Bobu:
From what I read, a potential divider set up does exactly the same, but with 2 resistors instead of a regulator. It also doesn't need a heatsink, as it doesn't generate any heat to clip off the voltage. Is that right?
As others have noted, current flowing through any non-superconductor will generate at least a little heat due to innate resistance. In fact, an easy way to make a small electric heater is to wire-up low value resistors in series and run a few amperes of current through them.
Also there's the fact that a voltage divider has a fixed proportion the input voltage is reduced by, not a fixed value it reduces the input voltage to. So if for some reason the 9 VDC is increased significantly, e.g. to 12 VDC, the output voltage will be too high to power an Arduino. By contrast a linear regulator will give you a known output for any voltage within the range of rated input voltages.
Edit: I forgot to mention that the resistors in the heater example would be 0.5 or 1 W resistors. 