- #1

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- 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

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