- #1

Data Base Erased

- 5

- 0

- TL;DR Summary
- I learned about transforming a vector to a covector using the metric tensor, but I feel I lack the basic understanding behind this.

For instance, using the vector ##A^\alpha e_\alpha##:

##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = g_{\mu \nu} (e^\mu, A^\alpha e_\alpha) e^\nu ##

##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = A^\alpha g_{\mu \nu} \delta_\alpha^\mu e^\nu = A^\mu g_{\mu \nu} e^\nu = A_\nu e^\nu ##.

In this case the intuition is I had a tensor that could take 2 vectors and give back a number. Since I fed it only with one vector, I still have a map that now takes one vector to a number.

When Carroll talks about this subject in his book he does

##\eta^{\mu \gamma} T^{\alpha \beta}_{\gamma \delta}## (the upper indexes come first) = ##T^{\alpha \beta \mu}_{\delta}##. Again, sorry for the bad typing, I have no idea how to order these indexes here.

In this example, one of the indexes of the metric is the same as one of the indexes of the tensor that's feeding it. How do I get his result?

##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = g_{\mu \nu} (e^\mu, A^\alpha e_\alpha) e^\nu ##

##g_{\mu \nu} e^\mu \otimes e^\nu (A^\alpha e_\alpha) = A^\alpha g_{\mu \nu} \delta_\alpha^\mu e^\nu = A^\mu g_{\mu \nu} e^\nu = A_\nu e^\nu ##.

In this case the intuition is I had a tensor that could take 2 vectors and give back a number. Since I fed it only with one vector, I still have a map that now takes one vector to a number.

When Carroll talks about this subject in his book he does

##\eta^{\mu \gamma} T^{\alpha \beta}_{\gamma \delta}## (the upper indexes come first) = ##T^{\alpha \beta \mu}_{\delta}##. Again, sorry for the bad typing, I have no idea how to order these indexes here.

In this example, one of the indexes of the metric is the same as one of the indexes of the tensor that's feeding it. How do I get his result?