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Thread: Solving the Einstein Field Equations - An Example

  1. #1 Solving the Einstein Field Equations - An Example 
    Administrator Markus Hanke's Avatar
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    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
    Last edited by Markus Hanke; 01-21-2013 at 04:04 PM.
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    Althought its implicit in the equations I think it would be handy to post the sign convetions that you're using as a footnote.
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    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?
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    Administrator Markus Hanke's Avatar
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    Quote Originally Posted by Popper View Post
    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.
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    Quote Originally Posted by Markus Hanke View Post
    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?
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  6. #6  
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    Quote Originally Posted by Popper View Post
    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.
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