What would the observer see if he could travel at the speed of light and shot a photon beam at the same time?
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Short answer: Nothing
Thoughts experiments like this are always problematic because you violate physics but still want a physical explanation. So we have to make the following assumptions: Somehow, the observer and the light emitting device are all massless.
From special relativity follows, that everything that moves slower than the speed of light will always be slower than the speed of light and everything that moves at the speed of light will always move at the speed of light (unless it ceases to exist).
Imagine that you and your friend are somewhere in empty space and the distance between you is closing. From your perspective, it might very well look like you are standing still and your friend is floating towards you. However, from his perspective it can look the same, i.e. he is standing still, and you are the one floating. There is no way to tell the difference, as long as you both are moving at a constant speed. The perspective where you are standing still is called your rest frame. By means of Lorentz transformations, a slow object can always be transformed into its rest frame. Lorentz transformations do transform not only the spatial coordinates but also time. This leads to an effect called time dilation. The faster an object is going, the slower it experiences time (you might have heard of the https://en.wikipedia.org/wiki/Twin_paradox). One way to understand this, that you always move at the speed of light, but it is divided into time and spatial dimension. The following might be a bit crude, but I hope it helps in illustrating. "If you are standing still in space, all your velocity is used to move you through time. If you now accelerate, you take part of your velocity and move it to the spatial dimension. So you now move less in time and therefore age slower." (You will not notice the difference in your frame though).
That being said: You can not find a rest frame for a light-like particle, such as a photon. All its speed is located in the spatial dimensions. Thus it does not "move through time"), so light like particles do not age.
Thus, the observer would be frozen in time and not able to observe anything.