Wave equation given a cosmological inflationary metric

[Nick2014]

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

 

[Chalnoth]

You'd need to use Einstein's equations with the stress-energy tensor of the scalar field on the right hand side.

 

[Nick2014]

You'd need to use Einstein's equations with the stress-energy tensor of the scalar field on the right hand side.

And then, to obtain that relation?

 

[bapowell]

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.

 

[Nick2014]

OK, thanks :)