Index Raising in Linearized General Relavitiy

[alex3]

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?

 

[George Jones]

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}

 

[alex3]

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.

 

[George Jones]

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.

 

[alex3]

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?

 

[George Jones]

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.

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.

 

[alex3]

Got it now, thank you very much!