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Thread: Solving the Einstein Field Equations

  1. #1 Solving the Einstein Field Equations 
    Moderator Moderator Markus Hanke's Avatar
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    Motivation

    Time and again I hear people saying that the field equations of GR are "way too difficult", "incomprehensible" and "way over my head". In this thread I wish to show that solving these equations is - although mathematically extremely tedious - fairly straightforward on a conceptual level. Also I have been asked more than once about how to do it by different people, so I would like to show the general procedure in this thread.


    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.


    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.


    Conclusion

    As has been shown, solving these equations is conceptually straightforward as one only needs to follow the prescribed steps; however, the actual algebra and analysis is tedious and time consuming, and increasingly so once one abandons some of the presumed symmetries, thereby complicating the ansatz.
    The next step up from here would be to perform the same calculation for non-vanishing stress-energy-momentum tensors, i.e. for the interior of a mass. Finally, one can allow a cosmological constant into the field equations. Each of these steps will rapidly complicate the resulting differential equations, which is why in many cases the solutions are possible only numerically, but not in closed algebraic form.


    References

    Fliessbach, Prof Torsten : Allgemeine Relativitätstheorie , Mannheim/Wien/Zuerich : BI-Wiss.-Verl. 1990



    A discussion thread for this sticky can be found here :

    SPLIT : Discussion of Solving Einstein Equations Sticky


    Last edited by Markus Hanke; October 1st, 2013 at 05:20 AM.
     

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  3. #2  
    Moderator Moderator Markus Hanke's Avatar
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    I have put all this together into a PDF, and added a calculation to show how to derive an expression for gravitational light deflection from the Schwarzschild metric. Here's the link :

    https://dl.dropbox.com/u/14461199/Li...ction%20SM.pdf

    Enjoy


     

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