# Question regarding the backreaction of a scalar field in curved spacetime

$${\mathcal L}_1 = \frac{1}{2} \sqrt{-g} \left( \partial_\mu \phi \partial^\mu \phi - m^2 \phi \right)$$

where $$g_{\mu \nu}$$ is a FRW metric and $$\phi$$ a scalar field.

In addition assume that for 'whatever' reason there is a deviation of the above Lagrangian such that an arbitrary potential $$V(\phi)$$ is introduced so that $${\mathcal L}_2 = \frac{1}{2} \sqrt{-g} \left( \partial_\mu \phi \partial^\mu \phi - m^2 \phi + V(\phi)\right).$$

Surely, having already a matter field interact with gravity should produce a backreaction so that the initial metric $$g_{\mu\nu} \rightarrow \tilde g_{\mu \nu}$$ while the introduction of the interacting terms will further aid this process.

My question is how can I calculate the above backreaction. In particular, I know that I need to compute the eom's of the field and the Einstein equations and solve the system of differential equations, but I can't figure out whether I should calculate the eom's while taking into account the initial metric or not.

So for example, the eom's would be equal to: $$\frac{\delta {\mathcal L}_2}{\delta \phi} \Bigg|_{g_{\mu \nu}} = 0 \quad or \quad \frac{\delta {\mathcal L}_2}{\delta \phi} \Bigg|_{\tilde g_{\mu \nu}} = 0 ???$$

If the latter is true, than I should first compute the new metric $$g'_{\mu \nu}$$ via the Einstein equation $$\delta G_{\mu \nu} \propto \delta T_{\mu \nu}$$ and then plug it in above?

However, the Einstein equation contain $$\phi$$ as well making the problem a cyclic process. In my mind it is sort of a chicken vs egg kind of problem where one should assume that the introduction of matter at the initial geometry forces the latter to change. With that in mind I could argue that the eom with $$g$$ should have the correct physical interpretation. What are your thoughts? What am I missing?

The general coupled system of gravity (metric) and the scalar field can be described by the Lagrangian (expressed in $$\hbar=c=1$$ natural units often used in cosmology) $$\mathcal{L} = \sqrt{-g}\left[\frac{1}{8 \pi G} R + \frac{1}{2} (g^{\mu\nu}\partial_\mu\phi \partial_\nu \phi + m^2 \phi^2 + V(\phi)) \right]$$ where $$R$$ is the Ricci scalar of the metric and $$G$$ Newton's constant. By varying with respect to $$g_{\mu\nu}$$ one obtains the Einstein equations $$R^{\mu\nu} - \frac{1}{2} R g^{\mu\nu} = 8 \pi G T^{\mu\nu}$$ where $$T^{\mu\nu}$$ is obtained by the variation of the scalar part of the Lagrangian density $$\mathcal{L}_{\phi}$$ with respect to the metric
$$T^{\mu\nu} = \frac{-2}{\sqrt{-g}} \frac{\delta \mathcal{L}_\phi}{\delta g_{\mu\nu}} = \partial^\mu \phi \partial^\nu \phi - \frac{1}{2} g^{\mu\nu}\left[g^{\alpha\beta}\partial_\alpha\phi \partial_\beta \phi + m^2 \phi^2 + V(\phi)\right]$$ The equations of motion for $$\phi$$ are obtained by varying with respect to $$\phi$$ and yield $$\frac{1}{\sqrt{-g}}\partial_\nu \left( g^{\mu\nu} \partial_\mu \phi \right) + m^2\phi + V'(\phi) = 0$$

The two sets of equations above apply to the self-consistent dynamics of any metric and scalar-field configurations. Generally, the Einstein equations and the equations of motion for $$\phi$$ are a coupled set of equations that are solved simultaneously. In general, the equations simply have to be solved all at the same time with the same $$\phi$$ and $$g_{\mu\nu}$$ appearing in both.

For weak couplings, one can sometimes solve iteratively, such as e.g. using a zeroth-order metric field $$g_{\mu\nu (0)}$$, solving for the dynamics of $$\phi$$ in this metric, and feeding this into the Einstein equations to find a corrected metric $$g_{\mu\nu(0)} + g_{\mu\nu(1)}$$. Then you can plug this metric into the evolution equations for $$\phi$$ and so on. However, the quality of the approximation you get by a couple of iterations will depend a lot on the setup.

For example, when you assume that the configuration of $$\phi$$ is isotropic and homogeneous in the sense of FLRW cosmology and that the metric is a FLRW metric with an undetermined scale factor $$a(t)$$, the Einstein equations boil down to a set of Friedmann equations for the evolution of $$a$$ with a contribution of the scalar $$T_{00}$$ to energy density $$\rho$$ and $$T_{ii}$$ to pressure $$p$$. Additionally, the equation of motion for the scalar reduces to $$\ddot{\phi} + 3 \frac{\dot{a}}{a} \phi + m^2 \phi + V'(\phi) = 0$$ Again, you see that the evolution of $$\phi$$ feeds into the metric function $$a$$, and the evolution of $$a$$ feeds into the evolution of $$\phi$$. The equations cannot be generally considered as separate.

If, for instance, you assume that $$\phi$$ represents something like the dark energy field (quintessence) or an inflaton during an era when these fields dominate the dynamics of the universe, an iterative procedure is not a good idea, you should solve the full coupled dynamics. On the other hand, if you want to use it during an era where the scalar field has only a weak impact on the dynamics (such as the radiation- or matter-dominated era), using an iterative approach should work reasonably well.

• The Wiki article that you have linked for seeing the variation wrt to $g_{\mu\nu}$, says under eq. (1) "The stationary-action principle then tells us that to recover a physical law, we must demand that the variation of this action with respect to the inverse metric be zero". Not sure why it says that the variation should be specifically wrt the inverse metric. I thought varying wrt to the covariant metric tensor would be fine too, effectively giving the same result. Commented Apr 22 at 18:21
• You can choose either the metric, or the inverse metric for the variation. They are fully dependent field variables, at least for non-degenerate metrics, so requiring the stationarity of the action with respect to one will imply the vanishing of the variation with respect to the other. (In essence, you will get the Einstein equations with indices either raised or lowered.)
– Void
Commented Apr 23 at 10:18