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The wave equation in Minkowski space can be given as $-\frac{\partial^2\phi}{\partial t^2}+\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}= 0$.

In a curved spacetime this can be re-written in the form $g^{\mu \nu}\nabla_{\mu}\nabla_{\nu}\phi = 0$, where $\nabla_{\mu}$ and $\nabla_{\nu}$ are covariant derivatives and $g^{\mu\nu}$ is a metric in $\it{M}^4$.

How can I determine the form of this equation in the spacetime with line element $ds^2 = -dt^2 + a(t)(dx^2 + dy^2 + dz^2)$?

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    $\begingroup$ MathOverflow is a platform for professional mathematicians about research-level mathematics. I think your question is more suitable for Physics.SE . In this case you were lucky, but in the future I would recommend asking questions like this there instead. $\endgroup$ Nov 25, 2021 at 16:30

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In view of the given line element, the metric can be represented as $g_{\mu \nu } =\mbox{diag} (-1,a(t),a(t),a(t))$, and therefore the wave equation is $\Delta \phi =0$ with the Laplacian $$ \Delta = \frac{1}{\sqrt{|g|} } \partial_{\mu } \sqrt{|g|} g^{\mu \nu } \partial_{\nu } = -\partial_{t}^{2} - \frac{3(\partial_{t} a(t))}{2a(t)} \partial_{t} + \frac{1}{a(t)} (\partial_{x}^{2} + \partial_{y}^{2} + \partial_{z}^{2} ) $$

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