special relativity - Is there any hope of inventing a mirror that can slow photons down?

Always interested in science, I’ve been fascinated by the consequences of the speed of light, and how the passage of time is relative to the speed of the observer. Imagine the effects if we could invent a mirror that slowed down the speed of light reflecting in it!

In other words, could we invent a mirror that slowed down light so much that you saw an image of you when you were years younger in the mirror?

Answer

We have the following equations from Einstein which hold good in all frames of reference: $$E^2 = (m_oc^2)^2 + (pc)^2 \space \space - (1)$$

$$p = m_ov\gamma = m_ov\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - (2)$$

where $E$ is the energy associated with the particle, $m_o$ is the rest mass of the particle, $c$ is the speed of light, $p$ is the momentum of the particle and $v$ is the velocity of the particle.

I'll make a claim that all massless particles must travel at the speed of light. If they don't, they aren't of any interest to us.

We all agree that photons have a zero rest mass. So what will happen if a photon was slowed down? Let's find out.

The equation $(1)$ reduces to the following as the rest mass is zero: $$E^2 = (pc)^2 - (3)$$

The interesting bit happens in the second equation. Is the momentum of the photon zero if its rest mass is zero? Not necessarily, the $\gamma$ can go to infinity if $v$ is equal to $c$. This would allow $p$ to take any value or in other words, you cannot obtain $p$ from equation $(2)$. But we have assumed that the photon was slowed down, so it is traveling at a speed lesser than $c$. This would mean that $\gamma$ is finite which in turn implies that the momentum of the photon is zero as there is no infinite which could fight the zero.

$$p = m_ov\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

As $m_o$ is zero, $v$ is finite, $\gamma$ is finite, $p$ turns out to be zero.

If $p$ is zero, then from equation $(3)$, we get $E$ to be zero.

If a particle has zero energy and zero momentum, how are we going to detect it? It will be a ghost particle.

Therefore, a device which slows photons down will never exist. If it manages to slow the photons down, we'll never see those photons ever again. The energy and momentum which the photons had will disappear.

The only way out is to hope that physics is wrong so that we might be able to slow down photons in the future.