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The cosmological models of Lemaitre-Tolman-Bondi describe spherically symmetric universes with isotropic but inhomogeneous space, i.e. there are concentric shells with different mean mass densities. The LTB line element is:

[tex]ds^2 = -dt^2 + \frac{(R'(t, r))^2}{1+2E(r) r^2}dr^2 + R^2(t, r)d\Omega^2[/tex]

R' denotes derivate of R wrt r. I had two questions. First, how can this metric be derived? (e.g. it would be nice to have similar arguments as used to derive the homogeneous and isotropic FRW metric). Second, how would the metric look like if the observer is not located at the centre of the spheres but at a slight distance of it (leading to anisotropy)?

[tex]ds^2 = -dt^2 + \frac{(R'(t, r))^2}{1+2E(r) r^2}dr^2 + R^2(t, r)d\Omega^2[/tex]

R' denotes derivate of R wrt r. I had two questions. First, how can this metric be derived? (e.g. it would be nice to have similar arguments as used to derive the homogeneous and isotropic FRW metric). Second, how would the metric look like if the observer is not located at the centre of the spheres but at a slight distance of it (leading to anisotropy)?

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