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This study investigates the properties of waves in relativistic extendedmagnetohydrodynamics (RXMHD), which includes Hall and electron thermal inertiaeffects. We focus on the case when the electron temperature isultrarelativistic, and thus, the electron thermal inertia becomes finite atnear the proton inertial scale. We derive the linear dispersion relation ofRXMHD and find that the Hall and electron thermal inertia effects couple withthe displacement current, giving rise to three superluminous waves in additionto the slow, fast, and Alfv\'en waves. We also show that the phase- andgroup-velocity surfaces of fast and Alfv\'en waves are distorted by the Halland electron thermal inertia effects. There is a range of scales where thegroup velocity of fast wave is smaller than that of the Alfv\'en and slowwaves. These findings are applicable to a region near the funnel base oflow-luminosity accretion flows where electrons can be ultrarelativistic.

Yet Another Modification of Relativistic Magnetohydrodynamic Waves: Electron Thermal Inertia