--- title: "The Schwarzschild solution: orbits, redshift and the horizon" description: "The Schwarzschild metric describes a spherical, uncharged vacuum region with zero cosmological constant. It predicts static-clock rate differences, orbital precession and light bending. In its black-h" canonical: https://lightmysky.com/learn/science/the-schwarzschild-solution-orbits-redshift-and-the-horizon-mt_dNZpRDeZUE source: https://lightmysky.com/learn/science/the-schwarzschild-solution-orbits-redshift-and-the-horizon-mt_dNZpRDeZUE.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`. # The Schwarzschild solution: orbits, redshift and the horizon The Schwarzschild metric describes a spherical, uncharged vacuum region with zero cosmological constant. It predicts static-clock rate differences, orbital precession and light bending. In its black-hole extension, the horizon is a causal boundary with finite curvature, unlike the central singularity. Subject: Science · Area: Astronomy & Astrophysics · Ages 23 to 24 Page: https://lightmysky.com/learn/science/the-schwarzschild-solution-orbits-redshift-and-the-horizon-mt_dNZpRDeZUE ## Ready when they can - Reads gravitational time dilation off the metric - Explains orbital precession as a departure from the inverse-square result - Separates a coordinate breakdown at the horizon from a genuine singularity ## Lesson: Clocks, orbits, and the edge of a black hole For a spherical, uncharged vacuum region with zero cosmological constant, the Schwarzschild metric is ds²=−f c²dt²+dr²/f+r²dΩ², where f=1−rₛ/r and rₛ=2GM/c². Here dΩ²=dθ²+sin²θ dφ². A static clock has fixed r>rₛ, θ and φ and obeys dτ=√f dt. The time coordinate is normalized to clocks at rest at infinity. An isolated Newtonian point mass gives closed elliptical bound orbits. Schwarzschild geometry adds a periapsis advance: the nearest point rotates forward each orbit. For a weak-field orbit, the advance is about 6πGM/[a(1−e²)c²] radians, where a is semimajor axis and e is eccentricity. Other bodies can also cause precession. **Example.** Compare two static clocks during the same coordinate-time interval. Each stays at fixed r, θ and φ. Their elapsed-time ratio is √[(1−rₛ/r_low)/(1−rₛ/r_high)]. At r_low=4rₛ and r_high=9rₛ, the ratio is √(27/32), below 1. The lower clock accumulates less proper time. This comparison does not describe orbiting or freely falling clocks. *(drawing: For light passing far outside the horizon, b is the impact parameter measured far away. This is the weak-deflection approximation.)* In the Schwarzschild black-hole extension, r=rₛ is an event horizon: signals inside cannot reach distant observers. The displayed coordinates fail there, but other coordinates cross it smoothly. A star whose surface lies outside rₛ does not have this black-hole horizon in its vacuum exterior. The horizon is not a solid wall, but smooth geometry does not promise a safe crossing. Tidal forces are finite there and can be large. In the ideal black-hole solution, curvature grows without bound as r approaches zero. That central singularity is different from the coordinate failure at rₛ. Within the isolated spherical model, relativistic precession follows from geodesic motion in the Schwarzschild metric; no extra force is added. For a real planetary orbit, first account for Newtonian perturbations from other bodies and the source's shape. A measured precession is not, by itself, proof of the Schwarzschild contribution. **Recap.** The Schwarzschild metric predicts static-clock rates, precession and light bending. A black-hole horizon is a causal boundary, not a curvature singularity or a promise of safe passage. *Original lesson, questions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Curvature and the Einstein field equation](https://lightmysky.com/learn/science/curvature-and-the-einstein-field-equation-mt_1CxE6kRDt_) - [Stellar Death: White Dwarfs, Neutron Stars and Black Holes](https://lightmysky.com/learn/science/stellar-death-white-dwarfs-neutron-stars-and-black-holes-mt_k1oUF3mJlu) ## Opens up - [Linearised gravity and gravitational waves](https://lightmysky.com/learn/science/linearised-gravity-and-gravitational-waves-mt_z0tMZDGMHi)