alex3
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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
<br /> R_{\mu\nu} =<br /> \partial_{\lambda} \Gamma^{\lambda}_{\phantom{k}\nu\mu} -<br /> \partial_{\nu} \Gamma^{\lambda}_{\phantom{k}\lambda\mu}<br />
with the Christoffel symbols given by
<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}\eta^{\alpha\beta}<br /> (<br /> h_{\mu\beta,\nu} +<br /> h_{\beta\nu,\mu} -<br /> h_{\mu\nu,\beta}<br /> )<br />
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:
<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}<br /> (<br /> \eta^{\alpha\beta}h_{\mu\beta,\nu} +<br /> \eta^{\alpha\beta}h_{\beta\nu,\mu} -<br /> \eta^{\alpha\beta}h_{\mu\nu,\beta}<br /> )<br /> \\<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}<br /> (<br /> h_{\mu\phantom{\alpha},\nu}^{\phantom{k}\alpha} +<br /> h^{\alpha}_{\phantom{\alpha}\nu,\mu} -<br /> h_{\mu\nu}^{\phantom{\mu\nu},\alpha}<br /> )<br />
i.e. the flat metric raises all \beta's to \alpha's.
However, the book gets this
<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}<br /> (<br /> h^{\alpha}_{\phantom{\alpha}\mu,\nu} +<br /> h^{\alpha}_{\phantom{\alpha}\nu,\mu} -<br /> h_{\mu\nu}^{\phantom{\mu\nu},\alpha}<br /> )<br />
So, the problem is in the first term: how come the book is able to swap the \alpha and \mu like that?
1. Relevant equations
Using Straumann, the Ricci tensor is given by
<br /> R_{\mu\nu} =<br /> \partial_{\lambda} \Gamma^{\lambda}_{\phantom{k}\nu\mu} -<br /> \partial_{\nu} \Gamma^{\lambda}_{\phantom{k}\lambda\mu}<br />
with the Christoffel symbols given by
<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}\eta^{\alpha\beta}<br /> (<br /> h_{\mu\beta,\nu} +<br /> h_{\beta\nu,\mu} -<br /> h_{\mu\nu,\beta}<br /> )<br />
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:
<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}<br /> (<br /> \eta^{\alpha\beta}h_{\mu\beta,\nu} +<br /> \eta^{\alpha\beta}h_{\beta\nu,\mu} -<br /> \eta^{\alpha\beta}h_{\mu\nu,\beta}<br /> )<br /> \\<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}<br /> (<br /> h_{\mu\phantom{\alpha},\nu}^{\phantom{k}\alpha} +<br /> h^{\alpha}_{\phantom{\alpha}\nu,\mu} -<br /> h_{\mu\nu}^{\phantom{\mu\nu},\alpha}<br /> )<br />
i.e. the flat metric raises all \beta's to \alpha's.
However, the book gets this
<br /> \Gamma^{\alpha}_{\phantom{k}\mu\nu}<br /> =<br /> \frac{1}{2}<br /> (<br /> h^{\alpha}_{\phantom{\alpha}\mu,\nu} +<br /> h^{\alpha}_{\phantom{\alpha}\nu,\mu} -<br /> h_{\mu\nu}^{\phantom{\mu\nu},\alpha}<br /> )<br />
So, the problem is in the first term: how come the book is able to swap the \alpha and \mu like that?