Hi, I have some elementary questions about geometry. I often find that I am perfectly able to do calculations, but sometimes I have the feeling I'm not totally understanding what I'm actually doing. Maybe this is familiar for some of you ;) Up 'till now I have some questions about quite different topics and maybe some other questions will pop up in my mind, I hope some of you can shine their light upon it. :)(adsbygoogle = window.adsbygoogle || []).push({});

1)

About the definition of the metric tensor: the coefficients are defined by

[tex]

g_{\mu\nu} = g(e_{\mu},e_{\nu}) = e_{\mu} \cdot e_{\nu}

[/tex]

where the dot is the standard inner product. It feels for me some kind of cheating to use the standard inner product to define a general inner product. How would this go for eg the minkowski tensor [tex] \eta_{\mu\nu}[/tex] ? What would the basis vectors be? For instance, [tex]\eta_{00} = -1 [/tex], so [tex]e_{0}\cdot e_{0} = -1 [/tex], so wouldn't we need imaginary components ?

2)

About viewing the integrand of an integral over a manifold as an n-form, and Stokes Theorem.

One identifies the volume element [tex]d^{4}x[/tex] with [tex] dx^{a}\wedge dx^{b} \wedge dx^{c} \wedge dx^{d}[/tex] . We know that the volume-element is a tensor density, but that wedge product looks like an honest tensor... If f is a scalar function, then

[tex]

\int f \sqrt{g} d^{4}x = \int f \sqrt{g} \varepsilon_{abcd} dx^{a}\wedge dx^{b} \wedge dx^{c} \wedge dx^{d} = \int f \epsilon

[/tex]

where [tex] \epsilon_{abcd} = \sqrt{g} \varepsilon_{abcd} [/tex] and [tex]\varepsilon[/tex] is the Levi Civita alternating symbol. So here I would say that the sqrt is a density, [tex]\epsilon[/tex] is a tensor, so the wedge product of coordinatefunctions is also a density. But if I express an n-form in an antisymmetrical basis as [tex]\omega = \omega_{abcd} dx^{a} \wedge dx^{b} \wedge dx^{c} \wedge dx^{d}[/tex], I know that the wedge product of coordinate functions is a tensor ( after all, it is an antisymmetric basis for tensors )... I'm overlooking something, but what ?

Also, if I have that

[tex]

\int f \sqrt{g} d^{4}x = \int d\omega

[/tex]

by looking at the integrand as an n-form, how can I solve this to find the 3-form [tex]\omega[/tex] ? I have the feeling I don't quite understand the connection between the Stokes theorem concerning n-forms and the Stokes theorem concerning vector densities Y;

[tex]

\int_{M} d\omega = \int_{\partial M} \omega

[/tex]

and

[tex]

\int_{M} \nabla_{\mu} Y^{\mu} d \Omega = \int_{\delta M} Y^{\mu}dS_{\mu}

[/tex]

3)

About looking at vectors as differential operators and one-forms as differentials.

I understand that one can look upon a vector as being a differential operator,

[tex]

X = X^{\mu}\partial_{\mu}

[/tex]

and that the basis vectors are given by

[tex]

e_{\mu} = \partial_{\mu}

[/tex]

But in my mind vectors have numerical values. What does a statement like

[tex]

dx^{\mu} (\partial_{\nu} ) = \delta ^{\mu}_{\nu}

[/tex]

mean? Is it appropriate to look upon it as if there is a one-to-one correspondence between numerical values and the operators themselves ? This also pops up if you consider the norm of a vector; how do I consider the norm of a basisvector if it is given by a differential operator? I'm feeling uncomfortable by giving the vector in that way a certain numerical value, like

[tex]

\partial_{\mu} = (1,0,0,0)

[/tex]

so I don't understand what it means to perform an innerproduct between 2 vectors expressed via differential operators.

It's bothering me for quite some time, so who can help me? :)

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# Some elementary geometry questions

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