  Scattering of Lyman Alpha (hereafter Lya) photons by neutral hydrogen gas in
a single outflowing 'supershell' around star forming regions often explains the
shape and offset of the observed Lya emission line from galaxies. We compute
the radiation pressure that is exerted by this scattered Lya radiation on the
outflowing material. We show that for reasonable physical parameters, Lya
radiation pressure alone can accelerate supershells to velocities in the range
v_sh=200-400 km/s. These supershells possibly escape from the gravitational
potential well of their host galaxies and contribute to the enrichment of the
intergalactic medium. We compute the physical properties of expanding
supershells that are likely to be present in a sample of known high-redshift
(z=2.7-5.0) galaxies, under the assumption that they are driven predominantly
by Lya radiation pressure. We predict ranges of radii r_sh=0.1-10 kpc, ages
t_sh=1-100 Myr, and energies E_sh=1e53-1e55 ergs, which are in reasonable
agreement with the properties of local galactic supershells. Furthermore, we
find that the radius, r_sh, of a Lya-driven supershell of constant mass depends
uniquely on the intrinsic Lya luminosity of the galaxy, L_alpha, the HI column
density of the supershell, N_HI, and the shell speed, v_sh, through the scaling
relation r_sh ~ L_alpha/(N_HI v_sh^2). We derive mass outflow rates in
supershells that reach ~10-100% of the star formation rates of their host
galaxies.
