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How Einstein field equation becomes the Poisson equation?

  1. Apr 9, 2012 #1
    I want to show that ∇2ϕ=ρ/2, which governs gravity in Newtonian physics?
    I found solution of this question in [General Relativity for Mathematicians, R.K.Sachs and H.Wu, 1997, page 112&271].
    Solution refer to optional exercise as follows:

    Let R^ be the (0, 4) –tensor field physically equivalent to the curvature tensor of (ℝ4,g) .
    (a) Show, that R^ corresponds to - (∂ϕ/∂xμ∂xν)eμ ⊗ eν. The tidal force tensor of Newtonian physics.
    (b) Show Ric(∂4, ∂4)= ∇2ϕ.
    (c) Show that , in the sense of the convections ,the Einstein tensor G becomes 2∇2ϕ .

    Please help me to understand how Einstein field equation becomes the Poisson equation?
     
  2. jcsd
  3. Apr 9, 2012 #2
    what exactly is [itex]\phi[/itex] and how is it related to g.
     
  4. Apr 9, 2012 #3
    in that book ,the author use a native spacetime (ℝ4,g) with g,

    g=-(1+2 ϕ)du4⊗ du4+(1-2 ϕ)[itex]\sum[/itex]duα⊗ duα ,where ϕ:ℝ3→ℝ will be interpreted as a time –independent Newtonian gravitational potential .
    (sigma from 1 to 3).
     
  5. Apr 9, 2012 #4
    Have you tried calculating the christoffel symbols and then just writing out local expressions for the curvature and Ricci curvature?
     
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