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One can see this reference for TBL space-times.

I would like to know how the explicit expression for the function called $G$ in equations $3.108,3.108,3.110$ in the above reference is obtained.

Also it would be nice to see some further references about TBL space-times.

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[ad] This site is about Mathematics. If you are interested in Physics, you may want to check out the Physics proposal. – kennytm Aug 22 '10 at 20:08
up vote 3 down vote accepted

It actually just comes from integration. Equation 3.106 from that book is

$$ \dot R^2 = \left( - \frac{\partial R}{\partial t} \right)^2 = \frac FR + f $$

rearranging the terms gives (note that $\dot R < 0$)

$$ dt = - \frac{dR}{\sqrt{\frac FR + f}} $$

Now perform the substitution $z = fR/F$, giving

$$ dt = -\frac F{f^{3/2}} \frac{dz}{\sqrt{1+\frac1z}}. $$

Integrating gives

$$ t - t_0 = \frac F{f^{3/2}} \left( \sinh^{-1}\sqrt z - \sqrt{z(z+1)}\right) $$

the rest is just algebra.

I am no expert on general relativity, but the common term of that TBL spacetime seems start with "Lemaitre-Tolman". There is a review article on arXiv[1] which might help.

[1]: Kari Enqvist (2008). Lemaitre–Tolman–Bondi model and accelerating expansion. General Relativity and Gravitation 40, 2–3, pp 451–466. DOI: 10.1007/s10714-007-0553-9.

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