Is there a way of powering an incandescent light bulb without AC mains?
Should I look into amplifiers?

Nope, amplifiers are for amplifying electrical signals. What you want is electrical power.

The solid state relay is useful. Control it from the Arduino. If you don't know how, search Google and this site for some examples.

Let the solid state relay switch on/off a low voltage (12 V perhaps) halogen bulb or a car (headlight) bulb (these are mostly halogen as well). Power it from a 12 V adapter or transformer with sufficient power (at least as much as the lightbulb needs).

If you cannot use a mains adapter use a 12 V (car) battery.