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Wave equation in curved spacetime 
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#1
Apr708, 01:00 PM

P: 197

Does anyone know how to derive the wave equation in curved spacetime?
[tex](g)^{1\over 2}\partial_\mu((g)^{1\over 2}g^{\mu \nu}\partial_\nu \phi) = 0 [/tex] A reference, or an outline of the derivation would be very helpful. Thanks. 


#2
Apr708, 02:59 PM

P: 197

It seems that just writing the d'Alembertian in covariant form
[tex]\Delta \phi = g^{\mu \nu}\phi_{;\mu \nu}=0[/tex] does the trick. This form is giving me the results I want, but I still don't know how to put it in the form written in my original post. 


#3
Apr708, 06:13 PM

P: 448

Given a vector [itex]V^\mu[/itex], can you think of any cute expressions for [itex]\nabla_\mu V^\mu[/itex]? (Hint: you can write the covariant fourdivergence of a vector in terms of the coordinate fourdivergence much like the expression in the first post.)



#4
Apr708, 07:37 PM

P: 303

Wave equation in curved spacetime
[addendeum: or is it [itex]\nabla_\mu V^\mu=\frac{\delta V^\mu}{\delta t}[/itex]?] Regards, Bill 


#5
Apr708, 11:01 PM

P: 448

[tex]\nabla_\mu V^\mu = \frac{1}{\sqrt{\textrm{det}g_{\rho\sigma}}}\partial_\mu(\sqrt{\textrm{d et}g_{\rho\sigma}}V^\mu)[/tex] 


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