Speed of light and headlight

Question

What will happen if you drive a car at the speed of light and keep headlights on?

Will it be behind the car or just work like normal?

Answer 1

This was a famous thought experiment Einstein himself claimed to have considered when he was a child that helped him work out special relativity later in life. The question is somewhat paradoxical in that you are using the notion of Galilean relativity to intuit that the light should never be able to stay in front of the car if the car is moving at the speed of light.

The solution to this paradox is that nothing with mass can attain the speed of light in vaccum, and in the reference frame of the car, light is still moving at $c$. However, in a material with some index of refraction, objects that emit light can move faster than light in the medium.

When this happens, a sort of shockwave forms and the phenomena of Cherenkov radiation occurs. This effect is used by neutrino detectors such as IceCube and Super-K to observe 'decaying' neutrinos.

Answer 2

Your question does not obey with the Einstein's special theory of relativity. An object cannot exceed the speed of light as its mass becomes infinite when it gets the speed of light.

Answer 3

Hope you have heard of Doppler's effect and know how to add velocities in Special relativity.

The answer to your question is, here is a beautiful answer which says that :

Within your car you observe no unusual effects. You can look at yourself in your mirror which is moving with the car and you will look the same as usual. Looking out of the window is a different matter. The light from your headlights will always go at the speed of light in your reference frame. It will strike any object in its path and be reflected back. Everything else will be coming towards you at nearly the speed of light, so the light reflected from it will be Doppler shifted to very high frequencies—towards the ultraviolet or beyond. If you have a suitable camera you could take a snapshot. The objects passing are contracted in length but because of the different times of passage for the light and effects of aberration, the snapshot will show the objects you pass as rotated.

So basically, c+c doesn't make 2c(the speed of headlight will still be c with respect to you), as the maximum possible speed in nature is c (the speed of light).