The equation of motion of a perfect fluid in GR is: $$(\rho+p)u^{\mu} \nabla_{\mu} u^{\nu} = -(g^{\mu\nu}+u^{\mu}u^{\nu})\nabla_{\mu}p$$ where $\rho$ is the energy density of the fluid and $p$ is the pressure. To get the Newtonian limit we are meant to work in an inertial frame so that the covariant derivative is substituted by a partial derivative. We are further meant to assume that the metric is $\eta^{\mu\nu}$, that the pressure is small $p/\rho<<1$, and that $u^{\mu}=(1, u_3^{i})+...$ with $|u_3|<<1$. The answer for $\nu=j$ is supposed to be: $$\rho(\partial_t+u^{i}\partial_i)u^j = -\partial_jp$$
But when I attempted to derive it, I got two extra terms on the RHS that I don't know how to get rid of.
This was my method: $$(\rho+p)u^{\mu} \nabla_{\mu} u^{\nu} = -(g^{\mu\nu}+u^{\mu}u^{\nu})\nabla_{\mu}p$$ $$\rho u^{\mu} \partial_{\mu} u^{j} = -(\eta^{\mu j}+u^{\mu}u^{j})\partial_{\mu}p$$ $$\rho(\partial_t u^j+u^{i}\partial_iu^j) = -(\eta^{tj}+u^j)\partial_tp-(\eta^{ij}+u^{i}u^j)\partial_ip=-u^j\partial_tp-\partial_jp-u^{i}u^j\partial_ip$$
The LHS looks like it's supposed to, but the right-hand side has these extra two terms $-u^j\partial_tp$ and $-u^{i}u^j\partial_ip$ that are not supposed to be there. I am not sure whether it is sufficient to say that these terms are very small compared to $-\partial_jp$ since they are 1st and 2nd order in $u^j$, or whether I made a mistake and the terms are just not supposed to be there.
