tensor density

diaoye

i have a question on antisymmetrical tensor density which i read on the internet
i read Geroch's "Geometrical Quantum Mechanics" but nothing changed.

here is the picture about this question

can anybody help me to solve it?

diaoye

this is not a homework question, it's just a question i read on the internet and cannot find the solution

K^2

What is your background? Why are you interested? Which part gives you difficulty?

You can't just paste a high level differential geometry question and ask how it's answered. The answer will depend very much on what it is that you want to know. Solution, by itself, will look to you just as cryptic as the question unless you are very, very close to understanding it already.

diaoye

i study general relativity by myself and did several exercises sucessifully, but only this i didn't understand. i can calculate tensors and the transformation of tensors, christoffel symbols, metrics etc. i read several books about general relativity, only Pauli's talked about tensor density, then i didnt pay much attention on it. Then i found this exercise which need the knowledge of tensor density that i could find few books talk about it, acctually only Pauli's and one on the internet.

K^2

You don't need to know anything special. It's all right there, really. You are given that [itex]g_{\mu \nu \alpha \beta}[/itex] is totally antisymmetric. That means that the matrix will change sign if you swap any two indices. For part one you just need to prove that any two matrices with such property are proportional to each other. That's just linear algebra.

The rest of it is pretty straight forward as well, so long as you know a bit about Levi-Cevita symbol, metric transformations, and covariant derivative. If you've worked with Christoffel symbol, you should have come across all that before.