Electromagnetic action in the expanding universe

In the static Minkowskian spacetime the action for the electromagnetic potential in the radiation gauge is given by $$\mathcal{S}=\frac{1}{2}\int~dt\int~d^{3}x\left(\left(\frac{\partial\textbf{A}}{\partial{t}} \right)^{2}+\textbf{A}\cdot\nabla^{2}\textbf{A} \right).$$

Now I want to rewrite this action in an expanding universe. Ofcourse, one way of doing so is to write the electromagnetic action in the expanding FRW background in the first place. But, in another way, I want to perform comoving transformation on the above action and compare it with the FRW result.

We know that in an expanding universe there is a relationship between physical coordinates $\textbf{r}$ and comoving coordinates $\textbf{x}$: $$\textbf{r}=a(\tau)~\textbf{x},$$ where $a(\tau)$ is the time-dependent scale factor and $\tau$ is the cosmic time; thus the time and space derivatives are not independent anymore: $$\nabla_{\textbf{r}}=a^{-1}\nabla_{\textbf{x}}$$ $$\left(\frac{\partial}{\partial{\tau}} \right)_{\textbf{r}}=\left(\frac{\partial}{\partial{\tau}} \right)_{\textbf{x}}-\frac{\dot{a}}{a}~\textbf{x}\cdot\nabla_{\textbf{x}}$$ My question is: Can I substitute these relations in the above electromagnetic action and find the resultant comoving action in the expanding universe? In other words, are $t$ and cosmic time $\tau$ the same? if not, what is the relation between them?

• Yes, your approach is sound. If you use a metric that is non-flat, include $\sqrt{|g|}$ in your measure of integration. Also you could use the fact that free EM field has conformal invariance. So the use of conformal time would reduce equations to that of flat space. – A.V.S. Apr 15 '18 at 18:29
• Thank you @A.V.S. I only consider flat space, but my main question is about the difference in times. Comoving transformation deals with cosmic time, but the time in electromagnetic action is proper time. How can I convince my professor that these two times are the same and performing comoving transformation on electromagnetic action is valid? – M Hoseini Apr 16 '18 at 5:33
• Even when the space is flat, space-time is not, so add $\sqrt{|g|}$. Two times are the same since $d\tau^2= g_{00} dt^2$, and $g_{00}=1$ in standard parametrization of FRW spacetime. – A.V.S. Apr 16 '18 at 18:58
• @A.V.S. you are right about FRW spacetime. But my problem is in a static flat space-time in which $g_{ij}=\delta_{ij}$. As I mentioned it in line4 we can write the action in the FRW metric in the first place. But in another approach I wrote it in non-expanding space-time and want to transform it to an expanding frame via comoving tranformation. My professor says the time present in action is proper time and comoving transformation deals with cosmic time which is different from proper time in this case. So I am really confused about times in this problem. Would you please elaborate more on this – M Hoseini Apr 17 '18 at 17:41
• Now I get it. You professor is right. You need to transform your action according to rules of general covariance. If you replace cartesian coordinates with expanding ones your new metric now acquires nonzero $g_{0i}$ components which would give contribution to $\sqrt{-g}$. Also, for a general curvilinear coordinates, I would recommend using gauge invariant lagrangian, since gauge fixing condition now has Christoffel symbols. – A.V.S. Apr 18 '18 at 4:39