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General Relativity-Variational approach

  1. Jun 4, 2010 #1
    1. The problem statement, all variables and given/known data

    I am given the following action S:
    S=1/2∫〖g^ab ((∇_a Ф ∇_b Ф)+(1/6 R_ab Ф^2) ) √(-g) dx^4 〗

    Where R_ab is the Ricci tensor, Ф is a scalar field and g^ab is the metric.

    I am asked to find the equation of motion for the field.

    2. Relevant equations
    I know I have to substitute my Lagrangian: L= 1/2 g^ab ((∇_a Ф ∇_b Ф)+(1/6 R_ab Ф^2) )

    into the Euler-Lagrange equations, but I don't know in which EL equation exactly!

    3. The attempt at a solution

    I mean should I use the EL equation where Ф and ∇_b Ф are my independent components? Or should I use the 2nd order EL equations where g_ab; g_ab,c & g_ab,cd are my independent components (as in the Einstein Hilbert action)?

    Thanks :)
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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