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Integral over metrics

  1. Jul 4, 2007 #1
    I'm going to be working in 4D, and I assume my metric is symmetric. So what I am interested in is an integral of the form [tex]\int {d^{10} \eta \delta (\eta - \bar \eta } )[/tex]. My question is what is the invariant volume element for an integral of this type? My guess is that it is [tex]\frac{{d^{10} \eta }}{{J^2 }}[/tex] where [tex]J[/tex] is the jacobian, but I don't know. This my logic:
    [tex]d\bar x^\alpha = \frac{{\partial \bar x^\alpha }}{{\partial x^\beta }}dx^\beta[/tex] and [tex]d^4 \bar x = \frac{{d^4 x}}{J}[/tex]. So since:[tex]\bar g^{\alpha \beta } = g^{\mu \tau } \frac{{\partial \bar x^\alpha }}{{\partial x^\mu }}\frac{{\partial \bar x^\beta }}{{\partial x^\tau }}[/tex] I figured [tex]d^{10} \bar g = \frac{{d^{10} g}}{{J^2 }}[/tex]. I would also like the requirements that [tex]g_{00} < 0[/tex], [tex]\det [g_{\alpha \beta } ] < 0[/tex], and [tex]\det [\gamma _{ij} ] > 0[/tex] where [tex]\det [g_{\alpha \beta } ] = g_{00} \det [\gamma _{ij} ][/tex]. Any suggestions or guesses?
     
  2. jcsd
  3. Jul 4, 2007 #2

    matt grime

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    A metric, by definition, satisfies d(x,y)=d(y,x).
     
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