651 problems

Let $(\mathcal{M},g)$ be a $3+1$-dimensional Lorentzian spacetime solving the Einstein field equations, arising as the maximal future development of generic, regular, asymptoticall…

Let $\Omega$ be a compact Riemannian $3$-manifold with boundary, and consider asymptotically flat extensions satisfying the natural Bartnik boundary constraints. The infimum of the…

Penrose conjecture. If $\Sigma$ is a past apparent horizon, then

Let $g$) be a smooth Riemannian metric on the closed hemisphere $\mathbb{S}^3_+$ satisfying the following conditions: its scalar curvature obeys ${\mathrm R}_g\geq 6$; its induced…

Let $(\mathcal{M},\mathbf{g})$ be the maximal development of generic asymptotically flat initial data for Einstein's vacuum field equations. Weak cosmic censorship conjecture. The…

Kerr stability conjecture. The maximal Cauchy development of any initial data set for the Einstein vacuum equations, sufficiently close to a subextremal Kerr initial data set in a…

Let the maximal globally hyperbolic development of generic initial data be given, and let a Cauchy horizon arise in its spacetime development. Christodoulou's strong cosmic censors…

A high-frequency limit is a spacetime obtained as the limit of a sequence of vacuum spacetimes. The massless Vlasov system is the Einstein–Vlasov system in which the Vlasov matter…

Horowitz–Myers conjecture. Roughly speaking, any complete initial data set for the Einstein equations with flat toroidal conformal infinity should have mass bounded below by the ma…

Gauge-invariant decomposition conjecture. There exist a tensor field ${\cal H}_{ab}$ and a vector field $X^a$ such that

Let ${{\mathscr M}}_{\epsilon}$ be a physical spacetime, let ${{\mathscr M}}$ be its background spacetime, and let $h_{ab}$ be a perturbative metric tensor pulled back from…

Let an expanding universe have a positive cosmological constant. Cosmic no-hair conjecture. All such universes asymptotically admit the de Sitter universe as a solution. This conje…

Let $B$ be a spatial region and let $C$ denote its circumferential radius. Thorne's hoop conjecture. A horizon will be present within $B$ if the mass within $B$ exceeds the mass of…

Let $g_{a,M}$ denote a member of the two-parameter family of Kerr black-hole spacetimes, with zero cosmological constant and $a<M$. Kerr stability conjecture. Any slightly perturbe…

Let $(\boldsymbol{\rm X},\boldsymbol{\rm g})$ be an asymptotically flat, stationary vacuum Einstein spacetime. Black hole uniqueness conjecture. The spacetime is the Kerr spacetime…

In solutions of Einstein's equations near a Big Bang singularity, asymptotically velocity term dominated (AVTD) behavior means that, with respect to a given coordinate system, the…

Strominger's antipodal conjecture. The two resulting functions should agree after composition with the antipodal map

Let $(\mathbb{R}^4,g)$ be a Ricci-flat pp-wave with metric … where $H\colon\mathbb{R}\times\mathbb{R}^2\to\mathbb{R}$. Ehlers–Kundt conjecture. The metric is geodesically complete…

Moncrief–Isenberg conjecture. Every compact Cauchy horizon in a smooth vacuum spacetime is a Killing horizon.

The strong cosmic censorship conjecture concerns the interiors of generic black-hole spacetimes and the possibility of extending their spacetime metrics beyond the Cauchy horizon.…

Let $(\mathcal{M}_{AdS},g_{AdS})$ denote the trivial anti-de Sitter solution, let $\mathcal{I}$ be its conformal boundary, and consider arbitrarily small perturbations of its initi…

An exotic $\mathbb{R}^{4}$ is a smooth 4-manifold homeomorphic but not diffeomorphic to the standard $\mathbb{R}^{4}$. The handlebody construction gives such an exotic smooth struc…

Let $n$ be the dimension of a spacelike hypersurface and let $\eta>0$ measure the loss from harmonic decay in the Carlotto–Schoen construction, which currently provides control in…

Hawking's chronology protection conjecture. The laws of physics prevent the formation of time machines or spacetime regions containing closed timelike curves.

The AdS instability conjecture. There exist arbitrarily small perturbations of Anti-de Sitter spacetime solving the vacuum Einstein equations with reflecting boundary conditions at…