### 5.1 The unequal-arm Michelson

The unequal-arm Michelson combination relies on the four measurements , , , and .
Note that the two combinations , represent the two synthesized two-way data
measured onboard spacecraft 1, and can be written in the following form,
where is the identity operator. Since in the stationary case any pairs of these operators commute,
i.e. , from Equations (50, 51) it is easy to derive the following expression for the
unequal-arm interferometric combination which eliminates :
If, on the other hand, the time-delays depend on time, the expression of the unequal-arm Michelson
combination above no longer cancels . In order to derive the new expression for the unequal-arm
interferometer that accounts for “flexing”, let us first consider the following two combinations of the
one-way measurements entering into the observable given in Equation (52):
Using Equations (53, 54) we can use the delay technique again to finally derive the following
expression for the new unequal-arm Michelson combination that accounts for the flexing effect:
As usual, and are obtained by cyclic permutation of the spacecraft indices. This expression is
readily shown to be laser-noise-free to first order of spacecraft separation velocities : it is
“flex-free”.