Notation and conventions

Unless otherwise and explicitly stated, we use geometrized units where $G = c = 1$ , 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.

$D$	Total number of spacetime dimensions (we always consider one timelike
	and $D − 1$ spacelike dimensions).
$L$	Curvature radius of (A)dS spacetime, related to the (negative) positive
	cosmological constant $Λ$ in the Einstein equations ( $G μν + Λg μν = 0$ )
	through $2 L = (D − 2)(D − 1 )∕ (2 \|Λ \|)$ .
$M$	BH mass.
$a$	BH rotation parameter.
$RS$	Radius of the BH’s event horizon in the chosen coordinates.
$ω$	Fourier transform variable. The time dependence of any field is $∼ e− iωt$ .
	For stable spacetimes, $Im (ω ) < 0$ .
$s$	Spin of the field.
$l$	Integer angular number, related to the eigenvalue $Alm = l(l + D − 3)$
	of scalar spherical harmonics in $D$ dimensions.
$a,b, ...,h$	Index range referred to as “early lower case Latin indices”
	(likewise for upper case indices).
$i,j, ...,v$	Index range referred to as “late lower case Latin indices”
	(likewise for upper case indices).
$gαβ$	Spacetime metric; greek indices run from 0 to $D − 1$ .
$α Γβγ$	$1 αμ = 2g (∂βgγμ + ∂γgμβ − ∂μgβγ)$ , Christoffel symbol associated with the
	spacetime metric $gαβ$ .
$R αβγδ$	$ρ ρ = ∂γΓ αδβ − ∂δΓ αγβ + Γ αγρΓ δβ − Γ αδρΓγβ$ , Riemann curvature tensor of the
	$D$ -dimensional spacetime.
$∇ α$	$D$ -dimensional covariant derivative associated with $Γ αβγ$ .
$γij$	Induced metric, also known as first fundamental form, on
	$(D − 1)$ -dimensional spatial hypersurface; latin indices run from 1 to $D − 1$ .
$Kij$	Extrinsic curvature, also known as second fundamental form, on
	$(D − 1)$ -dimensional spatial hypersurface.
$Γ ijk$	$(D − 1)$ -dimensional Christoffel symbol associated with $γij$ .
$ℛijkl$	$(D − 1)$ -dimensional Riemann curvature tensor of the spatial hypersurface.
$Di$	$(D − 1)$ -dimensional covariant derivative associated with $Γ i jk$ .
$Sn$	$n$ -dimensional sphere.

	Abstract
	Acronyms
	Notation and conventions
1	Prologue
2	Milestones
3	Strong Need for Strong Gravity
	3.1	Astrophysics
	3.2	Fundamental and mathematical issues
	3.3	High-energy physics
4	Exact Analytic and Numerical Stationary Solutions
	4.1	Exact solutions
	4.2	Numerical stationary solutions
5	Approximation Schemes
	5.1	Post-Newtonian schemes
	5.2	Spacetime perturbation approach
	5.3	The zero-frequency limit
	5.4	Shock wave collisions
6	Numerical Relativity
	6.1	Formulations of the Einstein equations
	6.2	Higher-dimensional NR in effective “3 + 1” form
	6.3	Initial data
	6.4	Gauge conditions
	6.5	Discretization of the equations
	6.6	Boundary conditions
	6.7	Diagnostics
7	Applications of Numerical Relativity
	7.1	Critical collapse
	7.2	Cosmic censorship
	7.3	Hoop conjecture
	7.4	Spacetime stability
	7.5	Superradiance and fundamental massive fields
	7.6	High-energy collisions
	7.7	Alternative theories
	7.8	Holography
	7.9	Applications in cosmological settings
8	Conclusions
	Acknowledgements
	References
	Footnotes
	Figures