Quasars are the most intrinsically luminous objects in the universe. Many of them fluctuate in brightness, often on a time scale of a day or so. How can the rapidity of these brightness changes be used to estimate an upper limit to the size of these objects? (Hint: Separated points cannot change in a coordinated way unless information is sent from one to the other.)

Astronomers call the amount of energy an object radiates each second its luminosity. If we have two objects of the same temperature but of different sizes, the larger one will emit more energy than the smaller one.
If two stars have the same temperature but one is more luminous than the other, the more-luminous star must have a larger surface area, therefore a larger radius than the dimmer star. Thus, if we know quasars temperature, we can infer its size from the amount of energy it radiates. To calculate the quasars radius, however, we need a mathematical relation between luminosity (L), temperature (T), and radius (R) - a relation known as Stefan-Boltzmann law.
L = 4πR²σT⁴
Here, σ is the Stefan’s constant.