This is really more maths than physics, but I think that's the sort of thing only physicists know...(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

I need to show that although [itex]V^{\beta}_{,\alpha}[/itex] and [itex]V^{\mu} \Gamma^{\beta}_{\mu \alpha}[/itex] don't transform like tensors, their sum, the covariant derivative, does.

2. Relevant equations

Regular transformation laws for tensors: [itex]T^{\alpha' \beta'}_{\gamma'\delta'} = \Lambda^{\alpha'}_{\alpha} \Lambda^{\beta'}_{\beta} \Lambda^{\gamma}_{\gamma'} \Lambda^{\delta}_{\delta'} T^{\alpha\beta}_{\gamma\delta}[/itex]

Also: [itex]T_{,\alpha} \equiv \frac{dT}{dx^{\alpha}}[/itex]

3. The attempt at a solution

This is reaaaaly lengthy so I'll post only what I've gotten after trying to transform the terms in the question (which takes long enough).

I cannot show that their sum is transformed like a tensor though!

What I get:

1. [itex]V^{\beta'}_{,\alpha'} = \Lambda^{\beta'}_{\beta} \Lambda^{\alpha}_{\alpha'} V^{\beta}_{,\alpha} + \Lambda^{\alpha}_{\alpha'} \Lambda^{\beta'}_{\beta, \alpha} V^{\beta}[/itex]

2. [itex]V^{\mu'}\Gamma^{\beta'}_{\mu'\alpha'} = \Lambda^{\mu'}_{\nu}V^{\nu} (

\Lambda^{\beta'}_{\beta} \Lambda^{\mu}_{\mu'} \Lambda^{\alpha}_{\alpha'} \Gamma^{\beta}_{\mu\alpha} + \Lambda^{\beta'}_{\beta} \Lambda^{\alpha}_{\alpha'} \Lambda^{\beta}_{\mu', \alpha}) [/itex]

Now, assuming that I properly played with the second expression, I get after some manipulations and usage of [itex] \Lambda^{\mu'}_{\nu} \Lambda^{\mu}_{\mu'} = \delta^{\mu}_{\nu}[/itex]:

2'. [itex]V^{\mu'}\Gamma^{\beta'}_{\mu'\alpha'} = \Lambda^{\beta'}_{\beta} \Lambda^{\alpha}_{\alpha'} V^{\mu}\Gamma^{\beta}_{\mu\alpha} + \Lambda^{\alpha}_{\alpha'} V^{\nu} \Lambda^{\beta'}_{\nu, \alpha} [/itex]

Now, it is clear that when we add these two equations, 1 and 2', the first parts are exactly the tensor transformations we want.

That means, that the second sum needs to somehow cancel out. But I don't see how/why it happens. Actually, the "extra parts" seem to be equal, so according to my "calculations", they add up. I cannot see where a minus sign could come from.

I either miss something here, or I did something wrong.

I'm sorry for not posting the whole way, it would just take forever.... I'd really really appreciate an answer!!!

Tomer.

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# Homework Help: Tensor Excercise.

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