In the case of the Sagnac variables , however, this is not the case as it is easy to understand on simple physical grounds. In the case of for instance, light originating from spacecraft 1 is simultaneously sent around the array on clockwise and counter-clockwise loops, and the two returning beams are then recombined. If the array is rotating, the two beams experience a different delay (the Sagnac effect), preventing the noise from cancelling in the combination.
In order to find the solution to this problem let us first rewrite in such a way to explicitly emphasize what it does: attempts to remove the same fluctuations affecting two beams that have been made to propagated clockwise and counter-clockwise around the array,same circuit. For LISA, the remaining laser phase noises in , , are several orders of magnitude below the secondary noises.
In the case of , however, the rotation of the array breaks the symmetry and therefore its uniqueness. However, there still exist three generalized TDI laser-noise-free data combinations that have properties very similar to , and which can be used for the same scientific purposes . These combinations, which we call , can be derived by applying again our time-delay operator approach.
Let us consider the following combination of the , measurements, each being delayed only once :
If the delay-times also change with time, the perfect cancellation of the laser noises is no longer achieved in the combinations. However, it has been shown in  that the magnitude of the residual laser noises in these combinations are significantly smaller than the LISA secondary system noises, making their effects entirely negligible.
The expressions for the Monitor, Beacon, and Relay combinations, accounting for the rotation and flexing of the LISA array, have been derived in the literature  by applying the time-delay iterative procedure highlighted in this section. The interested reader is referred to that paper for details.
A mathematical formulation of the “second generation” TDI, which generalizes the one presented in Section 4 for the stationary LISA, still needs to be derived. In the case when only the Sagnac effect is considered (and the delay-times remain constant in time) the mathematical formulation of Section 4 can be extended in a straight-forward way where now the six time-delays and must be taken into account. The polynomial ring is now in these six variables and the corresponding module of syzygies can be constructed over this enlarged polynomial ring . However, when the arms are allowed to flex, that is, the operators themselves are functions of time, the operators no longer commute. One must then resort to non-commutative Gröbner basis methods. We will investigate this mathematical problem in the near future.
© Max Planck Society and the author(s)