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GR metric gauge transformation, deduce 'generating' vector

  1. Apr 5, 2017 #1
    1. Problem

    ##g_{uv}'=g_{uv}+\nabla_v C_u+\nabla_u C_v##

    If ##g_{uv}' ## is given by ##ds^2=dx^2+2\epsilon f'(y) dx dy + dy^2##
    And ##g_{uv}## is given by ##ds^2=dx^2+dy^2##, Show that ## C_u=2\epsilon(f(y),0)##?

    2. Relevant equations

    Since we are in flatspace we have ##g_{uv}'=g_{uv}+\partial_v C_u+\partial_u C_v##

    3. The attempt at a solution

    Since ##g_{xx}=g'_{xx}## and ##g_{yy}=g'_{yy}##

    I get ##\partial_x C_x=\partial_y C_y=0##

    ##\implies C_x=c(y)##, ##c(y)## the constant of ##x## from integration, some function of ##y##
    ##C_y=k(x)## ##k(x)## some function of ##x##

    From the cross term ##g_{xy}'=g_{yx}'=\epsilon f'(y)## I get:

    ## \epsilon f'(y)=\partial_x C_y + \partial_y C_x ##

    So ## \epsilon f'(y)= k'(x) + c'(y) ##

    I have no idea what to do now..

    Many thanks in advance
     
  2. jcsd
  3. Apr 8, 2017 #2
    bump please. thank you very much.
     
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