Hello all,(adsbygoogle = window.adsbygoogle || []).push({});

I'm having a minor annoyance in proving an identity.

The identity is the following

[tex]

\star\text{d}\star A_p = \frac{(-)^{p(D-p+1)-1+t}}{(p-1)!}\nabla_\mu A^\mu_{\,\, \mu_1 \cdots \mu_{p-1}}\text{d}^\mu_1\wedge \cdots \wedge \text{d}^\mu_{p-1}

[/tex]

I'm stuck at the first step of proving this

[tex]\text{d}\star A_p = ?[/tex]

Because I'm in arbitrary dimension it's difficult to write out the full expression.

Not in the least because this can be written as

[tex]\text{d}(\star A_p) = \text{d}\left(\frac{1}{p! (D-p)!}A_{\mu_1\cdots \mu_p}\epsilon_{\nu_1\cdots \nu_{D-p}}^{\quad\quad\mu_1\cdots\mu_p} \text{d}^\nu_1\wedge \cdots \wedge \text{d}^\nu_{D-p}\right)[/tex]

If I immediately apply the definition of the exterior derivative to this I get a lot of ugliness.

I would need to introduce the metric ##p## times, once for each of the upper indices.

Then I need to use ##\nabla_\alpha \epsilon_{\mu_1\cdots \mu_D} = 0## to rewrite the partial in terms of the connection. The same happens for the metric factors.

Is there any way I can write this down in a clean way? If need be I can grind my way through but if there is an alternative I'm not going to do that (at first sight) useless exercise in index gymnastics.

Joris

PS.

The next step would be to write the identity as

[tex]

\star\text{d}\star A_p = \frac{(-)^{p(D-p+1)-1+t}\partial_\mu\left(\sqrt{|g|}A^{\mu\nu_1\cdots\nu_{p-1}}\right)}{(p-1)!\sqrt{|g|}}g_{\nu_1\mu_1}\cdots g_{\nu_{p-1}\mu_{p-1}}\text{d}^\mu_1\wedge \cdots \wedge \text{d}^\mu_{p-1}

[/tex]

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Exterior derivative of hodge dual

Loading...

Similar Threads - Exterior derivative hodge | Date |
---|---|

A Radial, exterior, outgoing, null geodesics in Schwarzschild | May 27, 2017 |

I Exterior Schwarzchild Solution/Gravitational Time Dilation | Jun 30, 2016 |

The exterior Schwarzschild spacetime | Dec 26, 2015 |

Exterior derivative identity in vacuum space-time | Jun 23, 2013 |

The Schwarzschild (Exterior) Solution | Jun 3, 2011 |

**Physics Forums - The Fusion of Science and Community**