Math 2.0

Relativity and gravitational theory Definition

Linearized gravity

Builds on Minkowski spaceEinstein field equationsStress–energy tensorWave equationGauge fixing

Definition

Let ημν=diag(−1,1,1,1)\eta_{\mu\nu} = \mathrm{diag}(-1, 1, 1, 1) be the Minkowski metric on ℝ4\R^4 in coordinates xμ=(ct,x,y,z)x^\mu = (ct, x, y, z). Consider metrics gμν=ημν+hμνg_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, where hμνh_{\mu\nu} is a symmetric tensor field with ∣hμν∣≪1\abs{h_{\mu\nu}} \ll 1; indices are raised and lowered with η\eta. Let TμνT_{\mu\nu} be the stress–energy tensor, taken to be of first order. Linearized gravity is the theory obtained by keeping only the terms of first order in hh in the Einstein field equations Gμν=8πGc4TμνG_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}. In terms of the trace-reversed perturbation hˉμν=hμν−12ημνh\bar h_{\mu\nu} = h_{\mu\nu} - \frac12 \eta_{\mu\nu} h, with h=ημνhμνh = \eta^{\mu\nu} h_{\mu\nu}, the linearized equations are −□hˉμν+∂μ∂ρhˉρν+∂ν∂ρhˉρμ−ημν∂ρ∂σhˉρσ=16πGc4Tμν,-\square \bar h_{\mu\nu} + \partial_\mu \partial^\rho \bar h_{\rho\nu} + \partial_\nu \partial^\rho \bar h_{\rho\mu} - \eta_{\mu\nu} \partial^\rho \partial^\sigma \bar h_{\rho\sigma} = \frac{16\pi G}{c^4} T_{\mu\nu}, where □=ημν∂μ∂ν\square = \eta^{\mu\nu} \partial_\mu \partial_\nu.

Remarks

The theory has the gauge freedom hμν↦hμν+∂μξν+∂νξμh_{\mu\nu} \mapsto h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu for any small vector field ξ\xi, which comes from infinitesimal changes of coordinates. Using it (gauge fixing), one can impose the harmonic (Lorenz, de Donder) gauge ∂μhˉμν=0\partial^\mu \bar h_{\mu\nu} = 0. The equations then become the wave equations □hˉμν=−16πGc4Tμν.\square \bar h_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}. The linearized equations imply ∂μTμν=0\partial^\mu T_{\mu\nu} = 0, the conservation law of flat space.

Example: a static point mass MM has T00=Mc2δ3(x)T_{00} = M c^2 \delta^3(x). In the harmonic gauge ∇2hˉ00=−16πGMδ3(x)/c2\nabla^2 \bar h_{00} = -16\pi G M \delta^3(x)/c^2 gives hˉ00=4GM/(rc2)\bar h_{00} = 4GM/(rc^2), the other components vanish, and so h00=2GM/(rc2)h_{00} = 2GM/(rc^2) and hij=2GMδij/(rc2)h_{ij} = 2GM \delta_{ij}/(rc^2). This matches the Schwarzschild metric in isotropic coordinates to first order in GM/(rc2)GM/(rc^2).

Non-example: near the horizon of a black hole of mass MM, at rr close to 2GM/c22GM/c^2, the perturbation is not small, and linearized gravity does not apply.

Reading path

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

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Einstein field equations, concept, no evidence levelEinstein field equati…conceptGauge fixing, concept, no evidence levelGauge fixingconceptMinkowski space, concept, no evidence levelMinkowski spaceconceptStress–energy tensor, concept, no evidence levelStress–energy tensorconceptWave equation, concept, no evidence levelWave equationconceptLinearized gravity, definition, no evidence levelLinearized gravitydefinition
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 field equations concept
  2. Gauge fixing concept
  3. Minkowski space concept
  4. Stress–energy tensor concept
  5. Wave equation concept
  6. Linearized gravity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Linearized_gravity
About this page
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Kinddefinition
WritingW0 stub
MSC83C25, 83C35
DefinesLinearized gravity
UsesMinkowski space, Einstein field equations, Stress–energy tensor, Wave equation, Gauge fixing
Changed3 November 2026

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