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

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.

💡 Research Summary

The paper investigates a novel class of black‑hole solutions obtained by taking the Carrollian limit (the ultra‑local limit c → 0 of the speed of light) of two families of static, spherically symmetric spacetimes: the ordinary Schwarzschild–(A)dS solution of General Relativity and its higher‑derivative counterpart, the Schwarzschild‑Bach–(A)dS solution of quadratic gravity. The latter contains an additional “Bach parameter” δ that measures the deviation from the pure Schwarzschild geometry. By applying the Carroll contraction to the metric, the authors obtain a degenerate spatial metric together with a distinguished vector field vμ that spans the kernel of the metric. The surface where the radial function f(r) vanishes (f(r)=0) becomes a Carroll extremal surface, playing the role of a horizon in the Carrollian geometry.

To study particle dynamics on these backgrounds the authors adopt a world‑line action that is invariant under Carroll diffeomorphisms. The action contains three independent couplings (g0, g1, g2) multiplying the only Carroll‑invariant scalars built from the spatial velocity, the temporal velocity and a constant term. Varying the action yields conserved energy E and angular momentum l, and the radial motion reduces to a one‑dimensional effective‑potential problem, \