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The 4-D Laplace equation and wave equation

  1. Feb 10, 2010 #1
    In relativity, the scalar wave equation in the coordinate system (x,y,z,ict)
    is
    [tex]\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}+\frac{\partial^2\phi}{\partial (ict)^2}=0[/tex]

    In 3D classical mechanics, the Laplace equation is:{when the coordinate system is (x,y,z)+t}
    [tex]\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}=0[/tex]

    And the ordinary wave equation is
    [tex]\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}=\frac{1}{v^2}\frac{\partial^2\phi}{\partial t^2}[/tex]

    the first equation is similar to the third,and the second equation has the same format of dimensions as the first.

    So, does this mean that a wave equation in 4 dimensions is a 4-D laplace equation in relativity,or something else,Thx very much.
     
  2. jcsd
  3. Feb 10, 2010 #2
    Yes! In four dimensional spacetime of Minkowski with signature (+,-,-,-), the wave equation is

    [tex]\square \phi=0[/tex]

    with [tex]\square = {\partial}^2_t-{\nabla}^2[/tex] being the famous D'Alembert operator, which is of course the field equation of Nordstrom's first theory of gravitation!

    AB
     
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