Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
[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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.