The Boyer-Finley equation, or SU (∞)-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionless limit of the 2d-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system i...

The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ = TdS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with δQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer...

We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational eld with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new eld equation generalizing the notion of Killing spinors. The solutions of this spinor eld equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Di...

We derive the ‘exact’ Newtonian limit of general relativity with a positive cosmological constant Λ. We point out that in contrast to the case with Λ ≤ 0, the presence of a positive Λ in Einsteins’s equations enforces, via the condition |Φ| 1, on the potential Φ, a range Rmax(Λ) r Rmin(Λ), within which the Newtonian limit is valid. It also leads to the existence of a maximum mass, Mmax(Λ). As a...

We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso–Cao soliton and the Lü–Page–Pope quasi-Einstein metrics on CP2]CP (in both cases the metrics are known explicitly). We also find...

We establish new existence and non-existence results for positive solutions of the Einstein–scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein–scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian con...

A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies the validity of the hypothesis. A more...