Einstein field equations

1 Introduction and Definition

The Einstein Field Equations are the fundamental equations of Einstein’s general theory of relativity. For a description of this physical theory and of the physical significance of solutions of these http://planetphysics.org/encyclopedia/TopicOnEquationsInMathematicalPhysics.htmlequations, please see http://planetphysics.org/PlanetPhysics. Here, we shall discuss the mathematical properties of these equations and their relevance to various branches of pure mathematics.

The Einstein field equations are a system of second orderPlanetmathPlanetmath coupled nonlinear partial differential equations for a Riemannian metricMathworldPlanetmath tensor on a Riemannian manifold. Let M be a differentiable manifold and let Tμν and gμν be symmetric tensor fields 11Throughout this entry, we shall use index notation for tensor fields because that is common in the literature (especially physics literature) and is convenient for computation of particular solutions. Moreover, we shall, fittingly enough, employ Einstein’s summation convention.. Further, assume that gμν is invertible on a dense subset of M and twice differentiablePlanetmathPlanetmath. (It is possible to relax the latter requirement by interpreting the equations distributionally.) Then the Einstein equations read as follows: 22In the physics literature, the coefficient of Tμν is written as 8πGc4, where G is the gravitational constant, c is the light velocity constant but, since we are interested in the purely mathematical properties of these equations, we shall set G=c=1 here, which may be accomplished by working in a suitable set of physical units. It might also be worth mentioning that, in physics, the tensor Tμν is the stress-energy tensor, which encodes data pertaining to the mass, energy, and momentum densities of the surrounding space. The number Λ is known as the cosmological constant because it determines large-scale properties of the universePlanetmathPlanetmath, such as whether it collapses, remains stationary, or expands.


Here, Gμυ=Rμυ-12gμυR is the Einstein Tensor, Rμυ is the Ricci tensor, and R=gμνRμν is the Ricci scalar, and gμν is the inversePlanetmathPlanetmathPlanetmath metric tensor.

One possibility is that the tensor field Tμν is specified and that these equations are then solved to obtain gμν. A noteworthy case of this is the vacuum Einstein equations, in which Tμν=0. Another possibility is that Tμν is given in terms of some other fields on the manifold and that the Einstein equations are augmented by differential equations which describe those fields. In that case, one speaks of Einstein-Maxwell equations, Einstein-Yang-Mills equations, and the like depending on what these other fields may happen to be. It should be noted that, on account of the Bianchi identity, there is an integrability condition μ(g)Tμν=0. (Here, (g) denotes covariant differentiation with respect to the Levi-Civita connectionMathworldPlanetmath of the metric tensor gμν.) When choosing Tμν, these conditions must be taken into account in order to guarantee that a solution is possible.

2 Diffeomorphism Invariance

Because they are constructed from tensors, the Einstein equations have an important invariance property. Suppose that gμν and Tμν satisfy the Einstein equations. Then, for any diffeomorphism f:MM, we also have that (f*g)μν and (f*T)μν also satisfy the Einstein equations. (Here, the notation f* denotes pullbackPlanetmathPlanetmath with respect to the diffeomorphism f.)

This fact means that we must be careful when talking about specifying solutions by boundary conditionsMathworldPlanetmath. Usually, when dealing with a differential equation, we would expect that we could specify a solution uniquely by providing enough boundary data. Here, however, this will not work since we could find a diffeomorphism which reduces to the identity near the boundary but differs from the identity elsewhere and use that to produce another solution which would satisfy the same boundary conditions. What one should do instead is to consier equivalence classesMathworldPlanetmathPlanetmath of solutions modulo diffeomorphism and only ask that boundary conditions specify solutions up to diffeomorphisms. As we shall see later, with such an understanding, one can indeed specify solutions in terms of initial data.

In order to adress this issue and to be able to treat the Einstein equations much as one would treat other differential equations, a common practise is to supplement the Einstein equations with auxiliary condidtions which serve to define a coordinate systemMathworldPlanetmath and hence single out a particular element of an equivalence class in diffeomorphism. While such auxiliary equations should ideally single out a representative for each equivalence class, in practise, one is content with considerably less — a particular choice auxiliary conditions might only work with some solutions or may only specify a subset of an equivalence class with more than one element.

Remarks: The major obstruction to the GR theory is that Einstein’s GR equations–although solvable in principle– are readily solvable only in special cases, with specified boundary conditions. The bigger problem is the difficulty of formulating quantum field theories (QFT) in a manner which is logically consistent with Einstein’s GR formulation so that a valid Quantum Gravity (QG) theory is formulated that yields results consistent with both GR and quantum theoriesPlanetmathPlanetmath in the presence of intense gravitational fields. So far, encouraging results have been obtained only for the limiting case of low intensity gravitational fields as in S. Weinberg’s algebraic approach to QFT and QG using supersymmetry and graded ‘Lie’ algebras or superalgebras.

3 Hyperbolic Formulations

4 Variational Principles

5 Alternative Formulations

An http://planetphysics.org/?op=getobj&from=objects&id=441alternative, more general formulation of GR and GR Field Equations would involve a categoricalPlanetmathPlanetmath framework, such as the category of pseudo-Riemannian manifolds, and/or the category of Riemannian manifolds, with, or without, a Riemannian metric. Expanding universes and black hole singularities, with or without hair, either with an event horizon, or ‘naked’, can be treated within such an unified categorical framework of Riemannian/ pseudo-Riemanian manifolds and their transformationsMathworldPlanetmathPlanetmath represented either as morphismsMathworldPlanetmath or by functorsMathworldPlanetmath and natural transformations between functors. Quantized versions in quantum gravity may also be available based on spin foams represented by time-dependent/ parameterized functors between spin networks including extremely intense, but finite, gravitational fields.

6 Global Structure

7 Initial Value Formulation

8 Special Solutions

8.1 Spatially Homogeneous Solutions

8.2 Solutions with Symmetries

8.3 Algebraically Special Solutions

8.4 Linearization

8.5 Singularities

8.6 Asymptotically Flat Solutions

8.7 Existence Theorems

Title Einstein field equations
Canonical name EinsteinFieldEquations
Date of creation 2013-03-22 15:02:34
Last modified on 2013-03-22 15:02:34
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 25
Author rspuzio (6075)
Entry type Topic
Classification msc 83C05
Synonym Einstein’s GR equations
Related topic CategoryOfRiemannianManifolds
Related topic PseudoRiemannianManifold
Related topic QuantumGeometry2
Related topic QuantumSpaceTimes
Related topic QuantumGravityTheories
Related topic RiemannianMetric
Related topic MathematicalProgrammesForDevelopingQuantumGravityTheories
Related topic MathematicalProgrammesForDevelopingQuantumGravityTheories