  We show that radiation dominated accretion discs are likely to suffer from a
``photon bubble'' instability similar to that described by Arons in the context
of accretion onto neutron star polar caps. The instability requires a magnetic
field for its existence. In an asymptotic regime appropriate to accretion
discs, we find that the overstable modes obey the remarkably simple dispersion
relation \omega^2 = -i g k F(B,k). Here g is the vertical gravitational
acceleration, B the magnetic field, and F is a geometric factor of order unity
that depends on the relative orientation of the magnetic field and the
wavevector. In the nonlinear outcome it seems likely that the instability will
enhance vertical energy transport and thereby change the structure of the
innermost parts of relativistic accretion discs.
