Notation and conventions
Unless otherwise and explicitly stated, we use geometrized units where
, so that energy and
time have units of length. Geometric objects are denoted with boldface symbols, whereas their components
are not. We also adopt the
convention for the metric. For reference, the following is a list of
symbols that are used often throughout the text.

Total number of spacetime dimensions (we always consider one timelike 

and spacelike dimensions). 

Curvature radius of (A)dS spacetime, related to the (negative) positive 

cosmological constant in the Einstein equations () 

through . 

BH mass. 

BH rotation parameter. 

Radius of the BH’s event horizon in the chosen coordinates. 

Fourier transform variable. The time dependence of any field is . 

For stable spacetimes, . 

Spin of the field. 

Integer angular number, related to the eigenvalue 

of scalar spherical harmonics in dimensions. 

Index range referred to as “early lower case Latin indices” 

(likewise for upper case indices). 

Index range referred to as “late lower case Latin indices” 

(likewise for upper case indices). 

Spacetime metric; greek indices run from 0 to . 

, Christoffel symbol associated with the 

spacetime metric . 

, Riemann curvature tensor of the 

dimensional spacetime. 

dimensional covariant derivative associated with . 

Induced metric, also known as first fundamental form, on 

dimensional spatial hypersurface; latin indices run from 1 to . 

Extrinsic curvature, also known as second fundamental form, on 

dimensional spatial hypersurface. 

dimensional Christoffel symbol associated with . 

dimensional Riemann curvature tensor of the spatial hypersurface. 

dimensional covariant derivative associated with . 

dimensional sphere. 