvature on the horizon. However, nonlinear perturbations typically do not evolve to extremal black holes and do not have growing curvature at late times. We show that a large class of nonlinear perturbations of an extremal planar anti-de Sitter black hole does have horizon curvature that grows unbounded in time. The late time behavior of the nonlinear evolution is found to be captured by a linearized analysis. We argue that the generic nonlinear perturbation behaves similarly.

Gary Horowitz

Maciej Kolanowski

United Kingdom

Jorge Santos

Frozen firewall: generic singularity formation on an extremal horizon

The Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) dilatonic black hole, a key solution in low-energy string theory, exhibits previously unexplored chaotic dynamics for charged test particles under electromagnetic influence. While characterizing such chaos requires high-precision numerical methods, our prior research identifies the explicit symplectic algorithm as the optimal integration tool for strongly curved gravitational systems. Leveraging the Hamiltonian formulation of the GMGHS black hole, we develop an optimized fourth-order symplectic algorithm, <math altimg='si28.svg'><mrow><mi>P</mi><mi>R</mi><msub><mi>K</mi><mn>6</mn></msub><mn>4</mn></mrow></math>. This algorithm enables a systematic investigation of chaotic motion using four distinct chaos indicators: the Shannon entropy, Poincar&#233; sections, the maximum Lyapunov exponent, and the Fast Lyapunov Indicator. Our results demonstrate a critical dependence of chaos on both electric charge ( <math altimg='si1.svg'><mi>Q</mi></math>, characterized by the Coulomb parameter <math altimg='si11.svg'><msup><mi>Q</mi><mo>*</mo></msup></math>) and magnetic charge ( <math altimg='si17.svg'><msub><mi>Q</mi><mi>m</mi></msub></math>). Specifically, in electrically charged backgrounds, order-to-chaos transitions occur with increasing <math altimg='si1.svg'><mi>Q</mi></math>. Conversely, in magnetically charged backgrounds, chaos emerges with increasing <math altimg='si17.svg'><msub><mi>Q</mi><mi>m</mi></msub></math>. When <math altimg='si1.svg'><mi>Q</mi></math> and <math altimg='si11.svg'><msup><mi>Q</mi><mo>*</mo></msup></math> vary simultaneously, the chaotic dynamics exhibit greater sensitivity to changes in <math altimg='si11.svg'><msup><mi>Q</mi><mo>*</mo></msup></math>. Especially, these findings establish the Shannon entropy as a robust chaos indicator within relativistic frameworks and provide novel insights into the dynamics of string-theoretic black holes.

China

Zhen-Meng Xu

Da-Zhu Ma

Kai Li

Chaos dynamics of charged particles near Gibbons-Maeda-Garfinkle-Horowitz-Strominger black holes

The Euclidean Einstein-Hilbert action is well-known to be unbounded below and thus to raise many questions regarding the definition of the gravitational path integral. A variety of works since the late 1980’s have suggested that this problem disappears when one fixes a foliation of the spacetime and imposes the corresponding gravitational constraints. However, we show here that this approach fails with various classes of boundary conditions imposed on the foliation: compact slices without boundary, asymptotically flat, or asymptotically locally anti-de Sitter slices. We also discuss the idea of fixing the scalar curvature and Wick-rotating the conformal factor, and show that it also fails to produce an action bounded from below.

Donald Marolf

Constraints are not enough

We show that the general charged, rotating black hole in five-dimensional Einstein-Maxwell theory has a singular extremal limit. Only the known analytic solutions with exactly zero charge or zero angular momenta have smooth extremal horizons. We also consider general black holes in five-dimensional Einstein-Maxwell-Chern-Simons theory, and show that they also have singular extremal limits except for one special value of the coefficient of the Chern-Simons term (the one fixed by supergravity). Combining this with earlier results showing that extremal black holes have singular horizons in four-dimensional general relativity with small higher derivative corrections, and in anti-de Sitter space with perturbed boundary conditions, one sees that smooth extremal horizons are indeed the exception and not the rule.

Smooth extremal horizons are the exception, not the rule

It was recently shown that (near-)extremal Kerr black holes are sensitive probes of small higher-derivative corrections to general relativity. In particular, these corrections produce diverging tidal forces on the horizon in the extremal limit. We show that adding a black hole charge makes this effect qualitatively stronger. Higher-derivative corrections to the Kerr-Newman solution produce tidal forces that scale inversely in the black hole temperature. We find that, unlike the Kerr case, for realistic values of the black hole charge large tidal forces can arise before quantum corrections due to the Schwarzian mode become important, so that the near-horizon behavior of the black hole is dictated by higher-derivative terms in the effective theory.

Grant Remmen

Sudden breakdown of effective field theory near cool Kerr-Newman black holes

In holography, the IR behavior of a quantum system at nonzero density is described by the near horizon geometry of an extremal charged black hole. It is commonly believed that for systems on S 3, this near horizon geometry is AdS2 × S 3. We show that this is not the case: generic static, nonspherical perturbations of AdS2 × S 3 blow up at the horizon, showing that it is not a stable IR fixed point. We then construct a new near horizon geometry which is invariant under only SO(3) (and not SO(4)) symmetry and show that it is stable to SO(3)-preserving perturbations (but not in general). We also show that an open set of nonextremal, SO(3)-invariant charged black holes develop this new near horizon geometry in the limit T → 0. Our new IR geometry still has AdS2 symmetry, but it is warped over a deformed sphere. We also construct many other near horizon geometries, including some with no rotational symmetries, but expect them all to be unstable IR fixed points.

Poland

A deformed IR: a new IR fixed point for four-dimensional holographic theories

We investigate the geometry near the horizon of a generic, four-dimensional extremal black hole. When the cosmological constant is negative, we show that (in almost all cases) tidal forces diverge as one crosses the horizon, and this singularity is stronger for larger black holes. In particular, this applies to generic nonspherical black holes, such as those satisfying inhomogeneous boundary conditions. Nevertheless, all scalar curvature invariants remain finite. Moreover, we show that nonextremal black holes have tidal forces that diverge in the extremal limit. Holographically, this singularity is reflected in anomalous scaling of the specific heat with temperature. Similar (albeit weaker) effects are present when the cosmological constant is positive, but not when it vanishes.

Almost all extremal black holes in AdS are singular

We study charged perturbations of the thermofield double state dual to a charged AdS black hole. We model the perturbation by a massless charged shell in the bulk. Unlike the neutral case, all such shells bounce at a definite radius, which can be behind the horizon. We show that the standard “shock wave” calculation of a scrambling time indicates that adding charge increases the scrambling time. We then give two arguments using the bounce that suggest that scrambling does not actually take longer when charge is added, but instead its onset is delayed. We also construct a boundary four point function which detects whether the shell bounces inside the black hole.

Henry Leung

Leonel Queimada

Ying Zhao

Bouncing inside the horizon and scrambling delays

We study a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with nonzero Hawking temperature. The implications for Hawking evaporation are discussed.