I am having some problem understanding the why in Minkowski spacetime, the continuity equation is written as $$\partial_\mu J^\mu=0.....................(*)$$
Physically, I know that $$\partial_t \rho=-\nabla\cdot\vec j$$
but isn't the Minkowski metric diag$\{1,-1,-1,-1\}$, such that $(*)$ reads $$\partial_t \rho=\nabla\cdot\vec j$$ ? What have I done wrong?
Thank you.
In (pseudo)Riemannian manifold, divergence, in most cases, is no long the usual form $\text{div}X=\partial_iX^i$. The divergence is defined as $$\text{div}X=\frac1{\sqrt g}\partial_i(\sqrt gX^i)$$ where $g$ is the determinant of metric matrix. So for Minkowski metric, $g=-1$ and thus $$\text{div}J=\frac1i\partial_\mu(iJ^\mu)=\partial_\mu J^\mu=\frac{\partial \rho}{\partial t}+\nabla\cdot j=0$$