Relativistic corrections for measuring Hubble's constant to 1% using stellar standard candles

Relativistic corrections are estimated for classical Cepheids and the Tip of the Red Giant Branch (TRGB stars), to enable future unbiased 1% measurements of Hubble's constant, $H_0$. We considered four effects: $K-$corrections, time-dilation, the apparent change of host dust extinction due to non-comoving reference frames, and the change of observed color due to redshift. Extinction-dependent $K-$corrections were computed using stellar atmosphere models applicable to giant stars for $0.005 < z < 0.03$ in HST, JWST, and 2MASS filters. The optical-NIR Wesenheit function advantageously combines filters with oppositely signed $K-$corrections and avoids complications due to host extinction. For TRGB stars, the JWST/NIRCAM F277W filter combines insensitivity to reddening with $K-$corrections $<1$% at Coma cluster distances. Missing corrections for host extinction due to circumgalactic or circumstellar material are discussed as potential systematics for TRGB distances although their impacts are insufficient to explain differences between $H_0$ based on Cepheid or TRGB supernova calibrations. All stellar standard candles require relativistic corrections to achieve an unbiased 1% $H_0$ measurement in the future. The combined relativistic correction involving $K$, redshift-Leavitt bias, and the redshift-dependence of the Wesenheit function yield an increase of the Cepheid-based $H_0$ by $0.45 \pm 0.05$ km/s/Mpc to $73.65 \pm 1.30$ km/s/Mpc and raises the tension with the {\it Planck} value from $4.2\sigma$ to $4.4\sigma$. For TRGB stars, we estimate a $\sim 0.5\%$ increase of $H_0$ reported by Freedman et al. (to $70.2\pm1.7$km/s/Mpc) and a small decrease by $-0.15\%$ for $H_0$ reported by Anand et al. (to $71.4 \pm 1.8$km/s/Mpc). The opposite sign of these corrections is due to different reddening systematics and reduces the difference between the studies by $\sim 0.46$km/s/Mpc.[abridged]