Understanding Tensor Bases: Solving Equations in General Relativity

[illuminatus33]

I found this discussion online:

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.

$\textbf{g}=g_{\mu \nu}\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu$

where

$g_{\mu \nu} \equiv \textbf{g}(\vec{\textbf{e}}_\mu , \vec{\textbf{e}}_\nu) = \vec{\textbf{e}}_\mu \cdot \vec{\textbf{e}}_\nu$

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

But I don't see how $\left\langle \tilde{\textbf{e}}^\mu , \vec{\textbf{e}}_\nu \right\rangle ={\delta^{\mu}}_\nu$ 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?

 

[dextercioby]

You need to use formula (6) to evaluate this <animal>

$$\tilde{e}^{\mu}\otimes\tilde{e}^{\nu}\left(\vec{e}_{\alpha},\vec{e}_{\beta}\right)$$

twhich takes you to the first formula in (18).

There's a tutorial about the LaTex code in the <Administrative> section of the forums. https://www.physicsforums.com/showthread.php?t=617567

 

[George Jones]

Take, for example the first expression.

$\textbf{g}=g_{\mu \nu}\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu$

Write $\textbf{g}$ as a general tensor expanded in terms of the basis $\left\{\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu\right\}$,

$$\textbf{g} = a_{\mu \nu} \tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^{\nu}.$$
What does $g_{\alpha \beta} = \textbf{g} \left( \textbf{e}_\alpha, \textbf{e}_\beta \right)$ then give you?

[edit]Didn't see dextercioby's similar answer.[/edit]

 

[illuminatus33]

You need to use formula (6) to evaluate this <animal>

$$\tilde{e}^{\mu}\otimes\tilde{e}^{\nu}\left(\vec{e}_{\alpha},\vec{e}_{\beta}\right)$$

twhich takes you to the first formula in (18).

There's a tutorial about the LaTex code in the <Administrative> section of the forums. https://www.physicsforums.com/showthread.php?t=617567

$$\tilde{e}^{\mu}\otimes\tilde{e}^{\nu}\left(\vec{e}_{\alpha},\vec{e}_{\beta}\right)$$ appears to give me $$\left\langle \tilde{e}^{\mu}, \vec{e}_{\alpha}\right\rangle \left\langle \tilde{e}^{\nu}, \vec{e}_{\beta}\right\rangle = {\delta^{\mu}}_{\alpha}{\delta^{\nu}}_{\beta}$$

So:

$g_{\mu \nu}{\delta^{\mu}}_{\alpha}{\delta^{\nu}}_{\beta}=g_{\alpha \beta}$

How does that demonstrate that $g_{\mu \nu}\tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu(\vec{A},\vec{B})=\textbf{g}(\vec{A},\vec{B})$? I'll have to think about this a spell. If is express $\vec{A}$ and $\vec{B}$ in terms of the basis, I will get the traditional contraction of two vectors with the metric tensor.

 

[sardar]

As a scientist, it is always important to fully understand the equations and concepts being discussed. In this case, the author is asking you to verify that equation (18) can be derived from equations (13) and (17). Let's break down these equations and see how they are related.

Equation (13) defines a tensor \textbf{g} as a mapping between two vectors \vec{\textbf{e}}_\mu and \vec{\textbf{e}}_\nu. This tensor is represented as a collection of tensors \tilde{\textbf{e}}^\mu \otimes \tilde{\textbf{e}}^\nu, which take two vectors as operands and return a scalar. This is essentially a way of representing the components of the tensor \textbf{g} in terms of the basis vectors \vec{\textbf{e}}_\mu and \vec{\textbf{e}}_\nu.

Equation (17) defines the inner product between two vectors \vec{\textbf{e}}_\mu and \vec{\textbf{e}}_\nu as the dot product of their components, g_{\mu \nu} = \vec{\textbf{e}}_\mu \cdot \vec{\textbf{e}}_\nu. This is essentially the same as the definition of the tensor \textbf{g} in equation (13), just written in a different form.

Now, to verify that equation (18) follows from (13) and (17), we need to use the definitions of the tensor \textbf{g} and the inner product to show that the two equations are equivalent. This can be done by using the properties of linearity and the fact that \vec{\textbf{e}}_\mu and \vec{\textbf{e}}_\nu are basis vectors, which means they are linearly independent and can span any vector space.

As for your question about formatting mathematical expressions on the forum, there are a few tutorials available online that can help you with this. You can also use LaTeX, a typesetting language, to format your equations. There are many resources available online to help you learn how to use LaTeX for mathematical expressions.