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The central ambition of the modern time delay cosmography consists indetermining 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 beenunderestimated. The most critical one probably comes from the degeneracyexisting between lens models that was first formalized by the well-knownmass-sheet transformation (MST). In this paper, we assess to what extent thesource position transformation (SPT), a more general invariance transformationwhich contains the MST as a special case, may affect the time delays predictedby a model. To this aim we use pySPT, a new open-source python package fullydedicated to the SPT that we present in a companion paper. For axisymmetriclenses, we find that the time delay ratios between a model and its SPT-modifiedcounterpart simply scale like the corresponding source position ratios, $\Delta\hat{t}/ \Delta t \approx \hat{\beta}/\beta$, regardless of the mass profileand the isotropic SPT. Similar behavior (almost) holds for non-axisymmetriclenses in the double image regime and for opposite image pairs in the quadrupleimage regime. In the latter regime, we also confirm that the time delay ratiosare not conserved. In addition to the MST effects, the SPT-modified time delaysdeviate 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 asinitially suspected. We also reflected upon the relevance of the SPT validitycriterion and present arguments suggesting that it should be reconsidered. Eventhough a new validity criterion would affect the time delays in a differentway, we expect from numerical simulations that our conclusions will remainunchanged.

Ambiguities in gravitational lens models: impact on time delays of the source position transformation