I'm trying to understand the way that the stress-energy tensor for a gravitational field is derived and I've run into a few problems. It seems that there are two main avenues which are kind of similar. One derivation involves looking at [tex]g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}[/tex] where [tex]\eta_{\mu\nu}[/tex] is the standard Minkowski metric, where we then allow second-order terms in the perturbation [tex]h_{\mu\nu}[/tex] instead of ignoring them like the linearized theory of gravitational radiation did. We then define the Einstein tensor as (here c and G equal 1) [tex]G_{\mu\nu}=8\pi(T_{\mu\nu}+t_{\mu\nu})[/tex] where [tex]t_{\mu\nu}[/tex] represents the stress-energy tensor for a gravitational field. Then we simply find the connection coefficients, Ricci scalar, Ricci tensor and Einstein tensor to second-order in [tex]h_{\mu\nu}[/tex] and through lots of tedious algebra we then find [tex]t_{\mu\nu}[/tex] in terms of [tex]h_{\mu\nu}[/tex] and voila, we have our stress-energy tensor for a gravitational field! The other derivation seems to look at a more general metric [tex]g_{\mu\nu}=\tilde{g}_{\mu\nu}+h_{\mu\nu}[/tex] where [tex]\tilde{g}_{\mu\nu}[/tex] is now an arbitrary curved background metric instead of being flat like the first derivation, and [tex]h_{\mu\nu}[/tex] is again just a metric perturbation. But this derivation imposes a restriction that the scale of curvature [tex]R[/tex] of the curved background metric is far greater than the wavelength [tex]\lambda[/tex] of the metric perturbation or equivalently that the frequency [tex]f[/tex] of the metric perturbation is far greater than that of the curved background metric's frequency [tex]f_B[/tex], thereby allowing us to clearly separate the two and think of the metric perturbation as a gravitational wave riding on top of the curved background. Once this is done we then just do what we did in the first derivation and thus find [tex]t_{\mu\nu}[/tex] in terms of [tex]h_{\mu\nu}[/tex]. Now, the first derivation said nothing about frequencies, wavelengths or background curvature, so I feel as though I am missing something very important. If anyone could enlighten me as to the proper way of deriving the stress-energy tensor of a gravitational field, and the reasons behind that derivation I would greatly appreciate it. Thanks! BTW sorry about the weird vertical layout, I don't know how to get it to display nicely!(adsbygoogle = window.adsbygoogle || []).push({});

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# Stress-Energy Tensor for Gravitational Field

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