Math 2.0

Relativity and gravitational theory Definition

Euclidean quantum gravity

Builds on Riemannian manifoldPath integral formulationWick rotationEinstein–Hilbert actionGibbons–Hawking–York boundary term

Definition

Let MM be a four-manifold with boundary ∂M\partial M, and let hh be a Riemannian metric on ∂M\partial M. For a Riemannian metric gg on MM inducing hh on ∂M\partial M, the Euclidean gravitational action is I[g]=−116πG∫M(R−2Λ)g d4x−18πG∫∂MKh d3x.I[g] = -\frac{1}{16 \pi G} \int_M (R - 2\Lambda) \sqrt{g} \, d^4x - \frac{1}{8 \pi G} \int_{\partial M} K \sqrt{h} \, d^3x. Here RR is the scalar curvature of gg, Λ\Lambda is the cosmological constant and KK is the trace of the extrinsic curvature of ∂M\partial M. The boundary term is the Gibbons–Hawking–York term. Euclidean quantum gravity defines amplitudes and partition functions by the formal path integral Z[h]=∫𝒟[g]  e−I[g]/ℏZ[h] = \int \mathcal{D}[g] \; e^{-I[g]/\hbar} in units c=1c = 1. The integral runs over Riemannian metrics gg on MM that induce hh on ∂M\partial M, up to diffeomorphism. It may also run over the topology of MM.

Remarks

The action II is −i-i times the Lorentzian Einstein–Hilbert action with boundary term after the Wick rotation t=−iτt = -i\tau. Example: the Euclidean Schwarzschild metric (1−2GMr)dτ2+(1−2GMr)−1dr2+r2dΩ2\left(1 - \frac{2GM}{r}\right) d\tau^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2 is smooth at r=2GMr = 2GM only if τ\tau has period β=8πGM\beta = 8 \pi G M (with ℏ=c=kB=1\hbar = c = k_B = 1). Taking this saddle point, Z≈e−IZ \approx e^{-I} with I=4πGM2I = 4\pi G M^2, gives the Hawking temperature 1/β1/\beta and the Bekenstein–Hawking entropy 4πGM24 \pi G M^2, a quarter of the horizon area in Planck units. Non-example: a Lorentzian metric is not in the domain of the integral, since its weight eiS/ℏe^{iS/\hbar} is oscillatory and not of the form e−I/ℏe^{-I/\hbar}. The action is unbounded below under conformal rescalings g↦Ω2gg \mapsto \Omega^2 g (the conformal factor problem), so the integral does not converge as written.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Einstein–Hilbert action, concept, no evidence levelEinstein–Hilbert acti…conceptGibbons–Hawking–York boundary term, concept, no evidence levelGibbons–Hawking–York …conceptPath integral formulation, concept, no evidence levelPath integral formula…conceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptWick rotation, concept, no evidence levelWick rotationconceptEuclidean quantum gravity, definition, no evidence levelEuclidean quantum gra…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. Einstein–Hilbert action concept
  2. Gibbons–Hawking–York boundary term concept
  3. Path integral formulation concept
  4. Riemannian manifold concept
  5. Wick rotation concept
  6. Euclidean quantum gravity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclidean_quantum_gravity
About this page
TagSE2KEW5QFH
URIm2://SE2KEW5QFH
Kinddefinition
WritingW0 stub
MSC83C45, 81S40
DefinesEuclidean quantum gravity
UsesRiemannian manifold, Path integral formulation, Wick rotation, Einstein–Hilbert action, Gibbons–Hawking–York boundary term
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.