- #1
Pentaquark5
- 17
- 2
Homework Statement
My textbook states:
Since the number of particles of dust is conserved we also have the conservation equation
$$\nabla_\mu (\rho u^\mu)=0$$
Where ##\rho=nm=N/(\mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z) m## is the mass per infinitesimal volume and ## (u^\mu) ## is the four velocity of the dust particles.
Homework Equations
$$ \nabla_\mu A^\nu=\partial_\mu A^\nu+\Gamma^\nu_{\;\; \mu \gamma} A^\gamma $$
The Attempt at a Solution
$$\nabla_\mu (\rho u^\mu)= \underbrace{m \partial_\mu n u^\mu}_{=0} + m n \underbrace{\partial_\mu u^\mu}_{=0}+\Gamma^\mu_{\;\;\mu \gamma} mnu^\gamma$$
Where the first underbrace is zero since the divergence of the particle number is zero, and the second underbrace is zero due to the partial derivative of the velocity.
I don't understand why the last term should be zero, however?