- #1

rezkyputra

- 5

- 2

## Homework Statement

Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it

## Homework Equations

Suppose we have

$$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C \phi $$

it should be proven that

$$\nabla_A \: X^{AB} =0$$

with ##\phi## is a scalar field

## The Attempt at a Solution

Naturally, we would expand the equations.

\begin{align}\nabla_A \: X^{AB} &= \nabla_A \left[ \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C \phi \right] \\

&= \nabla^B \phi \left[ \nabla_A \nabla^A \phi \right] + \nabla^A \phi \left[ \nabla_A \nabla^B \phi \right] - \frac{1}{2} g^{AB} \nabla_C \phi \left[\nabla_A \nabla^C \phi\right] - \frac{1}{2} g^{AB} \nabla^C \phi \left[\nabla_A \nabla_C \phi\right]

\end{align}

We know that the covariant derivative of a scalar is its partial derivative ## \nabla_A \phi = \partial_A \phi##

\begin{align}\nabla_A \: X^{AB} &= \partial^B \phi \left[ \nabla_A \partial^A \phi \right] + \partial^A \phi \left[ \nabla_A \partial^B \phi \right] - \frac{1}{2} g^{AB} \partial_C \phi \left[\nabla_A \partial^C \phi\right] - \frac{1}{2} g^{AB} \partial^C \phi \left[\nabla_A \partial_C \phi\right]

\end{align}

Now the 2nd Cov. Der. would depends on the christoffel symbol where

\begin{align}

\nabla_A \partial_B \phi &= \partial_A \partial_B \phi - \partial_C \phi \Gamma_{AB}^C \\

\nabla_A \partial^B \phi &= \partial_A \partial^B \phi + \partial^C \phi \Gamma_{AC}^B

\end{align}

so that

\begin{align}\nabla_A \: X^{AB} &= \partial^B \phi \left[ \partial_A \partial^A \phi + \partial^C \phi \Gamma_{AC}^A \right] + \partial^A \phi \left[ \partial_A \partial^B \phi + \partial^C \phi \Gamma_{AC}^B \right] \\

\nonumber

& - \frac{1}{2} g^{AB} \partial_C \phi \left[\partial_A \partial^C \phi + \partial^D \phi \Gamma_{AD}^C\right] - \frac{1}{2} g^{AB} \partial^C \phi \left[\partial_A \partial_C \phi - \partial_D \phi \Gamma_{AC}^D\right]

\end{align}

Now I'm stuck as to where should I go? How could t be proven that last (and quite long) equation be equal to zero?

Any help would be much appreciated

Thanks in advance