# Covariant and partial derivative of metric determinant

1. Apr 19, 2015

### the_doors

1. The problem statement, all variables and given/known data

is this statement is true : $\nabla_\mu \nabla_\nu \sqrt{g} \phi = \partial_\mu \sqrt{g} \partial_\nu \phi$

2. Relevant equations

3. The attempt at a solution

well we know $\nabla_\mu \sqrt{g} =0$ so it moves back : $\nabla_\mu \sqrt{g} \nabla_\nu \phi =\sqrt{g} \nabla_\mu \nabla_\nu \phi$ . but because metric determinant is not transfers like scalar we can not write $\partial_\mu$ instead of$\nabla_\mu$.

2. Apr 24, 2015