What would it look like to be in a spherical mirror? If you were in a spherical room with smooth mirror walls, what would it look like?
Of course it would be dark without a light source. So if I was to turn the light on, would the room get brighter and brighter OR would it remain at the same brightness.
How would you look like when you look into the mirror? Would you keep expanding due to the curved mirror and it reflecting back and forth? 
What are the physics principles that would govern what it would look like inside a spherical room with mirror walls?
 A: Your question can be basically divided into two parts. One, what will happen to the light, as would the room get brighter and brighter, or at the same brightness. The second is, how would you see your image in the mirror.
Since you simply said a light source it may be a continuous light source or a light source just for an instant. For simplicity, lets consider the instant light source first, which flashed just once and ended there. Then, that light beam should keep on reflecting right? See, whenever a reflection take place on the mirror, a portion of it is absorbed and during each reflection, the reflected ray is less intense than the incident ray. Even if the absorption by the mirror is so small, the number of reflections that take place in just one second is so so so high that the light would be completely absorbed just within a fraction of second. What now, in the continuous light? Each ray of light gets ended as in normal circumstances and DOESN'T keep bouncing back and forth. The sphere will just be lighted as usual. More intense the light source is, more intense would be the light inside it. It won't get more brightened as time passes. It is as simple as that.
Now, how would your image be, inside the mirror. Remember, a light ray won't get reflected back and forth for ever. Let it be the pure, white, source light ray, or the reflected, colorful image's light ray. Your reflection would just as you see yourself in a concave mirror. I don't think I should explain the image from concave mirror here. At the sides of your image, you would see distortions, which would be so complex to explain, because the reflections which caused them will be huge in number. When you look to the side, you would be able to receive rays coming towards you from the side correctly and hence, would see your image as in a concave mirror again. Then, the rest of the portion would seem distorted. 
That's all. Thank you for asking.
