  Gravitational wave (GW) galaxy lens reconstruction is a crucial step for many
GW lensing science applications. However, dark siren GW lensing (i.e. lensed GW
without observed electromagnetic (EM) counterpart) suffers from similarity
transformation degeneracy and mass-sheet degeneracy. We review these two
degeneracies and discuss their implications on GW-based lens reconstruction and
two well-known GW lensing science cases: the Hubble constant measurement and
test for modified GW propagation. Building upon previous works, our conclusions
are:1) GWs can only infer the scale-free lens mass model parameters, the
dimensionless source position, the GW luminosity distance and the time delay
scaling (a combination of Einstein radius, lens redshift, and cosmology).2)
Lens reconstruction (of singular isothermal ellipsoid lens) with only two GW
signals is unlikely to yield a complete lens model, while four (three) signals
can measure all the above parameters accurately (with large uncertainties).3)
The similarity transformation degeneracy causes the lens redshift/Einstein
radius/cosmology to be degenerate in dark siren measurements. Breaking the
degeneracy can be achieved by supplementing the GWs with EM observation of lens
redshifts/Einstein radius (source redshift is not required).4) The mass-sheet
degeneracy causes the GW luminosity distance to be entirely degenerate with a
constant mass sheet.5) Contrary to expectation, the Hubble constant is
degenerate with the mass-sheet even when supplemented with lens
reconstruction/redshift/Einstein radius and can only be lifted with lens galaxy
velocity dispersion measurement, while modified GW propagation test discussed
in prior literature is unaffected by the degeneracy. These properties highlight
the need for GW observations to be supplemented by EM observations, which could
become accessible through a lens archival search or a rapid EM follow-up.
