I found this discussion online:(adsbygoogle = window.adsbygoogle || []).push({});

http://web.mit.edu/edbert/GR/gr1.pdf

The author tell me to verify that eq. (18) follows from (13) and (17).

I'm not getting how that works on the basis of what he's given me so far. Take, for example the first expression.

[itex]\textbf{g}=g_{\mu \nu}\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu[/itex]

where

[itex]g_{\mu \nu} \equiv \textbf{g}(\vec{\textbf{e}}_\mu , \vec{\textbf{e}}_\nu) = \vec{\textbf{e}}_\mu \cdot \vec{\textbf{e}}_\nu[/itex]

From the definitions already given, [itex]\textbf{g}[/itex] is a tensor that maps two vectors into a scalar. [itex]\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu[/itex] is a collection of tensors taking two vectors as operands such that [itex]\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu(\vec{A},\vec{B})=A^\mu B^\nu[/itex]. I can use (12) in the article to contract those values with [itex]\vec{\textbf{e}}_\mu \cdot \vec{\textbf{e}}_\nu[/itex], wave my hands vigorously and claim linearity will allow me to treat that as [itex]\textbf{g}(\vec{A}, \vec{B})[/itex], which demonstrates the assertion.

But I don't see how [itex]\left\langle \tilde{\textbf{e}}^\mu , \vec{\textbf{e}}_\nu \right\rangle ={\delta^{\mu}}_\nu [/itex] does anything for me with the available definitions.

Am I missing something here?

BTW, is there a tutorial on how to format mathematical expression on the forum?

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

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

# Tensor bases

Loading...

Similar Threads - Tensor bases | Date |
---|---|

A Dark Matter and the Energy-Momentum Tensor | Monday at 8:09 PM |

I Tensor derivates of EM four-potential | Feb 14, 2018 |

I How to sum the first term in the Maxwell field Lagrangian | Feb 14, 2018 |

I Stress–energy pseudotensor of gravitation field for DE | Jan 30, 2018 |

I Non-coordinate bases | Nov 16, 2017 |

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