(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is not really part of some homework, but I'd rather post it here than annoy other people on the more general threads.

I'm trying to self study the topic of asymptotic flatness related to General Relativity. The textbook I'm using tries to explain the concept by introducing a flat spacetime M with a metric, in spherical coordinates, and then modifying it.

2. Relevant equations

It all starts with the metric:

[tex]g = dt^2-dr^2-r^2(d\theta^2+sin^2{\theta}d\phi^2)[/tex]

and expresses it in terms of a retarded system (u, r, theta, phi), where u=t-r:

[tex]g = du^2-2du dr-r^2(d\theta^2+sin^2{\theta}d\phi^2)[/tex]

In order to have points at infinity, it then introduces a new coordinate system [tex](u,\Omega,\theta,\phi)[/tex], where [tex]\Omega=r^{-1}[/tex], and gets:

[tex]g = \Omega^{-2} (\Omega^2du^2-2du d\Omega-d\theta^2-sin^2{\theta}d\phi^2)[/tex]

In this way, M gets extended to a new manifold [tex]\hat{M}[/tex], which has a well-defined boundary at infinity for [tex]\Omega=0[/tex].

The metric is not yet fine at the boundary, so it gets rescaled by defining a new metric:

[tex]\hat{g} = \Omega^2g = \Omega^2du^2-2du d\Omega-d\theta^2-sin^2{\theta}d\phi^2[/tex]

3. The attempt at a solution

So far, so good. Now the text says that (I report it literally here):

"from the form of the metric [tex]\hat{g}[/tex], we see that:

[tex]\hat{\nabla}_a\Omega\hat{\nabla}^a\Omega = \Omega^2[/tex]

and

[tex]\hat{\nabla}^a\Omega\hat{\nabla}_au = -1[/tex]"

and this really beats me! The best I was able to obtain, by some manipulation is:

[tex]\hat{\nabla}_a\Omega\hat{\nabla}^a\Omega=\hat{\nabla}_a\Omega\hat{\nabla}_b\Omega\hat{g}^{ab}=(\hat{\nabla}_a\Omega\hat{\nabla}_b\Omega)\Omega^{-2}g^{ab}=(\nabla_a\Omega\nabla_b\Omega)\Omega^{-2}g^{ab}=(\nabla_a\Omega\nabla^a\Omega)\Omega^{-2}[/tex]

from which one would get that:

[tex]\hat{\nabla}_a\Omega\hat{\nabla}^a\Omega=\nabla_a\Omega\nabla^a\Omega=0[/tex]

which is not true in general, but I can't see where I'm making a mistake...

Any help is really appreciated.

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# Homework Help: Asymptotic flatness

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