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We systematically searched for gravity- and Rossby-mode period spacingpatterns in Kepler eclipsing binaries with $\gamma$ Doradus pulsators. Thesestars provide an excellent opportunity to test the theory of tidalsynchronisation and angular momentum transport in F- and A-type stars. Wediscovered 35 systems that show clear patterns, including the spectroscopicbinary KIC 10080943. Combined with 45 non-eclipsing binaries with $\gamma$ Dorcomponents that have been found using pulsation timing, we measured theirnear-core rotation rates and asymptotic period spacings. We find that manystars are tidally locked if the orbital periods are shorter than 10 days, inwhich the near-core rotation periods given by the traditional approximation ofrotation (TAR) are consistent with the orbital period. Compared to the singlestars, $\gamma$ Dor stars in binaries tend to have slower near-core rotationrates, likely a consequence of tidal spin-down. We also find three stars thathave extremely slow near-core rotation rates. To explain these, we hypothesisethat unstable tidally excited oscillations can transfer angular momentum fromthe star to the orbit, and slow the star below synchronism, a process we referto as `inverse tides'.

The effect of tides on near-core rotation: analysis of 35 Kepler $\gamma$ Doradus stars in eclipsing and spectroscopic binaries