Einstein Field Equations
The Einstein field equations are the fundamental equations of general relativity, stating that the curvature of spacetime, captured by the Einstein tensor, is proportional to the energy and momentum of matter, captured by the stress-energy tensor.
Definition
The Einstein field equations are a set of ten coupled, nonlinear partial differential equations equating the Einstein curvature tensor (plus a cosmological-constant term) to the stress-energy tensor, thereby determining how matter and energy curve spacetime.
Scope
The area covers the form and meaning of the field equations, the Einstein tensor and the stress-energy tensor, their derivation from the Einstein-Hilbert action, the role of the cosmological constant, the conservation laws built into them, and the exact solutions, such as the Schwarzschild and Kerr metrics, obtained by imposing symmetry.
Sub-topics
Core questions
- What do the Einstein field equations say about the relationship between matter and geometry?
- How are the equations derived from a variational principle?
- Why are they difficult to solve, and how do symmetries make exact solutions possible?
Key concepts
- Einstein tensor
- Stress-energy tensor
- Einstein-Hilbert action
- Cosmological constant
- Bianchi identities and conservation
- Exact solutions
Key theories
- Einstein field equations
- The Einstein tensor, a specific combination of Ricci curvature and the metric, equals a constant times the stress-energy tensor, so that the distribution of energy and momentum determines spacetime curvature while local energy-momentum conservation is automatically built in.
- Einstein-Hilbert action
- Varying the integral of the Ricci scalar over spacetime, together with the matter action, yields the field equations, giving them a variational foundation analogous to the action principles of other physical theories.
Clinical relevance
Solving the field equations yields every quantitative prediction of relativistic gravity: the metrics describing black holes, the expanding-universe models of cosmology, the gravitational-wave templates used by detectors, and the strong-field environments around neutron stars and accreting compact objects.
History
Einstein arrived at the final field equations in November 1915 after several years of effort, with David Hilbert deriving them nearly simultaneously from an action principle; within months Schwarzschild found the first exact solution, and exact solutions with various symmetries have been catalogued ever since.
Debates
- Localization of gravitational energy
- Because the equivalence principle lets the gravitational field be transformed away locally, there is no agreed local tensor for gravitational energy density; only quasi-local and global definitions exist, an enduring conceptual subtlety of the theory.
Key figures
- Albert Einstein
- David Hilbert
- Karl Schwarzschild
- Roy Kerr
Related topics
Seminal works
- einstein1916
- mtw1973
Frequently asked questions
- Why are the Einstein equations so hard to solve?
- They are ten coupled, nonlinear partial differential equations in which the geometry both responds to and influences the matter, so closed-form solutions exist only under strong symmetry assumptions; general situations require numerical relativity on supercomputers.
- What is the cosmological constant doing in the equations?
- The cosmological constant is an allowed extra term proportional to the metric that acts like a uniform energy of empty space; introduced by Einstein for a static universe and later revived to explain cosmic acceleration, it is the simplest candidate for dark energy.