Index Raising in Linearized General Relavitiy

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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?
 
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Because h_{\mu \beta , \nu} is symmetric in \mu and \beta, h_{\mu\phantom{\alpha},\nu}^{\phantom{k}\alpha} = h^{\alpha}_{\phantom{\alpha}\mu,\nu}.


\eta^{\alpha\beta} h_{\mu\beta,\nu} = \eta^{\alpha \beta} h_{\beta \mu , \nu}
 
How do we know that h_{\alpha\beta} is symmetric? I can't see it mentioned anywhere. The only condition I see is \lvert h_{\alpha\beta}\rvert \ll 1.
 
alex3 said:
How do we know that h_{\alpha\beta} is symmetric? I can't see it mentioned anywhere. The only condition I see is \lvert h_{\alpha\beta}\rvert \ll 1.

h_{\alpha\beta} = g_{\alpha\beta} - \eta_{\alpha\beta}, and g and \eta are both symmetric.
 
Why do we assume g_{\alpha\beta} is symmetric then? Is that a property we assume of all metrics? I didn't think we did. Do we assume symmetry of g_{\alpha\beta} as it deviates only slightly from the Minkowski metric?
 
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alex3 said:
Why do we assume g_{\alpha\beta} is symmetric then? Is that a property we assume of all metrics?[/itex]

In standard general relativity, yes.

alex3 said:
I didn't think we did. Do we assume symmetry of g_{\alpha\beta} as it deviates only slightly from the Minkowski metric?

No, a symmetric g can differ substantially from the Minkowski metric.

If the metric weren't symmetric, then it would not always have a tangent space isomorphic to Minkowski spacetime. If a metric tensor field is not symmetric, then there exists at least one point (event) at which the metric tensor for the tangent space is not symmetric.
 
Got it now, thank you very much!
 
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