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We derive the conductivity tensor for axisymmetric perturbations of a hot,collisionless, and charge-neutral plasma in the shearing sheet approximation.Our results generalize the well-known linear Vlasov theory for uniform plasmasto differentially rotating plasmas and can be used for wide range of kineticstability calculations. We apply these results to the linear theory of themagneto-rotational instability (MRI) in collisionless plasmas. We showanalytically and numerically how the general kinetic theory results derivedhere reduce in appropriate limits to previous results in the literature,including the low frequency guiding center (or "kinetic MHD") approximation,Hall MHD, and the gyro-viscous approximation. We revisit the cold plasma modelof the MRI and show that, contrary to previous results, an initiallyunmagnetized collisionless plasma is linearly stable to axisymmetricperturbations in the cold plasma approximation. In addition to theirapplication to astrophysical plasmas, our results provide a useful frameworkfor assessing the linear stability of differentially rotating plasmas inlaboratory experiments.

Linear Vlasov Theory in the Shearing Sheet Approximation with Application to the Magneto-Rotational Instability