  Length scales probed by large scale structure surveys are becoming closer to
the horizon scale. Further, it has been recently understood that
non-Gaussianity in the initial conditions could show up in a scale dependence
of the bias of galaxies at the largest distances. It is therefore important to
include General Relativistic effects. Here we provide a General Relativistic
generalization of the bias, valid both for Gaussian and non-Gaussian initial
conditions. The collapse of objects happens on very small scales, while
long-wavelength modes are always in the quasi linear regime. Around every
collapsing region, it is therefore possible to find a reference frame that is
valid for all times and where the space time is almost flat: the Fermi frame.
Here the Newtonian approximation is applicable and the equations of motion are
the ones of the N-body codes. The effects of long-wavelength modes are encoded
in the mapping from the cosmological frame to the local frame. For the linear
bias, the effect of the long-wavelength modes on the dynamics is encoded in the
local curvature of the Universe, which allows us to define a General
Relativistic generalization of the bias in the standard Newtonian setting. We
show that the bias due to this effect goes to zero as the squared ratio of the
physical wavenumber with the Hubble scale for modes longer than the horizon, as
modes longer than the horizon have no dynamical effects. However, the bias due
to non-Gaussianities does not need to vanish for modes longer than the Hubble
scale, and for non-Gaussianities of the local kind it goes to a constant. As a
further application, we show that it is not necessary to perform large N-body
simulations to extract information on long-wavelength modes: N-body simulations
can be done on small scales and long-wavelength modes are encoded simply by
adding curvature to the simulation and rescaling the coordinates.
