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It is well accepted that nothing with mass can travel at the speed of light in vacuum, $c$. Now let's say that a light wave is traveling through a medium that slows the wave's speed significantly (less than $c$). Since we are able to accelerate objects to over 99% the speed of light (via LHC), could we potentially create a situation in which we could view a stationary light wave as Einstein first wondered during his childhood?

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Yes, it is possible.

Cherenkov radiation, also known as Vavilov–Cherenkov radiation, is electromagnetic radiation emitted when a charged particle (such as an electron) passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. The characteristic blue glow of an underwater nuclear reactor is due to Cherenkov radiation.

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Source: Cherenkov radiation - Wikipedia

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