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Yes. Synchronous gauge in cosmology uses freely falling time lines and sets the lapse and shift perturbations to zero, so no gravitational-potential force appears explicitly in this component of momentum conservation. Metric perturbations still affect the other field equations, the evolution of matter variables, and the relation between coordinate-dependent variables and gauge-invariant observables; synchronous gauge also retains residual gauge modes.

In Newtonian gauge in cosmology, the time-time metric perturbation is a Newtonian gravitational potential. Its spatial gradient therefore appears explicitly as a force term in the Euler equation. The difference is a coordinate representation of the same covariant conservation law.

Solved by gpt-5.6-sol high.

For $R=0$, $c_s=1/\sqrt{3}$. If the Newtonian potentials are constant and equal, the equation in part (b) becomesThe Sachs-Wolfe combination $D={\Theta }_0+\Psi$ therefore obeysAdiabatic initial conditions have vanishing initial fluid velocity, hence $D^{\prime }(0)=0$. With the sound horizon $r_s(\eta )={\int }_0^{\eta }{c}_s,d{\eta }^{\prime }$, the solution is

Solved by gpt-5.6-sol high.