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Thread: Can you form the differential geometry in the EFEs from a geodesic and the initial conditions of the stress energy tensor?

  1. #1 Can you form the differential geometry in the EFEs from a geodesic and the initial conditions of the stress energy tensor? 
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    So suppose you have a geodesic for a small massed particle(say a proton), and are given the initial conditions of the stress energy of the system(at t=0 for simplicity). We do not, however, know the differential form of the Tensor components(R_ab, g_ab, and T_ab). Is it possible to derive a specific solution or set of solutions for the differential geometry given this information? I've been playing around with the math and it seems very difficult.


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    KJW
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    Quote Originally Posted by frumpydolphin View Post
    So suppose you have a geodesic for a small massed particle(say a proton), and are given the initial conditions of the stress energy of the system(at t=0 for simplicity). We do not, however, know the differential form of the Tensor components(R_ab, g_ab, and T_ab). Is it possible to derive a specific solution or set of solutions for the differential geometry given this information? I've been playing around with the math and it seems very difficult.
    It is very difficult. The best I can offer is the ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) which is a Hamiltonian formulation of general relativity.

    It is important to note that any solution will involve the freedom to choose the coordinate system.


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    Quote Originally Posted by KJW View Post
    Quote Originally Posted by frumpydolphin View Post
    So suppose you have a geodesic for a small massed particle(say a proton), and are given the initial conditions of the stress energy of the system(at t=0 for simplicity). We do not, however, know the differential form of the Tensor components(R_ab, g_ab, and T_ab). Is it possible to derive a specific solution or set of solutions for the differential geometry given this information? I've been playing around with the math and it seems very difficult.
    It is very difficult. The best I can offer is the ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) which is a Hamiltonian formulation of general relativity.

    It is important to note that any solution will involve the freedom to choose the coordinate system.
    Yes choosing the coordinate system is the most daunting task because we all know solving the EFE's with "ugly" coordinates(going from metric tensor backwards) can be quite the pain in the ass. Thankyou though!
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    KJW
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    Quote Originally Posted by frumpydolphin View Post
    Quote Originally Posted by KJW View Post
    It is important to note that any solution will involve the freedom to choose the coordinate system.
    Yes choosing the coordinate system is the most daunting task because we all know solving the EFE's with "ugly" coordinates(going from metric tensor backwards) can be quite the pain in the ass. Thankyou though!
    Actually, the point I was making was that because the Einstein Field Equations are tensor equations, their solution cannot specify a unique metric or energy-momentum tensor at future times given only the geometry at some initial time. Specifying a unique metric or energy-momentum tensor at future times require that the coordinate system at those future times be fixed, and this cannot be done by the Einstein Field Equations and initial geometry alone.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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