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For my physics research course I am supposed to get the equation of motion from a given action $S$:

$$ S = \frac{-1}{2} \int d^4x \sqrt{-g} (g^{\mu\nu} \frac{\partial \phi}{\partial x^\mu} \frac{\partial \phi}{\partial x^\nu} - m^2\phi^2).\tag{1} $$

I take the variations according to the following rules:

$$ \phi \rightarrow \phi + \delta \phi $$ $$ \partial_\mu \phi \rightarrow \partial_\mu \phi + \delta(\partial_\mu \phi) = \partial_\mu \phi + \partial_\mu (\delta \phi). $$

And I find that the solution is:

$$ \frac{\delta S}{\delta \phi} = -g^{\mu\nu} \partial_\mu (\frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi)}) -g^{\mu\nu} \partial_\nu (\frac{\partial\mathcal{L}}{\partial(\partial_\nu\phi)}) - 2m^2 \frac{\partial\mathcal{L}}{\partial\phi} = 0. $$

But my professor said the solution has also need to contain metric. How can I get to right solution? I'm an undergrad student trying to improve myself in theory bu this is kinda hard for me.

Qmechanic
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AK4ST
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1 Answers1

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Okay, so you have the following action:

$$ S = -\frac{1}{2} \int d^4x \sqrt{-g} (g^{\mu\nu} \frac{\partial \phi}{\partial x^\mu} \frac{\partial \phi}{\partial x^\nu} - m^2\phi^2) $$

The Equations of Motion (EoMs) when varying an action with respect to a scalar field yield $$\partial_{\mu}\Big(\frac{\delta \mathcal{L}}{\delta (\partial_{\mu}\phi)}\Big)- \frac{\delta \mathcal{L}}{\delta \phi}=0$$ where the Lagrangian density is given by $$\mathcal{L}=-\frac{1}{2}\sqrt{-g} \Big(g^{\mu\nu} \frac{\partial \phi}{\partial x^\mu} \frac{\partial \phi}{\partial x^\nu} - m^2\phi^2\Big)$$ Therefore, one needs to calculate the two functional derivatives $$\frac{\delta \mathcal{L}}{\delta (\partial_{\mu}\phi)} \hspace{3em}\text{and}\hspace{3em} \frac{\delta \mathcal{L}}{\delta \phi}$$

  1. First, the variation with respect to the derivatives of the scalar field $$\frac{\delta \mathcal{L}}{\delta (\partial_{\mu}\phi)}= -\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi$$

  2. Then, the variation with respect to the scalar field itself $$\frac{\delta \mathcal{L}}{\delta \phi}= \sqrt{-g}m^2\phi$$

Then, you substitute into the EoM and figure out the final version of the classical equation of motion for the scalar field in curved space.

schris38
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