  The central ambition of the modern time delay cosmography consists in
determining the Hubble constant $H_0$ with a competitive precision. However,
the tension with $H_0$ obtained from the Planck satellite for a spatially-flat
$\Lambda$CDM cosmology suggests that systematic errors may have been
underestimated. The most critical one probably comes from the degeneracy
existing between lens models that was first formalized by the well-known
mass-sheet transformation (MST). In this paper, we assess to what extent the
source position transformation (SPT), a more general invariance transformation
which contains the MST as a special case, may affect the time delays predicted
by a model. To this aim we use pySPT, a new open-source python package fully
dedicated to the SPT that we present in a companion paper. For axisymmetric
lenses, we find that the time delay ratios between a model and its SPT-modified
counterpart simply scale like the corresponding source position ratios, $\Delta
\hat{t}/ \Delta t \approx \hat{\beta}/\beta$, regardless of the mass profile
and the isotropic SPT. Similar behavior (almost) holds for non-axisymmetric
lenses in the double image regime and for opposite image pairs in the quadruple
image regime. In the latter regime, we also confirm that the time delay ratios
are not conserved. In addition to the MST effects, the SPT-modified time delays
deviate in general no more than a few percent for particular image pairs,
suggesting that its impact on time-delay cosmography seems not be as crucial as
initially suspected. We also reflected upon the relevance of the SPT validity
criterion and present arguments suggesting that it should be reconsidered. Even
though a new validity criterion would affect the time delays in a different
way, we expect from numerical simulations that our conclusions will remain
unchanged.
