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About poisson's equation

  1. Nov 22, 2008 #1
    I have read that Einstein's formulation of gravity [tex]G_{ab}=\frac{8 \pi G}{c^4}T_{ab}[/tex] is analogous to the differential form of Newton's version [tex]\nabla ^2 \phi = 4 \pi G \rho[/tex] with the metric tensor and energy-momentum tensor in the modern form playing the same roles as gravitational potential and density in the classical one, respectively.

    My question: why did the 4 become an 8?
     
  2. jcsd
  3. Nov 22, 2008 #2

    HallsofIvy

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    In simple, though rough terms, Newton's version is in a 3 dimensional space and measures the field at the surface of a sphere and that area is given by 4\pi r^2.

    Einstein's equation is in a 4 dimensional space and measures the field at the surface of a 4-sphere and that area is given by 8\pi r^3.
     
  4. Nov 22, 2008 #3
    I may be miscalculating but I'm getting the 'surface volume' of a 3-sphere to be [tex]2 \pi ^2 R^3 [/tex] (instead of [tex]8 \pi R^3[/tex]).
     
    Last edited: Nov 22, 2008
  5. Nov 22, 2008 #4

    tiny-tim

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    Hi snoopies622! :smile:

    (have a pi: π and a rho: ρ :wink:)
    Yes, according to http://en.wikipedia.org/wiki/4-sphere#Volume_of_the_n-ball, the 4-ball has volume π2r4/2, and surface area 2π2r3

    (the surface area is always n/r times the volume of the n-ball)
    The mathematical reason:

    we require R00 = 4πGρ.

    But R00 = constant(T00 - 1/2 T g00),

    and T00 - 1/2 T g00 = ρc4 - 1/2 ρc4,

    so the constant must be 8πG :smile:

    (i got this from http://en.wikipedia.org/wiki/Einstein_Field_Equations_(EFE)#The_correspondence_principle :redface:)

    … but I'd still like someone to explain the physical significance of this! :rolleyes:
     
  6. Nov 22, 2008 #5
    Thanks, tiny-tim. I'll spend some time looking through that Wikipedia derivation.
     
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