- #1

- 27

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_{μν}over arbitrary background metric g

_{μν}whit the restriction

[tex]\left| \dfrac{h_{\mu\nu}}{g_{\mu\nu}} \right| << 1[/tex]

Following Michele Maggiore Gravitational Waves vol 1 I correctly expressed the Chirstoffel symbol in terms of the perturbation (equation 1.205)

[tex] \tilde\Gamma^\alpha_{\mu\nu} = \Gamma^\alpha_{\mu\nu} + \dfrac{1}{2}(\nabla_\mu h_\nu^\alpha + \nabla_\nu h_\mu^\alpha - \nabla^\alpha h_{\mu\nu})[/tex]

After that I correctly obtain the perturbed Rieman tensor

[tex] \tilde R^\alpha_{\beta\mu\nu} = \partial_\mu \tilde \Gamma^\alpha_{\beta\nu} - \partial_\nu \tilde \Gamma^\alpha_{\beta\mu} + \tilde \Gamma^\alpha_{\tau\mu} \tilde \Gamma^\tau_{\beta\nu} - \tilde \Gamma^\alpha_{\tau\nu} \tilde \Gamma^\tau_{\beta\mu} = R^\alpha_{\beta\mu\nu} + \underbrace{R^{(1)} + R^{(2)}}_{\delta R} [/tex]

Where the first part of the remaining terms is in agreement with equation 1.206

[tex] 2R^{(1)} = \nabla_\mu \nabla_\beta h^\alpha_\nu + \nabla_\nu \nabla^\alpha h_{\beta\mu} - \nabla_\nu \nabla_\beta h^\alpha_\mu - \nabla_\mu \nabla^\alpha h_{\beta\nu} [/tex]

But the second is problematic,

**my**expression:

[tex] 2R^{(2)} =( \nabla_\mu \nabla_\nu - \nabla_\nu \nabla_\mu ) h^\alpha_\beta = R^\alpha_{\tau\mu\nu} h^\tau_\beta - R^\tau_{\beta\mu\nu} h^\alpha_\tau [/tex]

**Maggiore**expression, which I suppose has a typo

[tex] 2R^{(2)} = h^{\alpha\tau} R_{\tau\beta\mu\nu} + h_{\beta}^\tau R^\alpha_{\tau\mu\nu} [/tex]

After getting the contracted Rieman tensor to obtain Ricci tensor

[tex] \tilde R_{\beta\nu} = \tilde g^{\alpha\mu} \tilde R_{\alpha\beta\mu\nu} = ( g^{\alpha\mu} - h^{\alpha\mu} )( R_{\alpha\beta\mu\nu} + \delta R) = g^{\alpha\mu}\delta R - h^{\alpha\mu}R_{\alpha\beta\mu\nu} [/tex]

Then here are is the first part of the remaining terms

[tex] 2g^{\alpha\mu}R^{(1)} = \nabla_\mu \nabla_\beta h^{\mu}_\nu + \underbrace{\nabla_\nu \nabla_\mu h^{\mu}_{\beta}}_{-R_{\tau\nu}h^\tau_\beta + R_{\tau\beta\mu\nu}h^{\mu\tau} + \nabla_\mu\nabla_\nu h^\mu_\beta} - \nabla_\nu \nabla_\beta h^\mu_\mu - \nabla^\mu \nabla_\mu h_{\beta\nu} [/tex]

and the second part of the remaining terms,

**my**expression

[tex] 2g^{\alpha\mu}R^{(2)} = R_{\tau\nu}h^{\tau}_{\beta} - R_{\tau\beta\mu\nu}h^{\mu\tau} [/tex]

**Maggiore**expression

[tex] 2g^{\alpha\mu}R^{(2)} = R_{\tau\beta\mu\nu}h^{\tau\mu} + R_{\tau\nu}h^{\tau}_\beta [/tex]

Now taking into account the underbrace rewriting of ∇

_{ν}∇

_{μ}h

^{μ}

_{β}in R(1) and focus only on the terms containing hR in δR it follows

using

**my**expressions

[tex] 2g^{\alpha\mu}R^{(2)} = R_{\tau\nu}h^{\tau}_{\beta} - R_{\tau\beta\mu\nu}h^{\mu\tau} -R_{\tau\nu}h^\tau_\beta + R_{\tau\beta\mu\nu}h^{\mu\tau} = 0 [/tex]

thus I am left with one contraction between the perturbation and background Rieman tensor

[tex] \tilde R_{\beta\nu} = R_{\beta\nu} + R^{(1)} - h^{\alpha\mu}R_{\alpha\beta\mu\nu} [/tex]

but using

**Maggiore**expression

[tex] 2g^{\alpha\mu}R^{(2)} = R_{\tau\beta\mu\nu}h^{\tau\mu} + R_{\tau\nu}h^{\tau}_\beta -R_{\tau\nu}h^\tau_\beta + R_{\tau\beta\mu\nu}h^{\mu\tau} = 2R_{\tau\beta\mu\nu}h^{\tau\mu} [/tex]

thus

[tex] \tilde R_{\beta\nu} = R_{\beta\nu} + R^{(1)} [/tex]

Which suggests that there is no typo... or some other way to rewrite my expression... I tried the first Bianchi identity but with no success...

I am totally stumped on this for the past week and any help will be appreciated.