- #1
alex3
- 44
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I'm reading a few textbooks (Straumann, Schutz, Hartle) on GR and am a little confused working through a small part of each on linearized GR.
1. Relevant equations
Using Straumann, the Ricci tensor is given by
[tex]
R_{\mu\nu} =
\partial_{\lambda} \Gamma^{\lambda}_{\phantom{k}\nu\mu} -
\partial_{\nu} \Gamma^{\lambda}_{\phantom{k}\lambda\mu}
[/tex]
with the Christoffel symbols given by
[tex]
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}\eta^{\alpha\beta}
(
h_{\mu\beta,\nu} +
h_{\beta\nu,\mu} -
h_{\mu\nu,\beta}
)
[/tex]
2. The problem
My problem is that the book is confusing me on the next equality. This what I expected when applying the flat metric:
[tex]
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}
(
\eta^{\alpha\beta}h_{\mu\beta,\nu} +
\eta^{\alpha\beta}h_{\beta\nu,\mu} -
\eta^{\alpha\beta}h_{\mu\nu,\beta}
)
\\
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}
(
h_{\mu\phantom{\alpha},\nu}^{\phantom{k}\alpha} +
h^{\alpha}_{\phantom{\alpha}\nu,\mu} -
h_{\mu\nu}^{\phantom{\mu\nu},\alpha}
)
[/tex]
i.e. the flat metric raises all [itex]\beta[/itex]'s to [itex]\alpha[/itex]'s.
However, the book gets this
[tex]
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}
(
h^{\alpha}_{\phantom{\alpha}\mu,\nu} +
h^{\alpha}_{\phantom{\alpha}\nu,\mu} -
h_{\mu\nu}^{\phantom{\mu\nu},\alpha}
)
[/tex]
So, the problem is in the first term: how come the book is able to swap the [itex]\alpha[/itex] and [itex]\mu[/itex] like that?
1. Relevant equations
Using Straumann, the Ricci tensor is given by
[tex]
R_{\mu\nu} =
\partial_{\lambda} \Gamma^{\lambda}_{\phantom{k}\nu\mu} -
\partial_{\nu} \Gamma^{\lambda}_{\phantom{k}\lambda\mu}
[/tex]
with the Christoffel symbols given by
[tex]
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}\eta^{\alpha\beta}
(
h_{\mu\beta,\nu} +
h_{\beta\nu,\mu} -
h_{\mu\nu,\beta}
)
[/tex]
2. The problem
My problem is that the book is confusing me on the next equality. This what I expected when applying the flat metric:
[tex]
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}
(
\eta^{\alpha\beta}h_{\mu\beta,\nu} +
\eta^{\alpha\beta}h_{\beta\nu,\mu} -
\eta^{\alpha\beta}h_{\mu\nu,\beta}
)
\\
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}
(
h_{\mu\phantom{\alpha},\nu}^{\phantom{k}\alpha} +
h^{\alpha}_{\phantom{\alpha}\nu,\mu} -
h_{\mu\nu}^{\phantom{\mu\nu},\alpha}
)
[/tex]
i.e. the flat metric raises all [itex]\beta[/itex]'s to [itex]\alpha[/itex]'s.
However, the book gets this
[tex]
\Gamma^{\alpha}_{\phantom{k}\mu\nu}
=
\frac{1}{2}
(
h^{\alpha}_{\phantom{\alpha}\mu,\nu} +
h^{\alpha}_{\phantom{\alpha}\nu,\mu} -
h_{\mu\nu}^{\phantom{\mu\nu},\alpha}
)
[/tex]
So, the problem is in the first term: how come the book is able to swap the [itex]\alpha[/itex] and [itex]\mu[/itex] like that?