Determining cosmological-model-independent $H_0$ and post-Newtonian parameter with time-delay lenses and supernovae

Strong gravitational lensing provides a natural opportunity to test General Relativity (GR). We propose a model-independent method for simultaneous constraining on Hubble constant ($H_0$) and post-Newtonian parameter (${\gamma_{\rm{PPN}}}$) using strong lensing systems and observational SNe Ia. The time-delay measurements from strong lesning can directly determine the Hubble constant, and the lens distance inferred from the spectroscopic measurement of the stellar kinematics of the deflector galaxy can help us to constrain the post-Newtonian parameter. We seek the Pantheon dataset and reconstruct unanchored distances using Gaussian process regression to achieve the cosmological model-independent GR testing instead of assuming a specific model, which can reduce possible bias on GR testing and measurement of Hubble constant. Combining the reconstructed unanchored distances and the four H0LiCOW lenses datasets, our results are $H_0=72.9^{+2.0}_{-2.3} {\mathrm{~km~s^{-1}~Mpc^{-1}}}$ and ${\gamma_{\rm{PPN}}}=0.89^{+0.17}_{-0.15}$. All the lenses show that there is no obvious evidence to support GR deviation within observational uncertainties. In the subsequent analysis, we consider a ratio of distance ${D_{\Delta t}}/{D^{'}_{d}}$ method to further avoid the influence of $H_0$ on GR testing. The results show that, except J1206 within the $\sim1.2\sigma$ observational uncertainty, the remaining 3 lenses support GR holds within the $1\sigma$ observational uncertainties.