Plebański's heavenly equations

E811622

equation in general relativity integrable system nonlinear partial differential equation

Plebański's heavenly equations are a set of nonlinear differential equations in general relativity that describe self-dual (heavenly) solutions of Einstein’s field equations, particularly important in the study of complex and integrable gravitational geometries.

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Observed surface forms (1)

Surface form	Occurrences
Plebański heavenly equations	1

Statements (50)

Predicate	Object
instanceOf	equation in general relativity ⓘ integrable system ⓘ nonlinear partial differential equation ⓘ
appliesTo	Ricci-flat metrics ⓘ complexified space-time ⓘ four-dimensional manifolds ⓘ
connectedTo	Penrose twistor correspondence NERFINISHED ⓘ complex analytic methods in relativity ⓘ heavenly spaces ⓘ
coordinateChoice	complex null coordinates ⓘ heavenly coordinates ⓘ
definedOn	potential function (heavenly potential) ⓘ
describes	anti-self-dual conformal structures via complex conjugation ⓘ complex gravitational geometries ⓘ heavenly metrics ⓘ self-dual solutions of Einstein's field equations ⓘ self-dual vacuum metrics ⓘ
field	differential geometry ⓘ general relativity NERFINISHED ⓘ integrable systems ⓘ mathematical physics ⓘ
hasVersion	first heavenly equation NERFINISHED ⓘ general heavenly equation ⓘ mixed heavenly equation NERFINISHED ⓘ second heavenly equation NERFINISHED ⓘ
introducedBy	Jerzy Plebański NERFINISHED ⓘ
introducedInContextOf	self-dual gravity ⓘ
invariantUnder	holomorphic coordinate transformations ⓘ
mathematicalType	system of nonlinear PDEs ⓘ
namedAfter	Jerzy Plebański NERFINISHED ⓘ
property	integrable ⓘ nonlinear ⓘ second-order ⓘ
relatedTo	Einstein's field equations NERFINISHED ⓘ Kähler geometry NERFINISHED ⓘ complex Monge–Ampère equation NERFINISHED ⓘ heavenly metrics of Petrov type D ⓘ heavenly metrics of Petrov type N ⓘ hyperkähler geometry ⓘ self-dual Yang–Mills equations ⓘ twistor theory NERFINISHED ⓘ
solutionSpace	anti-self-dual vacuum metrics via complex conjugation ⓘ self-dual conformal structures ⓘ
studiedIn	complex general relativity ⓘ integrable geometry of four-manifolds ⓘ
timePeriod	1970s ⓘ
usedFor	classification of self-dual metrics ⓘ construction of self-dual vacuum solutions ⓘ generating exact solutions in complex general relativity ⓘ study of integrable gravitational instantons ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jerzy Plebański → notableWork → Plebański's heavenly equations ⓘ

Jerzy Plebański → hasConcept → Plebański's heavenly equations ⓘ

this entity surface form: Plebański heavenly equations