Exact Solutions to General Relativity & Einstein's Field Eqns.

[shounakbhatta]

Hello,

Can anybody tell me what is meant by exact solution to General Relativity or exact solution to Einstein's field equation.

-- Shounak

 

[martinbn]

It means explicit solution i.e. the components of the metric can be written down explicitly in terms of well known functions in some coordinates.

 

[HomogenousCow]

The rumored exact solutions...

 

[shounakbhatta]

Thanks for the reply. But in general is it that the metric cannot be written down in co-ordinates? Is it a special case?

 

[martinbn]

Thanks for the reply. But in general is it that the metric cannot be written down in co-ordinates? Is it a special case?

There are local coordinates and the metric will have its components in terms of the coordinates, but in general the functions involved will not be the ones that have names say polynomials, exponentials...

 

[Nugatory]

Thanks for the reply. But in general is it that the metric cannot be written down in co-ordinates? Is it a special case?

No, it is always possible to to write the metric down "in coordinates". But doing this requires solving the Einstein Field Equation for the specified conditions... And sometimes those equations are so hairy that no one has found a solution and we're forced to use numeric methods instead.

Note that I said "specified conditions", not "specified coordinates". If you can solve the EFE for a given situation using one coordinate system, a coordinate transformation will get you the same solution expressed in another coordinate system.