Proper-Time Hypersurface of Non-Relativistic Matter Flows: Galaxy Bias
  in General Relativity
  We compute the second-order density fluctuation in the proper-time
hypersurface of non-relativistic matter flows and relate it to the galaxy
number density fluctuation in general relativity. At the linear order, it is
equivalent to the density fluctuation in the comoving synchronous gauge, in
which two separate gauge conditions coincide. However, at the second order, the
density fluctuations in these gauge conditions differ, while both gauge
conditions represent the proper-time hypersurface. Compared to the density
fluctuation in the temporal comoving and the spatial C-gauge conditions, the
density fluctuation in the commonly used gauge condition ($N=1$ and
$N^\alpha=0$) violates the mass conservation at the second order. We provide
their physical interpretations in each gauge condition by solving the geodesic
equation and the nonlinear evolution equations of non-relativistic matter. We
apply this finding to the second-order galaxy biasing in general relativity,
which complements the second-order relativistic description of galaxy
clustering in Yoo & Zaldarriaga (2014).
