# Thread: Solving the Einstein Field Equations - An Example

1. Motivation

To give an example of the general methodology one would follow to find a solution to the Einstein Field Equations. The specific example used yields the exterior Schwarzschild metric.

Definitions

Recall the definitions of the basic entities used in the equations.

Einstein Field Equations ( without cosmological constant ) :

(1) Ricci tensor :

(2) Christoffel symbols :

(3) Contracted Christoffel symbols :

(4) Ansatz

Every solution of the field equations requires an ansatz; in this thread we will look at the simplest possible solution of the equations, which is the vacuum solution of a spherically symmetric gravitational field for a static mass. The solution is called the Schwarzschild Metric. The spherical symmetry and the condition that mass and resulting field are static leads to the following simple ansatz :

(5) with two as yet unspecified functions A(r) and B(r). Our task will be to find these two functions from the field equations. EDIT : Note that I am using the sign convention (+,-,-,-) for this ansatz.

Field Equations

In a vacuum ( ) the Einstein Field Equations (1) reduce to

(6) which is a set of partial differential equations for the unknown functions A(r) and B(r).

Calculating the Christoffel Symbols

The elements of the Christoffel symbols which do not vanish are         Calculating the Ricci Tensor

The non-vanishing elements of the Ricci tensor are thus    Solving the Equations

From the above we obtain the system of equations    We now write and, doing some algebra, we obtain from this We also know that the gravitational field vanishes at infinity, i.e for we obtain  and therefore Now we can insert this into the remaining equations :  One can easily verify that these two differential equations are solved by  with an integration constant a. This constant is determined by the condition that the solution of the field equation must reduce the usual Newton's law at infinity; therefore Putting all this back into the ansatz (5) gives us the solution of the Einstein field equation we were looking for : This is called the Exterior Schwarzschild Metric, and its form is the simplest possible vacuum solution to the original field equations without cosmological constant.

References
Fliessbach, Prof Torsten : Allgemeine Relativitätstheorie , Mannheim/Wien/Zuerich : BI-Wiss.-Verl. 1990  2. Althought its implicit in the equations I think it would be handy to post the sign convetions that you're using as a footnote.  3. Althought its implicit in the equations I think it would be handy to post the sign convetions that you're using as a footnote.

Question: Why do you refer to a as a constant of integration?  4. Originally Posted by Popper Althought its implicit in the equations I think it would be handy to post the sign convetions that you're using as a footnote.
Ok, I have inserted that as an "edit note" into the ansatz section.

Question: Why do you refer to a as a constant of integration?
This constant is introduced as a result of integrating the differential equations for R11 and R22. In actual fact I will need to check the calculation, because that being a second order differential equation there should be a second constant of integration; not sure where that's gone. Don't have time right now, but need to look at that in the next few days. It's been a good while since I have done the maths for this solution.  5. Originally Posted by Markus Hanke This constant is introduced as a result of integrating the differential equations for ....
Yeah. I realized that right after hit submit but was unable to go back and edit it out.

I it possible to put a delete slection in so we can delete a post that we thought better of posting and want to get rid of it?  6. Originally Posted by Popper Yeah. I realized that right after hit submit but was unable to go back and edit it out.

I it possible to put a delete slection in so we can delete a post that we thought better of posting and want to get rid of it?
There is a "Delete" function available to the moderators, but not to general forum members, and I prefer to keep it that way since that function lends itself to misuse ( trust me, I know from experience ). You do however have an "Edit" function, so you can just overwrite what you have written by a little note like "Please ignore" or "Irrelevant Post" if you so wish.

To be honest it is not such a bad question - we should leave it on for other, less mathematically versed, members in case they are wondering about that same thing.  Posting Permissions
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