Part III: Einstein Field Equations
Einstein's field equations relate the curvature of spacetime to the distribution of matter and energy. This is the core of general relativity: "Matter tells spacetime how to curve, and spacetime tells matter how to move."
Part Overview
The Einstein field equations are where is the Einstein tensor (geometry) and is the stress-energy tensor (matter/energy). These 10 coupled nonlinear partial differential equations determine how spacetime curves in response to matter. We derive them from the equivalence principle and the Einstein-Hilbert action, explore their limits, and learn to solve them.
Key Topics
- • Equivalence principle: gravity = acceleration locally
- • Einstein field equations:
- • Stress-energy tensor: energy density, momentum density, stress
- • Einstein-Hilbert action and variational derivation
- • Weak field limit: recovering Newtonian gravity
- • Post-Newtonian approximations and experimental tests
6 chapters | The heart of GR | From principle to equations
Chapters
Chapter 1: Equivalence Principle
The foundation of GR: locally, gravity is indistinguishable from acceleration. Weak equivalence principle (universality of free fall). Einstein's gedankenexperiment: the elevator. Why gravity must curve spacetime. Strong equivalence principle: all local physics is the same in freely falling frames. Experimental tests.
Chapter 2: Einstein Field Equations
The field equations: where is the Einstein tensor and is the cosmological constant. Why this form? Consistency with energy-momentum conservation. Structure: 10 equations for 10 metric components, but gauge freedom reduces independent equations.
Chapter 3: Stress-Energy Tensor
encodes energy density, momentum density, momentum flux, and stress. Perfect fluid: . Electromagnetic field: . Vacuum energy: . Conservation: .
Chapter 4: Einstein-Hilbert Action
Deriving the field equations from the action principle. The Einstein-Hilbert action: . Varying with respect to the metric gives the field equations. The Gibbons-Hawking-York boundary term. Action principle unifies GR with quantum field theory formalism.
Chapter 5: Weak Field Limit
Linearized gravity: where . The linearized field equations become wave equations. Recovering Newtonian gravity: where . Gravitational waves as ripples in spacetime. Gauge choices: harmonic gauge, transverse-traceless gauge.
Chapter 6: Newtonian Limit
The rigorous derivation of Newtonian gravity from GR. Assumptions: weak fields, slow velocities (), static sources. The geodesic equation reduces to . Post-Newtonian corrections: perihelion precession, light bending, Shapiro delay. Experimental verification of GR.
Course Navigation
Prerequisites:
- • Part I: Differential Geometry
- • Part II: Curvature of Spacetime
- • Variational calculus and Lagrangian mechanics