Math 2.0

Relativity and gravitational theory Definition

Exact solutions in general relativity

Builds on Lorentzian manifoldEinstein field equationsStress–energy tensorEnergy conditions

Definition

Fix a matter model: a list of matter fields on a 4-manifold, together with their field equations. The model also gives their stress–energy tensor TT in terms of the fields and the metric. An exact solution of general relativity for this model is a 4-dimensional Lorentzian manifold (M,g)(M, g) with matter fields on MM such that:

  1. gg and TT satisfy the Einstein field equations Gab+Λgab=8πTabG_{ab} + \Lambda g_{ab} = 8\pi T_{ab}, for a fixed constant Λ\Lambda;
  2. the matter fields satisfy their own field equations on (M,g)(M, g).

Remarks

Example: the Schwarzschild metric g=−(1−2M/r)dt2+(1−2M/r)−1dr2+r2dΩ2g = -(1 - 2M/r) dt^2 + (1 - 2M/r)^{-1} dr^2 + r^2 d\Omega^2 on r>2Mr > 2M is an exact solution with no matter and Λ=0\Lambda = 0, since its Ricci tensor is zero. Another example is the Reissner–Nordström metric with a Coulomb electromagnetic field, which solves the Einstein–Maxwell equations. Non-example: a gravitational plane wave of linearised gravity solves the linearised equations, but in general not the full Einstein equations, so it is an approximate solution only. Without a matter model the notion is empty: any metric solves the equations with TT defined as (G+Λg)/8π(G + \Lambda g)/8\pi. For this reason one also asks that TT be physically reasonable, for instance that it satisfies the energy conditions.

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
Energy conditions, concept, no evidence levelEnergy conditionsconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptPseudo-Riemannian manifold, concept, no evidence levelPseudo-Riemannian man…conceptRicci curvature, concept, no evidence levelRicci curvatureconceptScalar curvature, concept, no evidence levelScalar curvatureconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptStress–energy tensor, concept, no evidence levelStress–energy tensorconceptTensor field, concept, no evidence levelTensor fieldconceptLorentzian manifold, definition, no evidence levelLorentzian manifolddefinitionEinstein field equations, definition, no evidence levelEinstein field equati…definitionExact solutions in general relativity, definition, no evidence levelExact solutions in ge…definition
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. Energy conditions concept
  2. Lorentzian manifold concept
  3. Pseudo-Riemannian manifold concept
  4. Ricci curvature concept
  5. Scalar curvature concept
  6. Signature of a quadratic form concept
  7. Stress–energy tensor concept
  8. Tensor field concept
  9. Lorentzian manifold definition
  10. Einstein field equations definition
  11. Exact solutions in general relativity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Exact_solutions_in_general_relativity
About this page
TagKCVJ91K4WD
URIm2://KCVJ91K4WD
Kinddefinition
WritingW0 stub
MSC83C15, 83C20, 83C05
DefinesExact solutions in general relativity
UsesLorentzian manifold, Einstein field equations, Stress–energy tensor, Energy conditions
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.