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 equations, please see 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 order coupled nonlinear partial differential equations for a Riemannian metric tensor on a Riemannian manifold. Let $M$ be a differentiable manifold and let $T_{\mu \nu}$ and $g_{\mu \nu}$ be symmetric tensor fields ¹. Further, assume that $g_{\mu \nu}$ is invertible on a dense subset of $M$ and twice differentiable. (It is possible to relax the latter requirement by interpreting the equations distributionally.) Then the Einstein equations read as follows: ² $$ G_{\mu \nu} = \Lambda g_{\mu \nu} + 8 \pi T_{\mu \nu $$ Here, $G_{\mu\upsilon}=R_{\mu\upsilon}-\frac{1}{2}g_{\mu\upsilon}R$ is the Einstein Tensor, $R_{\mu\upsilon}$ is the Ricci tensor, and $R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $g^{\mu\nu}$ is the inverse metric tensor.

One possibility is that the tensor field $T_{\mu \nu}$ is specified and that these equations are then solved to obtain $g_{\mu \nu}$ . A noteworthy case of this is the vacuum Einstein equations, in which $T_{\mu \nu} = 0$ . Another possibility is that $T_{\mu \nu}$ 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 $\nabla_\mu (g) T^{\mu \nu} = 0$ . (Here, $\nabla (g)$ denotes covariant differentiation with respect to the Levi-Civita connection of the metric tensor $g_{\mu \nu}$ .) When choosing $T_{\mu \nu}$ , these conditions must be taken into account in order to guarantee that a solution is possible.

Diffeomorphism Invariance

Because they are constructed from tensors, the Einstein equations have an important invariance property. Suppose that $g_{\mu \nu}$ and $T_{\mu \nu}$ satisfy the Einstein equations. Then, for any diffeomorphism $f \colon M \to M$ , we also have that $(f^* g)_{\mu \nu}$ and $(f^* T)_{\mu \nu}$ also satisfy the Einstein equations. (Here, the notation $f^*$ denotes pullback with respect to the diffeomorphism $f$ .)

This fact means that we must be careful when talking about specifying solutions by boundary conditions. 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 classes 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 system 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 theories 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.

Hyperbolic Formulations

Variational Principles

Alternative Formulations

An alternative, more general formulation of GR and GR Field Equations would involve a categorical 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 transformations represented either as morphisms or by functors 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.

Global Structure

Initial Value Formulation

Special Solutions

Spatially Homogeneous Solutions

Solutions with Symmetries

Algebraically Special Solutions

Linearization

Singularities

Asymptotically Flat Solutions

Existence Theorems

Footnotes

...http://planetmath.org/encyclopedia/Field.html ¹

Throughout 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.

... follows:²

In the physics literature, the coefficient of $T_{\mu \nu}$ is written as $\frac{8\pi G}{c^4}$ , 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 $T_{\mu \nu}$ is the stress-energy tensor, which encodes data pertaining to the mass, energy, and momentum densities of the surrounding space. The number $\Lambda$ is known as the cosmological constant because it determines large-scale properties of the universe, such as whether it collapses, remains stationary, or expands.