Math 2.0

Relativity and gravitational theory Spacetime geometry and causal structure Theorem

Goldberg–Sachs theorem

Builds on Lorentzian manifoldWeyl tensorPetrov classificationCongruence of geodesicsNewman–Penrose formalismVacuum solution

Statement

Let (M,g)(M, g) be a four-dimensional Lorentzian manifold that is a vacuum solution, so that its Ricci tensor vanishes. Let U⊆MU \subseteq M be an open set on which the Weyl tensor is nowhere zero, and let ll be a null vector field on UU. Then the following are equivalent.

  1. ll is tangent to a congruence of null geodesics that is shear-free.
  2. ll is at each point a repeated principal null direction of the Weyl tensor.

In particular, the Weyl tensor is algebraically special on UU if and only if UU carries a shear-free congruence of null geodesics. Here algebraically special refers to the Petrov classification.

Remarks

In the Newman–Penrose formalism with ll as first tetrad vector, clause 1 reads κ=σ=0\kappa = \sigma = 0 and clause 2 reads Ψ0=Ψ1=0\Psi_0 = \Psi_1 = 0. The proof uses the Bianchi identities written in this formalism. In vacuum they link the derivatives of Ψ0\Psi_0 and Ψ1\Psi_1 to κ\kappa and σ\sigma multiplied by the Weyl scalars, and this forces the two conditions to hold together. For example, in the Schwarzschild spacetime the radial ingoing and outgoing light rays form two shear-free geodesic congruences. The Weyl tensor there is of type D, with these two repeated principal null directions. The vacuum hypothesis is needed in this form; there are generalisations that only ask for weaker conditions on the Ricci tensor.

Going deeper

For a student

The theorem turns an algebraic condition on the curvature into a geometric condition on a family of light rays. It is a main tool to find algebraically special exact solutions, because one can start from an ansatz with a shear-free null geodesic congruence. The Kerr metric and the Robinson–Trautman metrics belong to this class.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Congruence of geodesics, concept, no evidence levelCongruence of geodesi…conceptEigenvector, concept, no evidence levelEigenvectorconceptLight cone, concept, no evidence levelLight coneconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptNewman–Penrose formalism, concept, no evidence levelNewman–Penrose formal…conceptRicci curvature, concept, no evidence levelRicci curvatureconceptVacuum solution, concept, no evidence levelVacuum solutionconceptWeyl tensor, concept, no evidence levelWeyl tensorconceptSchwarzschild metric, example, no evidence levelSchwarzschild metricexamplePetrov classification, statement, no evidence levelPetrov classificationstatementGoldberg–Sachs theorem, statement, no evidence levelGoldberg–Sachs theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Congruence of geodesics concept
  2. Eigenvector concept
  3. Light cone concept
  4. Lorentzian manifold concept
  5. Newman–Penrose formalism concept
  6. Ricci curvature concept
  7. Vacuum solution concept
  8. Weyl tensor concept
  9. Schwarzschild metric example
  10. Petrov classification statement
  11. Goldberg–Sachs theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Goldberg%E2%80%93Sachs_theorem
About this page
Tag44Z5GQTCXW
URIm2://44Z5GQTCXW
Kindstatement, theorem
WritingW0 stub
MSC83C20, 83C60, 53B30
StatusThis copy of the site has no build data. Run m2 status 44Z5GQTCXW.
AboutWeyl tensor, Petrov classification
UsesLorentzian manifold, Weyl tensor, Petrov classification, Congruence of geodesics, Newman–Penrose formalism, Vacuum solution
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Changed3 November 2026

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