Exact solutions of Einstein's field equations

Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energy-momentum tensor (by the field equations). Although no generally agreed definition exists, an exact solution of the Einstein field equation is usually taken to mean a Lorentz metric which solves the field equation given an energy-momentum tensor. Often the requirement of symmetry conditions on the metric, initial and boundary conditions on the spacetime or algebraic restrictions on the Riemann tensor are imposed in obtaining exact solutions. Some well-known and popular exact solutions include: #Schwarzschild metric (which describes the spacetime geometry around a spherical mass) #Kerr metric (which describes the geometry around a rotating spherical mass) #Reissner-Nordstrom metric (which describes the geometry around a charged spherical mass) #Kerr-Newman metric (which describes the geometry around a charged-rotating spherical mass) #Robertson-Walker metric (which is an important model of an expanding universe) #pp-wave spacetimes (which include various types of gravitational waves) #G�del metric (which describes a rotating universe which permits time travel) #wormhole metrics (which serve as theoretical models for time travel) #Alcubierre metric (which serves as a theoretical model of space travel) Solutions (1), (2), (3) and (4) also include black hole. == Techniques == Exact solutions may be found by using a variety of techniques, many of them being algebraic ones. Certain classifications of tensors are useful in studying exact solutions, for example the Petrov classification of the Weyl tensor and the Segre classification of the energy-momentum tensor. There is also a method of classifying the Riemann tensor algebraically. Other techniques involve the study of symmetries in general relativity, where, for example, certain vector fields are imposed on the spacetime and the resulting metrics are to be found. In the study of exact solutions, and in particular, of symmetries in general relativity, it is often useful to decompose the Riemann tensor into its trace and trace-free parts. This is accomplished by taking the definition of the Weyl tensor in terms of the Riemann tensor and the Ricci tensor and making the Riemann tensor the subject of the formula. In four dimensions, this gives:

R_{abcd}=\frac{R}{6}G_{abcd}+E_{abcd}+C_{abcd}

where the Weyl tensor is the trace-free part (as it satisfies

C^a{}_{bad}=0)

and the tensors

G

and

E

have the following components: :

G_{abcd}=g_{a[c}g_{d]b}

E_{abcd}=\tilde{R}_{a[c}g_{d]b}+\tilde{R}_{b[d}g_{c]a}

where

\tilde{R}_{ab}

are the components of the trace-free Ricci tensor. The Petrov and Segre classifications may be used in conjunction with this expression for the Riemann tensor to write a list of all possible Riemann tensors. Exact solutions for which the energy-momentum tensor is identically zero in the region under consideration are termed vacuum solutions and represent the gravitational field in a region of spacetime where there are no material gravitational sources. Strictly speaking, as the gravitational field can do work (in moving planets around the Sun, for example), the field possesses energy (although determining the precise location of this energy in the field is still a problem) and therefore by

E=m c^2

has an effective mass which thereby creates another gravitational field - this feedback effect has the consequence that the gravitational field produced by two bodies is not the sum of the gravitational fields of the individual bodies. This is one of the major difficulties in finding exact solutions of the field equations, and quite often simplifying assumptions such as linearising the field equations are made. ==References== *Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. and Herlt, E. ''Exact Solutions of Einstein's Field Equations (2nd edn.)'' (2003) CUP ISBN 0521461367 * Adler, R., Bazin, M. and Schiffer, M. ''Introduction to General Relativity (2nd edn.)'' (1975) McGraw-Hill New York ISBN 0-07-000423-4 General relativity Equations

Exact solutions of Einstein's field equations

I've renamed the page 'exact solutions' to 'Exact solutions of Einstein's field equations' as was requested and agreed upon by several editors. --- User:Mpatel 11:29, 12 Jun 2005 (UTC) == Segre and Petrov classifications == For anyone interested in such things, I've created a page on the Segre classification of rank two symmetric tensors and on the Petrov classification of the Weyl tensor. ---User:Mpatel 17:37, 14 Jun 2005 (UTC)

See other meanings of words starting from letter:

E

EA | EB | EC | ED | EF | EG | EH | EI | EJ | EK | EL | EM | EN | EO | EP | ER | ES | ET | EU | EW | EX | EY | EZ |

Words begining with Exact_solutions_of_Einstein\'s_field_equations:

Exact_solutions_of_Einstein's_field_equations
Exact_solutions_of_Einstein's_field_equations