Math 2.0

Relativity and gravitational theory Definition

Hamiltonian constraint

Builds on ADM formalismInitial value formulation of general relativityScalar curvatureFirst class constraint

Definition

Let Σ\Sigma be a 3-manifold with a Riemannian metric hijh_{ij} and a symmetric tensor KijK_{ij}, and let πij=h (Kij−Khij)\pi^{ij} = \sqrt{h} \, (K^{ij} - K h^{ij}) with K=hijKijK = h^{ij} K_{ij}, as in the ADM formalism. In units with 16πG=116 \pi G = 1, the Hamiltonian constraint of vacuum general relativity is the equation ℋ=1h(πijπij−12π2)−h R(h)=0,\mathcal{H} = \frac{1}{\sqrt{h}} \Bigl( \pi^{ij} \pi_{ij} - \tfrac12 \pi^2 \Bigr) - \sqrt{h} \, R(h) = 0, where R(h)R(h) is the scalar curvature of hh and π=hijπij\pi = h_{ij} \pi^{ij}. In terms of KK it reads R(h)+K2−KijKij=0.R(h) + K^2 - K_{ij} K^{ij} = 0. With matter of energy density ρ\rho measured by the normal observer, and with G=1G = 1, the right-hand side becomes 16πρ16 \pi \rho.

Remarks

The constraint is the normal-normal component of the Einstein equations: 2Gabnanb=R(h)+K2−KijKij2 G_{ab} n^a n^b = R(h) + K^2 - K_{ij} K^{ij} for the unit normal nn, by the Gauss equation. It is one of the constraint equations of the initial value problem; the other three form the momentum constraint. Example: the spatial Schwarzschild data h=(1+m/(2r))4δh = (1 + m/(2r))^4 \delta on ℝ3∖{0}\R^3 \setminus \set{0} with K=0K = 0 satisfy it. Indeed, for h=u4δh = u^4 \delta one has R(h)=−8u−5ΔuR(h) = -8 u^{-5} \Delta u, and u=1+m/(2r)u = 1 + m/(2r) is harmonic. Non-example: the round 3-sphere of radius aa with K=0K = 0 fails it, since R(h)=6/a2≠0R(h) = 6/a^2 \ne 0. The Hamiltonian constraint and the momentum constraint are first class constraints in the sense of Dirac.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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ADM formalism, concept, no evidence levelADM formalismconceptFirst class constraint, concept, no evidence levelFirst class constraintconceptInitial value formulation of general relativity, concept, no evidence levelInitial value formula…conceptMatrix, definition, no evidence levelMatrixdefinitionPseudo-Riemannian manifold, concept, no evidence levelPseudo-Riemannian man…conceptRicci curvature, concept, no evidence levelRicci curvatureconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionScalar curvature, definition, no evidence levelScalar curvaturedefinitionHamiltonian constraint, definition, no evidence levelHamiltonian constraintdefinition
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  1. ADM formalism concept
  2. First class constraint concept
  3. Initial value formulation of general relativity concept
  4. Matrix definition
  5. Pseudo-Riemannian manifold concept
  6. Ricci curvature concept
  7. Square matrix definition
  8. Trace of a matrix definition
  9. Scalar curvature definition
  10. Hamiltonian constraint definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hamiltonian_constraint
About this page
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MSC83C05, 70H45, 83C45
DefinesHamiltonian constraint
UsesADM formalism, Initial value formulation of general relativity, Scalar curvature, First class constraint
Changed3 November 2026

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