Dynamics of Cats

That is a 1.96 kg cube of 3/4 gold, 1/4 platinum.

The actual LISA spacecraft, in a very real sense, is three of those, separated by 5 million kms.

At about $600 per ounce, that is a little over $100,000 of precious metal. Each of these is surrounded by about a 2 m "saucer", which is the auxilliary spacecraft.

Somewhat amusingly, depending on how you do your accounting, the ~ 300kg of epoxy, metal, optics and electronics wrapped around the gold/platinum cube is the expensive part. Per unit mass the chunk of gold is the cheapest part of the spacecraft!

The cube material was basically chosen to be dense, non-corrosive and to have very low magnetic susceptibility.
The cubes are in free falling orbits, and the surrounding spacecraft maneuvers continuously to center on the cubes. The cubes are actually in tight vacuum enclosures (the vacuum of space is not quite good enough, mainly because of random volatile outgassing by the spacecraft); optical sensors and capacitors track the position of the cube and low powered FEEP thrusters maneuver the disk module to center on the cube.
For the purpose of tracking wiggles in spacetime, a 2m saucer in solar orbit is not in good enough a free fall orbit, radiation pressure and solar wind drag is a measurable gross perturbation on the orbit.

Small stable lasers are emitted by each module to the other two, 5 million km away, where they are detected by 40cm telescopes and the light is allowed to interfere.
That is the measurement done by the spacecraft, the time variation in the pattern of interference fringes made by 3 lasers shone in an equilateral triangle, tracking the position of the cubes.

The interference pattern is measured with an accuracy of about one part in 10 million in phase, sampled about once per second. This gives an effective measurement of the relative change in separation in time scales from 1 second to 1 hour of a few hundred picometers. That is less than the width of an atom change, over 5 million km. (On longer time scales the separation drifts by thousands of km, but that shows up as DC or very low frequency drift, quite distinct from the milliHertz oscillations that are the primary science measurement).

That's it. Simple really. Then you send the measurements (the choice of which variables to measure and transmit is slightly non-trivial, you can form different orthogonal combinations of variables and to some extent choose what to optimise in measuring). Track the separation as a function of time for several years - nominal mission is 5 years, extended mission is 10 years, limited by FEEP thruster fuel supply.

And that is how we expect to see, among other things the final stages of merger of supermassive black holes, emitting a million times more energy that the brightest supernova in a few seconds, with almost all the energy going into rippling space time itself.

There will be several other sources, including compact binaries in the Milky Way, and "local" supermassive black holes eating low mass stellar black holes. There will also probably be surprises, serendipitous sources that no one predicted. And, a good source (we hope for S/N > 1000 sources) will allow very strong tests of strong field gravity - we will test General Relativity in the extreme.

Next: some of the atrophysics of the sources...