Figure 2

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"Exploring New Physics Frontiers Through Numerical Relativity"

Vitor Cardoso and Leonardo Gualtieri and Carlos Herdeiro and Ulrich Sperhake

	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

Figure 2: Illustration of two hypersurfaces of a foliation $Σt$ . Lapse $α$ and shift $βμ$ are defined by the relation of the timelike unit normal field $μ n$ and the basis vector $∂t$ associated with the coordinate $t$ . Note that $⟨dt,αn ⟩ = 1$ and, hence, the shift vector $β$ is tangent to $Σt$ .