I know that the LIGO test masses are considered to be in free fall. However, they are supported by pendula. I thought that free fall means the only acting force is gravity - doesn't the tension in the pendulum wire make the test masses not in free fall? Where is the flaw in my logic?
They are free to move (in “free fall”) only in the plane of the interferometer. That means they are not in in free fall with respect to earths gravity, but they are in free fall with respect to the gravity of gravitational waves travelling parallel to the plane of the interferometer.
Imagine a pendulum that had been inclined to 90 degrees. The mass would (immediately after release) be in "free fall" vertically, but not horizontally. A pendulum at 0 degree inclination would be in "free fall" horizontally, but not vertically.