Wave equation for Schwarzschild metric

[Arman777]

Homework Statement

Wave equation for Schwarzschild metric

Relevant Equations

GR equations

I am trying to find the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for $$ds^2 = (1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$$

I have did some calculations by using

$$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$

and I have found

$$\nabla_{\mu}\nabla^{\mu} \Phi = [g^{tt}\partial^2_t + 2(\frac{1}{r} - \frac{M}{r^2})\partial_r + g^{rr}\partial^2_r + \frac{cot(\theta)}{r^2}\partial_{\theta} + g^{\theta \theta}\partial^2_{\theta} + g^{\phi \phi}\partial^2_{\phi}]\Phi$$

but I am not sure that is this true or it can be further simplified ? Any ideas

 

[Haorong Wu]

Hello, @Arman777 . I am recently studying this problem. I would suggest Spacetime and geometry by Sean M. Carroll, especialy in pages 395 to 400.

Will I violate any rules in this forum if I just suggest a book?

 

[vanhees71]

Of course not. Of course you violate rules, if you link to some illegal download-link to a copyrighted book.

 

[Haorong Wu]

Of course not. Of course you violate rules, if you link to some illegal download-link to a copyrighted book.

Thanks! Sometimes I hesitate to answer in homework forum, because not only there are too many professors, but I am afraid I would give the answer directly which would violate the rules.

 

[vanhees71]

Don't worry about the "professors". They are also just human beings. Not giving directly the answer is also my problem with the homework forums, but it's of course much better to give only hints first and let the student find the solution him or herself.