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The KG equation in curved geometries has the following form: $$\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}~g^{\mu \nu}\partial_\nu\phi) + m^2\phi = 0,$$

where $g$ is the determinant of the metric tensor $g^{\mu \nu}$ and $m$ the mass of the field. Is it possible to reexpress the last equation as the usual KG equation in flat spacetimes defining a covariant derivative $\nabla_\mu$ in such a way that the equation reads:

$$(\square + m^2)\phi(x^\mu) = 0?$$

I don't arrive at the desired form, any help or hint?

Qmechanic
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T. ssP
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1 Answers1

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A covariant derivative acting upon a scalar will reduce to a partial derivative. So you will have $$\nabla_{\mu} \phi = \partial_{\mu} \phi$$ The second covariant derivative will now act on a covector $\partial_{\mu}\phi$ so you will have $$\Box \phi = \nabla^{\mu}\nabla_{\mu}\phi = \partial^{\mu}\partial_{\mu}\phi - g^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}\partial_{\alpha}\phi.$$

Noone
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