# Dervation of the first-order Klein-Gordon equation

How to derive the first-order perturbed Klein-Gordon equation:

$$\square \phi=\left[\frac{1}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g}g^{\mu\nu} \partial_{\nu} \right) \right]\phi=0$$

For a first-order perturbed metric:

$$g_{00} = -a^2 ( 1+ 2 \Phi), ~~~ g_{0i} = 0 , ~~~~ g_{ij} = a^2 (1-2 \Psi).$$

Here is my trial:

First : $$\sqrt{-g}= \sqrt{a^2 (1 +2 \Phi) (1 -2 \Psi)^3 }$$

Also the scalar field: $$\phi$$ is perturbed : $$\phi(x,t) \to \phi_0(t) + \phi_1(x,t)$$. And both $$\Phi$$ and $$\Psi$$ are functions of (x,t).

The KG equation becomes:

$$\partial_0 \sqrt{-g} g^{00} \partial_0 \phi + \partial_i \sqrt{-g} g^{ij} \partial_j \phi=0$$

Perturbe $$g^{\mu\nu}$$ and $$\phi$$, the equation becomes:

$$\partial_0 \sqrt{-g} (g^{00}+ \delta^{(1)} g^{00} ) \partial_0 (\phi_0 + \phi_1 ) + \partial_i \sqrt{-g} (g^{ij}+ \delta^{(1)} g^{ij} ) \partial_j (\phi_0 + \phi_1 ) =0$$

Yields:

$$\partial_0 \sqrt{-g} (-a^{-2} + -a^{-2}/2 \Phi^{-1} ) \partial_0 (\phi_0 + \phi_1 ) + \partial_i \sqrt{-g} (a^{-2} - a^{-2}/2 \Psi^{-1}) \partial_j (\phi_0 + \phi_1 ) =0$$

Now I’m confuced how to substite by $$\sqrt{-g}$$ and get only the first order terms .

So any help to deal with the terms under the square root and complete the derivation.

I should get a result similar to equation (3.4) paper.

Any help is appreciated!

In your case, one has $$\sqrt{-g}=\sqrt{a^8(1-2\Phi)(1-2\Psi)^3}\approx a^4\left(1-\Phi-3\Psi\right).$$