If not swamped by other features (such as slower decay or an exponential instability), these
rates will be visible at late times, as in the known 1/t [6] and 1/
√
v [7] tails of massless
perturbations of extremal Kerr. If not, they should still be identifiable at intermediate
times, as in the transient 1/t decay occurring for charged perturbations of extremal Kerr-
Newman [8]. At the very least, they can be identified with spectral analysis, as they
are associated with a calculable special frequency in each example, often the superradiant
bound. The special rates of 1/t and 1/
√
v occur over large swaths of parameter space in
our analysis, and we therefore expect that these rates will appear much more generally
than the known examples, functioning as “calling cards” for an extremal black hole.
The universality can be traced to the shared AdS
2
factor in extremal near-horizon
geometries [3] and is simply understood in holographic terms. Each angular mode features
a special frequency near which the dynamics are governed by a field in AdS
2
with some
scaling dimension h. Elementary symmetry considerations force the bulk-boundary and
boundary-boundary propagators to scale with exponents h and 2h, respectively,
DG
B∂
= −hG
B∂
, DG
∂∂
= −2hG
∂∂
, (1.1)
where D is the action of an infinitesimal dilation. The key observation is that these
propagators encode the effects of AdS
2
on externally sourced perturbations (initial data of
compact support away from the horizon), since the AdS
2
boundary functions as the gateway
between near and far regions. In particular, G
∂∂
governs fields that propagate in and out
of the near-horizon region, corresponding to off-horizon properties of externally sourced
fields, while G
∂B
governs propagation only in to the near-horizon region, corresponding to
on-horizon properties. The relative factor of 2 in eq. (1.1) accounts for the first result above,
while the bound Re[h] ≥ 1/2 on AdS
2
scaling dimensions accounts for the second. The
third result, the Aretakis instability, is also a direct consequence of the symmetry (1.1) [9].
We flesh out these arguments in section 2 below.
While the symmetry argument captures the essence of our results, it is far from the
whole story. In particular, the argument assumes that holographic propagators can be
defined for some exponent h, which is possible only for certain choices of AdS
2
boundary
conditions (typically Dirichlet). In black hole perturbation problems, the AdS
2
boundary
conditions are determined by the physics of the far region, and we are not free to adjust
them to satisfy our holographic urges. In fact, in many important cases, such as for Kerr
black holes, these conditions are such that dynamics in pure AdS
2
would not even be well-
posed! After presenting the symmetry argument in section 2, we go on to tell the full story
in sections 3–5 in terms of the range of boundary conditions that can arise in practice. We
delineate the parameter space where the basic results 1–3 survive, giving conditions that
can be checked for any particular perturbation problem of interest. We give an example of
applying the formalism in section 6.
Some of the main features and results of our analysis have been noticed before in the
holographic condensed matter literature. In particular, beginning with ref. [10], it was
recognized that AdS
2
scaling behavior emerges at frequencies near the chemical potential,
giving rise to power laws in the dual theory. However, this body of literature has not, to our
knowledge, considered decay on the horizon or discussed the growth of derivatives (Aretakis
– 2 –
