We report on a test of Lorentz invariance performed by comparing the resonance frequencies of one stationary optical resonator and one continuously rotating on a precision air bearing turntable. Special attention is paid to the control of rotation induced systematic effects. Within the photon sector of the Standard Model Extension, we obtain improved limits on combinations of 8 parameters at a level of a few parts in $10^{-16}$. For the previously least well known parameter we find $tilde kappa_{e-}^{ZZ} =(-1.9 pm 5.2)times 10^{-15}$. Within the Robertson-Mansouri-Sexl test theory, our measurement restricts the isotropy violation parameter $beta -delta -frac 12$ to $(-2.1pm 1.9)times 10^{-10}$, corresponding to an eightfold improvement with respect to previous non-rotating measurements.
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