List of Footnotes

1		http://relativity.livingreviews.org/Articles/subject.html
2		That is, the existence of mass currents in opposite spatial directions (in between the horizons) and at relativistic velocities in the centre of energy frame.
3		This bound has an interesting story. Kip Thorne, and others after him, attribute the conjecture to Freeman Dyson; Freeman Dyson denies he ever made such a conjecture, and instead attributes such notion to his 1962 paper [157 *], where he works out the power emitted by a binary of compact objects. (We thank Gary Gibbons and Christoph Schiller for correspondence on this matter.)
4		Such fragmentation may however not be a counterexample to the spirit of the cosmic censorhip conjecture, if black strings do not form in generic collapse situations. One hint that this may indeed be the case comes from the Dyson–Chandrasekhar–Fermi instability of higher-dimensional cylindrical matter configurations [174 *]: if cylindrical matter configurations are themselves unstable it is unlikely that their collapse leads to black strings.
5		In the context of quantum gravity, it has ben shown that including a fundamental minimal length, a solution exists in which an interior regular solution is matched to the exterior Kerr metric. Such configuration, however, is a “regularized” BH rather than a description of stars [706 ].
6		Here, [ $D−1] 2$ denotes the integer part of $D−-1 2$ .
7		Numerical solutions of axially symmetric, rotating NSs in GR have been derived by several groups (see [330 ] and the Living Reviews article [728 ], and references therein), and in some cases their codes have been made publically available [729 , 117 ]. These solutions are used to build initial data for NR simulations of NS-NS and BH-NS binary inspiral and merger.
8		If matter or energy is present, there is a stress-energy tensor which is also perturbed, $(0) Tμν = Tμν + δTμν$ . If $Tμν$ describes a fluid, its perturbation can be described in terms of the perturbations of the thermodynamic quantities characterizing the fluid and of the matter velocity. We will only consider vacuum spacetimes here.
9		Strictly speaking, the signature represents the signs of the eigenvalues of the metric: $gαβ$ has $1$ negative and $D − 1$ positive eigenvalues even when the timelike coordinate is replaced in terms of one or two null coordinates.
10		Decompositions in terms of null foliations have to our knowledge not been studied yet, although there is no evident reason that speaks against such an approach.
11		…but beware! For many realistic types of matter, novel effects – such as shocks – can hamper an efficient evolution. These have to be handled with care and would require a review of its own.
12		We remark that the dimensional reduction here discussed is different from Kaluza–Klein dimensional reduction [463 , 283 ], an idea first proposed about one century ago, which in recent decades has attracted a lot of interest in the context of SMT. Indeed, a crucial feature of Kaluza–Klein dimensional reduction is spacetime compactification, which does not occur in our case.
13		There is a length-squared factor multiplying the exponential which we set to unity.
14		Note that Ref. [700 *] chooses $z$ instead of $y$ .
15		Generalizations to higher dimensions have been studied for Einstein gravity [836 ] and for five-dimensional Gauss–Bonnet gravity [814 ].
16		“Petrov type D” is a class of algebraically special spacetimes, which includes in particular the Schwarzschild and Kerr solutions.
17		The electric and magnetic part of the Weyl tensor may be interpreted as describing tidal effects and differential dragging of inertial frames, respectively, which has been employed to visualize spacetimes in terms of so-called “Frame-Drag vortexes” and “Tidal Tendexes” [593 , 579 , 578 , 833 ].
18		http://relativity.livingreviews.org/Articles/subject.html
19		Cosmic censorship does not apply to cosmological singularities, i.e., Big Bang or Big Crunch.
20		The $D = 5$ analysis in [701 , 700 ] has been revised in yet unpublished work, obtaining better agreement with the linear results in [273 ], cf. Ref. [694 ].
21		Floating orbits would manifest themselves in observations by depleting the inner part of accretion disks of stars and matter, and modifying the emitted gravitational waveform.
22		This quote, usually attributed to Carl Sagan, was published in Seeking Other Worlds in Newsweek magazine (April 15, 1977), a tribute to Carl Sagan by D. Gelman, S. Begley, D. Gram and E. Clark.

	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