Einstein field equations

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.html

equations, 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 order

coupled nonlinear partial differential equations for a

Riemannian metric

tensor on a

Riemannian manifold

. Let

M

be
a

differentiable manifold

and let

T

μ

⁢

ν

and

g

μ

⁢

ν

be

symmetric
tensor

fields

1

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.

. Further, assume that

g

μ

⁢

ν

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

2

In the physics literature, the coefficient of

T

μ

⁢

ν

is
written as

8

⁢

π

⁢

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

μ

⁢

ν

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

universe

, such as whether it collapses, remains

stationary

, or expands.

G

μ

⁢

ν

=

Λ

⁢

g

μ

⁢

ν

+

8

⁢

π

⁢

T

μ

⁢

ν

Here,

G

μ

⁢

υ

=

R

μ

⁢

υ

-

1

2

⁢

g

μ

⁢

υ

⁢

R

is the

Einstein Tensor

,

R

μ

⁢

υ

is the

Ricci tensor

, and

R

=

g

μ

⁢

ν

⁢

R

μ

⁢

ν

is the Ricci scalar, and

g

μ

⁢

ν

is the

inverse

metric tensor

.

One possibility is that the tensor field

T

μ

⁢

ν

is specified and that
these equations are then solved to obtain

g

μ

⁢

ν

. A noteworthy case of
this is the

vacuum Einstein equations

, in which

T

μ

⁢

ν

=

0

.
Another possibility is that

T

μ

⁢

ν

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

)

⁡

T

μ

⁢

ν

=

0

. (Here,

∇

⁡

(

g

)

denotes covariant
differentiation with respect to the

Levi-Civita connection

of the metric
tensor

g

μ

⁢

ν

.) When choosing

T

μ

⁢

ν

, 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

g

μ

⁢

ν

and

T

μ

⁢

ν

satisfy

the Einstein equations. Then, for any

diffeomorphism

f

:

M

→

M

, we also have that

(

f

*

⁢

g

)

μ

⁢

ν

and

(

f

*

⁢

T

)

μ

⁢

ν

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.

3

Hyperbolic Formulations

4

Variational Principles

5

Alternative Formulations

An

http://planetphysics.org/?op=getobj&from=objects&id=441

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.

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

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