- #1

space-time

- 218

- 4

_{μν}as well as its inverse G

^{μν}in a coordinate basis. I tried converting the inverse into an orthonormal basis using a technique for this that was taught to me on another thread long ago. When I tried using said technique, I got a very strange result. One of my basis vectors (e

_{3}to be precise) ended up having an imaginary term. Needless to say, this was wrong, as I looked up what the orthonormal basis inverse Einstein tensor G

^{μν}should be on Wiki (http://en.wikipedia.org/wiki/Gödel_metric) , and it did not match up to what I had derived. I decided that I needed to study to see if there was any other technique for converting to an orthonormal basis. My studies led me to two wiki pages:

http://en.wikipedia.org/wiki/Gödel_metric (this is the one from above)

http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

Now according to the 2nd wiki, to derive a basis vector e

_{a}(this should have an arrow over it, but I don't know how to put one there), the formula is:

e

_{a}= e

_{a}

^{j}∂

_{xj}

An example of the usage of this formula is shown in the 1st wiki:

e

_{0}= ω√2 * ∂

_{t}

Now here is where my problem comes in:

1. First of all, the above example contains within it ∂

_{t}. I see where the ω√2 term comes in. However, what exactly are they differentiating with respect to time? The metric tensor for this metric doesn't contain a single t term in it and neither does the term ω√2 . Differentiating anything that I just mentioned with respect to time would result in a basis vector e

_{0}= 0.

2. Their 3rd basis vector e

_{3}= 2ω(e

^{-x}∂

_{z}- ∂

_{t}). Why does the calculation of this basis vector involve a both a time derivative and a spatial derivative when the other basis vectors only involved derivatives with respect to one coordinate?

Thank you. P.S: In case you would like to see the metric tensor:

g

_{00}= -1/2ω

^{2}

g

_{03}and g

_{30}= -e

^{x}/2ω

^{2}

g

_{11}= 1/2ω

^{2}

g

_{22}= 1/2ω

^{2}

g

_{33}= -e

^{2x}/4ω

^{2}