Einstein field equations

E1603

equations of general relativity fundamental physical law system of equations tensor equation

The Einstein field equations are the core mathematical framework of general relativity, relating the curvature of spacetime to the distribution of matter and energy.

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All labels observed (10)

Label	Occurrences
Einstein field equations canonical	38
Friedmann equations	3
Einstein equations	1
Einstein field equations (via κ = 8πG/c^4)	1
Einstein field equations formulation	1
Einstein field equations with cosmological constant	1
Einstein field equations with negative cosmological constant	1
Einstein’s 1915 field equations of general relativity	1
Friedmann equations in cosmology	1
vacuum Einstein equations	1

Statements (49)

Predicate	Object
instanceOf	equations of general relativity ⓘ fundamental physical law ⓘ system of equations ⓘ tensor equation ⓘ
admitsSolution	FLRW cosmological models ⓘ surface form: Friedmann–Lemaître–Robertson–Walker metric Kerr metric ⓘ Minkowski space-time ⓘ surface form: Minkowski spacetime Reissner–Nordström metric ⓘ Schwarzschild black hole ⓘ surface form: Schwarzschild metric de Sitter spacetime ⓘ
basedOnPrinciple	equivalence principle ⓘ general covariance ⓘ
constant	Newtonian gravitational constant G ⓘ cosmological constant \Lambda ⓘ speed of light c ⓘ
contrastsWith	quantum gravity theories ⓘ
describes	relationship between spacetime curvature and energy-momentum ⓘ
difficulty	analytical solutions are rare ⓘ
domain	classical gravitation ⓘ
field	general relativity ⓘ
foundationOf	modern cosmology ⓘ
hasVacuumForm	G_{\mu\nu} + \Lambda g_{\mu\nu} = 0 ⓘ R_{\mu\nu} = 0 (for \Lambda = 0 and vacuum) ⓘ
includes	cosmological constant ⓘ
introducedBy	Albert Einstein ⓘ
invariantUnder	general coordinate transformations ⓘ
numberOfIndependentEquations	10 ⓘ
order	second-order partial differential equations ⓘ
predicts	black holes ⓘ gravitational lensing ⓘ gravitational time dilation ⓘ gravitational waves ⓘ
publishedIn	Annalen der Physik ⓘ
reducesTo	Newtonian gravity in the weak-field limit ⓘ
relates	geometry of spacetime to matter and energy ⓘ
spacetimeDimensionAssumed	4 ⓘ
symbolicallyWrittenAs	G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} ⓘ
tensorRank	2 ⓘ
type	nonlinear equations ⓘ
usedIn	astrophysics ⓘ cosmological models ⓘ gravitational wave modeling ⓘ numerical relativity ⓘ
uses	Einstein tensor ⓘ Ricci curvature tensor ⓘ Ricci scalar ⓘ metric tensor ⓘ stress–energy tensor ⓘ
yearProposed	1915 ⓘ

How these facts were elicited

The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.

Instruction

You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10.

# Requirements
- If you don't know the subject at all, return an empty list.
- If the subject is not a named entity, return an empty list.
- Include at least one triple where predicate is "instanceOf".
- Do not get too wordy.
- Separate several objects into multiple triples with one object.

Input

Subject: Einstein field equations
Description of subject: The Einstein field equations are the core mathematical framework of general relativity, relating the curvature of spacetime to the distribution of matter and energy.

