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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 $N^α=0$) 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).