Einstein field equations

1 Introduction and Definition

The Einstein Field Equations are the fundamental equations of Einstein’sgeneral theory of relativity. For a description of this physical theory and ofthe physical significance of solutions of these http://planetphysics.org/encyclopedia/TopicOnEquationsInMathematicalPhysics.htmlequations, please seehttp://planetphysics.org/PlanetPhysics. Here, we shall discuss the mathematical properties of theseequations 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 $M$ bea differentiable manifold and let $T_{\mu \nu }$ and $g_{\mu \nu }$ be symmetrictensor fields . 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 equationsread as follows:

$$G_{\mu \nu }=\mathrm{\Lambda }{g}_{\mu \nu }+8\pi T_{\mu \nu }$$

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 theinverse metric tensor.

One possibility is that the tensor field $T_{\mu \nu }$ is specified and thatthese equations are then solved to obtain $g_{\mu \nu }$. A noteworthy case ofthis is the vacuum Einstein equations, in which $T_{\mu \nu }=0$.Another possibility is that $T_{\mu \nu }$ is given in terms of some otherfields on the manifold and that the Einstein equations are augmented bydifferential equations which describe those fields. In that case, one speaksof Einstein-Maxwell equations, Einstein-Yang-Mills equations, and the likedepending on what these other fields may happen to be. It should be notedthat, on account of the Bianchi identity, there is an integrability condition${\nabla }_{\mu }(g)T^{\mu \nu }=0$. (Here, $\nabla (g)$ denotes covariantdifferentiation with respect to the Levi-Civita connection of the metrictensor $g_{\mu \nu }$.) When choosing $T_{\mu \nu }$, these conditions mustbe 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 animportant invariance property. Suppose that $g_{\mu \nu }$ and $T_{\mu \nu }$ satisfy the Einstein equations. Then, for any diffeomorphism $f:M\to M$, we also have that ${(f^{*}g)}_{\mu \nu }$ and ${(f^{*}T)}_{\mu \nu }$ alsosatisfy the Einstein equations. (Here, the notation $f^{*}$ denotes pullbackwith respect to the diffeomorphism $f$.)

This fact means that we must be careful when talking about specifying solutionsby boundary conditions. Usually, when dealing with a differential equation,we would expect that we could specify a solution uniquely by providing enoughboundary data. Here, however, this will not work since we could find adiffeomorphism which reduces to the identity near the boundary but differsfrom the identity elsewhere and use that to produce another solution whichwould satisfy the same boundary conditions. What one should do instead is toconsier equivalence classes of solutions modulo diffeomorphism and onlyask that boundary conditions specify solutions up to diffeomorphisms. As weshall see later, with such an understanding, one can indeed specify solutionsin terms of initial data.

In order to adress this issue and to be able to treat the Einstein equationsmuch as one would treat other differential equations, a common practise is tosupplement the Einstein equations with auxiliary condidtions which serve todefine a coordinate system and hence single out a particular element of anequivalence class in diffeomorphism. While such auxiliary equations shouldideally single out a representative for each equivalence class, in practise,one is content with considerably less — a particular choice auxiliaryconditions might only work with some solutions or may only specify asubset 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 biggerproblem is the difficulty of formulating quantum field theories (QFT) in a mannerwhich is logically consistent with Einstein’s GR formulation so that a valid Quantum Gravity (QG) theoryis formulated that yields results consistent with both GR and quantum theories in the presence of intensegravitational fields. So far, encouraging results have been obtained only for the limiting caseof low intensity gravitational fields as in S. Weinberg’s algebraic approach to QFT and QG using supersymmetry andgraded ‘Lie’ algebras or superalgebras.

3 Hyperbolic Formulations

4 Variational Principles

5 Alternative Formulations

Anhttp://planetphysics.org/?op=getobj&from=objects&id=441alternative, 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 withinsuch an unified categorical framework of Riemannian/ pseudo-Riemanian manifolds and theirtransformations represented either as morphisms or by functors and natural transformationsbetween functors. Quantized versions in quantum gravity may also be available based onspin foams represented by time-dependent/ parameterized functors between spin networksincluding 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