Referenced by (49)

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

Albert Einstein → knownFor → Einstein field equations ⓘ

FLRW cosmological models → governedBy → Einstein field equations ⓘ

this entity surface form: Friedmann equations

theory of relativity → coreConcept → Einstein field equations ⓘ

Newtonian gravitational constant G → appearsInEquation → Einstein field equations ⓘ

this entity surface form: Einstein field equations (via κ = 8πG/c^4)

black hole no-hair theorem → assumes → Einstein field equations ⓘ

The Mathematical Theory of Black Holes → mainSubject → Einstein field equations ⓘ

Ricci curvature tensor → appearsIn → Einstein field equations ⓘ

Einstein tensor → appearsInEquation → Einstein field equations ⓘ

Schwarzschild Penrose diagram → basedOn → Einstein field equations ⓘ

Arthur Geoffrey Walker → hasResearchInterest → Einstein field equations ⓘ

Riemann curvature tensor → usedIn → Einstein field equations ⓘ

Die Grundlage der allgemeinen Relativitätstheorie → introduces → Einstein field equations ⓘ

Israel–Carter–Robinson uniqueness theorems → involves → Einstein field equations ⓘ

general relativity → coreEquation → Einstein field equations ⓘ

Christoffel symbols → usedIn → Einstein field equations ⓘ

this entity surface form: Einstein field equations formulation

Ricci scalar → appearsIn → Einstein field equations ⓘ

Oppenheimer–Volkoff limit → basedOn → Einstein field equations ⓘ

Lorentzian geometry → providesFrameworkFor → Einstein field equations ⓘ

Einstein–Rosen bridge → basedOnSolutionOf → Einstein field equations ⓘ

"Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie" → topic → Einstein field equations ⓘ

subject surface form: Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie

Einstein–Maxwell equations → basedOn → Einstein field equations ⓘ

Einstein–Maxwell equations → relatedTo → Einstein field equations ⓘ

this entity surface form: Einstein equations

On the Curvature of Space → basedOnTheory → Einstein field equations ⓘ

On the Curvature of Space → relatedTo → Einstein field equations ⓘ

this entity surface form: Friedmann equations

Lemaître–Tolman metric → satisfies → Einstein field equations ⓘ

Gravitation (with Charles Misner and Kip Thorne) → subject → Einstein field equations ⓘ

Tolman–Ehrenfest effect → relatedTo → Einstein field equations ⓘ

Gravitation and Cosmology → subject → Einstein field equations ⓘ

anti-de Sitter space → isSolutionOf → Einstein field equations ⓘ

this entity surface form: Einstein field equations with negative cosmological constant

The Foundation of the General Theory of Relativity → describes → Einstein field equations ⓘ

Planck units → simplifiesEquation → Einstein field equations ⓘ

this entity surface form: Friedmann equations

Jerzy Plebański → studies → Einstein field equations ⓘ

Einstein’s 1907 equivalence principle paper → relatedTo → Einstein field equations ⓘ

this entity surface form: Einstein’s 1915 field equations of general relativity

textbook "A First Course in General Relativity" → topic → Einstein field equations ⓘ

subject surface form: A First Course in General Relativity

A First Course in General Relativity → topic → Einstein field equations ⓘ

cosmic censorship conjecture → relatedTo → Einstein field equations ⓘ

Ricci calculus → usedIn → Einstein field equations ⓘ

David C. Robinson → studiesTopic → Einstein field equations ⓘ

Einstein–Hilbert action → yieldsViaVariation → Einstein field equations ⓘ

f(R) gravity → affects → Einstein field equations ⓘ

this entity surface form: Friedmann equations in cosmology

Tolman–Oppenheimer–Volkoff equation → basedOn → Einstein field equations ⓘ

Hawking–Penrose singularity theorems → basedOn → Einstein field equations ⓘ

Raychaudhuri equation → relatedTo → Einstein field equations ⓘ

Einstein–Rosen 1935 paper → uses → Einstein field equations ⓘ

stress–energy tensor → appearsInEquation → Einstein field equations ⓘ

Einstein–Yang–Mills equations → basedOn → Einstein field equations ⓘ

Einstein–Yang–Mills equations → generalizes → Einstein field equations ⓘ

this entity surface form: vacuum Einstein equations

Einstein static universe → satisfies → Einstein field equations ⓘ

this entity surface form: Einstein field equations with cosmological constant