Carroll black holes in (A)dS and their higher-derivative modifications

TL;DR

This work develops Carrollian (ultra-local) limits of Schwarzschild-(A)dS and Schwarzschild-Bach-(A)dS spacetimes in quadratic gravity and analyzes both geodesic dynamics and thermodynamics in these Carrollian settings. The authors derive Carrollian geodesics using a Carroll-invariant worldline action, identify extremal (horizon) surfaces, and show that nearly tangential particles wind around these surfaces with behavior that depends on the cosmological constant for the $(A)dS$ case and diverges for the Bach-extended case, where higher-curvature terms induce infinite windings. Thermodynamically, they compute Carrollian entropies and temperatures via Carroll Wald formulas and show that entropy diverges and the specific heat diverges in the strict Carroll limit, characterizing the black holes as incompressible thermodynamic systems with a sign-insensitive negative specific heat. For the Carroll Schwarzschild-(A)dS family, windings are finite and controlled by $oldsymbol{ extLambda}$, while in the Carroll Schwarzschild-Bach-(A)dS sector the windings are infinite near the extremal surface, highlighting the strong impact of Bach curvature on Carrollian horizon physics. The results point to a thermodynamic correspondence with Lorentzian black holes and open avenues for rotating/charged Carroll black holes and richer phase structure in higher-derivative theories.

Abstract

We define the Carrollian black holes corresponding to the limit of Schwarzschild-(A)dS spacetime and its higher-derivative counterpart known as Schwarzschild-Bach-(A)dS spacetime, which is also a static spherically symmetric vacuum solution of quadratic gravity. By analyzing motion of massive particles in these geometries, we found that: In the case of Schwarzschild-(A)dS, a (nearly) tangential particle from infinity will wind around the extremal surface with a finite number of windings depending on the impact parameter and the cosmological constant. In Schwarzschild-Bach-(A)dS, a particle passing close enough to the extremal surface will have an infinite number of windings; hence, it will not escape to asymptotic infinity as in Schwarzschild-(A)dS. We also calculate the thermodynamical quantities for such black holes and argue that it is analogous to an incompressible thermodynamical system with divergent entropy when the temperature goes to zero (in the strict Carroll limit). We then define a divergent specific heat that can be positive, negative, or zero.

Figures (2)

• Figure 1: In the case of Carroll Schwarzschild-Bach black hole (the left picture), in the region near the extremal surface i.e. where the black hole's metric is defined, a particle will undergo an infinite number of windings around the extremal surface for any impact parameter $b \approx r_0$. The trajectory takes the shape of an infinitely winding spiral which infinitesimally approaches the extremal surface but never touches it. That is in contrast with Carroll Schwarzschild-(A)dS (the right picture where the green lines represent the trajectory of a particle near Carroll Schwarzschild-dS black hole, the red lines represent the trajectory near Carroll Schwarzschild, and the blue lines represent the trajectory near Carroll-Schwarzschild-AdS black hole) where the winding number is finite but differ depending on the value of the cosmological constant if we fix all the other variables. In this case, a positive cosmological constant, the number of windings decrease in comparison to Carroll Schwarzschild (the particle gets ejected earlier), while in the case of a negative cosmological constant the number of windings increase (the particle gets ejected later).
• Figure 2: This graph represents the impact parameter as a function of the winding number in the cases of Carroll Schwarzschild (red), Carroll Schwarzschild-dS (green) and Carroll Schwarzschild-AdS (blue) with $M=1$ and $\sqrt{|\Lambda|}=0.02$. For a given value of the impact parameter $b>2$, we see that Carroll Schwarzschild-dS winds less than Carroll Schwarzschild which winds less than Carroll Schwarzschild-AdS, which is consistent with the conclusions above. For $b=2$, the winding number can be arbitrarily high. This coincides with the result in Ciambelli:2023tzb in the case of Carroll Schwarzschild. And although $b=2$ corresponds to a tangential particle in the cases of Carroll Schwarzschild and Carroll Schwarzschild-AdS, it is not the case for Schwarzschild-dS where a tangential particle has $b=1.99998$. Thus, the graph is not monotonically decreasing like the others instead it has a local minimum corresponding to a tangential particle, then $\phi$ goes up and stabilizes outside the extremal surface. This means that a tangential particle winds 2.48 times while the arbitrary high winding number occurs just outside the extremal surface. This is a result of the fact that at these particular values of the parameters, we have two horizons i.e. it is expected to have two winding numbers (one around each horizon) for some values of the impact parameter. On the other hand, for large impact parameter, we have low winding number in all cases as expected.