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Wave equation given a cosmological inflationary metric

  1. Mar 10, 2015 #1
    Hi everybody!
    Can you explain me how I can obtain wave equation given a metric? For example, if I have this metric $$g_{μν}=diag(−e^{2a(t)},e^{2b(t)},e^{2b(t)},e^{2b(t)})$$, how can derive the relation $$\frac{1}{\sqrt{g}}\partial _t(g^{00}\sqrt{g}\partial _t \phi)+\frac{1}{\sqrt{g}}g^{ii}\partial ^2 \phi$$ where ##\phi=\phi (t)## is a scalar field? Moreover, from this equation the professor has derived a Bessel's equation in the form u¨+ταu=0. I don't understand... Thanks
  2. jcsd
  3. Mar 10, 2015 #2


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    You'd need to use Einstein's equations with the stress-energy tensor of the scalar field on the right hand side.
  4. Mar 11, 2015 #3
    And then, to obtain that relation?
  5. Mar 11, 2015 #4


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    That will give you the relation you've written down in your post. To get the Bessel equation, simply substitute in the metric you've been provided.
  6. Mar 11, 2015 #5
    OK, thanks :)
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