# Energy of photons in perfect mirror box with FRW metric

In FRW metric, distance is described by \begin{align} ds^2 = dt^2 - a(t)^2[d\chi^2 + S_k(\chi)^2 d\Omega^2]\\ =dt^2 - a(t)^2\gamma_{ij}dx^i dx^j \end{align} where $$a$$ is the scale factor. Now by using homogeneity, \begin{align} \partial_i P^\mu=0 \end{align} and Christoffel symbols, \begin{align} \Gamma^i_{jk}, \Gamma^0_{ij}=a \dot{a} \gamma_{ij}, \Gamma^i_{0j}=\frac{\dot{a}}{a}\delta^i_j \end{align} the equation \begin{align} E\frac{dP^\mu}{dt}+\Gamma^\mu_{ij}P^iP^j+2E\,\Gamma^\mu_{0j}P^j=0 \end{align} \begin{align} for\; \mu=0,\,E\frac{dE}{dt}+a\dot{a}\gamma_{ij}P^iP^j=E\frac{dE}{dt}+\frac{\dot{a}}{a}P_{phy}^2=0,\,thus\,P_{phy}\propto \frac{1}{a} \end{align} is obtained, where the $$P_{phy}$$ is the physical momentum.
Since photons has no rest mass, it can be said that energy is also inversely proportional to the scale factor $$a$$.

Now consider the box that is constructed with perfect mirror. Initially photons of total energy amount of 100J are placed inside the box. After the time passes, the scale factor has been doubled. If you open the box and measure the total amount of energy, will the value be halved, 50J?

Additionally, if a particle with no physical momentum is placed, will its mass will be conserved?

If the box has a fixed size, then the energy of the photons will be conserved. If the box is expanding or contracting, then the photons will lose or gain energy, respectively. Thermodynamically, the process can be framed as adiabatic expansion/compression of the photon gas, which has adiabatic index $$\gamma=4/3$$. As the box volume $$V$$ changes, the temperature of the gas scales as $$T\propto V^{1-\gamma}$$. You can also calculate the evolution of each photon within the framework of classical mechanics, as I explained here, with the result being that the photon momentum scales as $$p\propto r^{-1}$$, where $$r$$ is the length scale of the box. You can directly apply these relationships to massive particles as well, if you simply set the adiabatic index ($$\gamma=5/3$$ for a nonrelativistic monatomic gas) or the momentum-energy relationship appropriately.

I've avoided talking about the FLRW metric so far because the expanding comoving coordinates are just a distraction when talking about systems that do not comove with them. (Note that "expansion of space" is not physical; it's just coordinates.) However, you could explicitly consider a box of fixed (proper) radius $$r\ll c/H$$ (where $$H=\dot a/a$$) within the context of the comoving coordinates. The photon crosses the box in time $$\Delta t=2r/c$$, so that within the frame of the comoving observers, its momentum drops by $$\Delta p_\mathrm{crossing}=-Hp\Delta t=-2Hp r/c$$ due to the cosmological redshift. (This is just the $$\mathrm{d}p/\mathrm{d}t=-Hp$$ that gives rise to $$p\propto 1/a$$, but I took advantage of the $$r\ll c/H$$ assumption to avoid integrating it.) However, comoving observers at distance $$r$$ recede at speed $$Hr$$, so with respect to them, the walls of the box are moving inward at that speed. Reflection off the wall blueshifts the photon by momentum $$\Delta p_\mathrm{reflection}=2Hp r/c$$, resulting in a net zero change in momentum over each half-cycle.

That argument is valid at subhorizon scales, $$r\ll c/H$$. If you try to extend it to boxes of size $$r \sim c/H$$ using the FLRW metric, you might trick yourself into thinking that a redshift due to cosmic expansion emerges at those scales. The difficulty here is that at horizon scales, the notion of "a box of fixed size" is not straightforward and generally depends on which spatial surfaces you choose for your coordinate system. For example, the size of a Born rigid box would not be constant when measured along the conventional FLRW spatial surfaces.

• Thanks for your comment. However this leads me to new question. In subhorizon scales, energy of photons do decrease due to the cosmological redshift, while blueshift by reflection exactly cancels out. Then what if our universe expands intermittently?(I know this sounds ridiculous, but let just say "if".) In other words, the Hubble parameter H=0 in some time intervals. Wouldn't that prevents blueshift from reflection, and make total energy decrease in result? Commented Jul 21 at 21:33
• @littlegiant I think you are right that if the universe suddenly "stopped" right before the reflection and "restarted" right after, there would be a net redshift. In comoving coordinates this is clear, and in physical (non-expanding) coordinates, the interpretation is that before reflection, the light is redshifted by the gravitational impulse that stops the expansion, and after reflection, it is redshifted again by the gravitational impulse that restarts the expansion. (Realistically, the time scale for $H$ to change is of order $1/H$.)
– Sten
Commented Jul 21 at 22:03
• Thank you so much! It really helped, and I appreciate that. Commented Jul 21 at 22:19
• Doesn't technically the "expansion" of the universe inside such a box cause Unruh effect? If it does, then the universe should actually deposit (a very small amount of) additional energy in the box. Commented Jul 22 at 0:03
• Yeah, e.g. the fact that the walls of the box need to accelerate should lead to Unruh radiation in the frame of the box. That's negligible in the subhorizon limit but might become important close to horizon scales. Another consideration for a horizon-scale box is that the time scale for the decrease in mass density inside the box becomes comparable to the cycle period of each photon. The non-adiabatic time variation of the "gravitational potential" would lead to net blueshifts or redshifts.
– Sten
Commented Jul 22 at 2:02

Great question. Thing to note is that expansion of the universe is inertial, scale factor is not constant. Scale factor inside your box is 0. So no, your photons don't lose energy and all mass is conserved as interactions with Higgs field and or strong potential energy is constant. Edit: if the box itself is expanding or expandable than your photons would lose energy, but only when they reflect not in between the mirrors.