--- title: "Curvature and the Einstein field equation" description: "The Riemann tensor measures the leading change from parallel transport around an infinitesimal loop. The Einstein field equation relates curvature to stress-energy. Geometry also depends on initial an" canonical: https://lightmysky.com/learn/science/curvature-and-the-einstein-field-equation-mt_1CxE6kRDt_ source: https://lightmysky.com/learn/science/curvature-and-the-einstein-field-equation-mt_1CxE6kRDt_.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Curvature and the Einstein field equation The Riemann tensor measures the leading change from parallel transport around an infinitesimal loop. The Einstein field equation relates curvature to stress-energy. Geometry also depends on initial and boundary data, and the weak, slow-motion limit recovers Newtonian gravity. Subject: Science · Area: Astronomy & Astrophysics · Ages 23 to 24 Page: https://lightmysky.com/learn/science/curvature-and-the-einstein-field-equation-mt_1CxE6kRDt_ ## Ready when they can - Describes curvature operationally through transport around a closed loop - Names the two sides of the field equation and what each represents - Identifies the weak, nearly static, slow-motion Newtonian limit and states its Poisson equation ## Lesson: How matter and geometry shape each other Use the parallel transport rule from the previous topic to carry a vector around a small closed loop. Its leading change depends on the loop's oriented area and the starting vector. The Riemann tensor records this local curvature information. One unchanged vector or one unchanged loop does not prove that every curvature component vanishes. With the cosmological constant set to zero, the field equation is Gμν=(8πG/c⁴)Tμν. Here Gμν=Rμν−(1/2)Rgμν is the Einstein tensor, built from curvature. The stress-energy tensor Tμν describes energy density, momentum, pressure and stresses. The symbol G without indices is Newton's gravitational constant. Loop rotation is evidence of curvature when the vector has been parallel-transported. Rotating an arrow by hand is not the same test. For a sufficiently small loop, the leading mismatch scales with its area. To recover local curvature information, compare mismatches for chosen vectors and oriented loop areas. *(drawing: For a nearly static, weak field and slow matter, the field equation recovers Newton's Poisson equation. Φ is potential; ρ is mass density.)* Here Φ is gravitational potential, meaning potential energy per unit mass; U=mΦ. Newton's Poisson equation ∇²Φ=4πGρ relates spatial variation of Φ to mass density ρ. In Cartesian coordinates, ∇²Φ sums its second derivatives in the three spatial directions. Matching the weak, nearly static, slow-motion limit fixes the coupling to 8πG/c⁴ in our convention. A weak field alone is not enough. Near a compact object or at relativistic speeds, use the relativistic equations instead. Stress-energy constrains geometry, but specifying it does not by itself select one metric. Initial or boundary data are also needed. Different vacuum gravitational-wave data can give different geometries even with the same local stress-energy. Matter moving through the resulting geometry must also obey its equations of motion. In vacuum, Tμν=0 and, with zero cosmological constant, Rμν=0. This does not require the full Riemann tensor to vanish. The empty region outside a spherical mass can have tidal curvature. Contraction pairs one upper and one lower tensor index and sums over that pair. Ricci curvature is such a contraction of Riemann curvature. Setting that sum to zero does not erase every Riemann component. **Recap.** Small transport loops probe Riemann curvature. The field equation relates geometry to stress-energy. Initial data also matter, and weak, slow gravity recovers Newton's equation. *Original lesson, questions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Parallel transport, Christoffel symbols and the geodesic equation](https://lightmysky.com/learn/science/parallel-transport-christoffel-symbols-and-the-geodesic-equation-mt_bUEL2R91Vy) - [Gravitational Potential and Escape Speed](https://lightmysky.com/learn/science/gravitational-potential-and-escape-speed-mt_drqTwCD9W8) ## Opens up - [The Schwarzschild solution: orbits, redshift and the horizon](https://lightmysky.com/learn/science/the-schwarzschild-solution-orbits-redshift-and-the-horizon-mt_dNZpRDeZUE) - [Linearised gravity and gravitational waves](https://lightmysky.com/learn/science/linearised-gravity-and-gravitational-waves-mt_z0tMZDGMHi)