Dear all,(adsbygoogle = window.adsbygoogle || []).push({});

I was reading "Nature of space and time" By Penrose and Hawking pg.13,

> If $$\rho=\rho_0$$ at $$\nu=\nu_0$$, then the RNP equation

>

> $$\frac{d\rho}{d\nu} = \rho^2 + \sigma^{ij}\sigma_{ij} + \frac{1}{n} R_{\mu\nu} l^\mu l^\nu$$

implies that the convergence $$\rho$$ will become infinite at a point $$q$$ within an affine parameter distance$$\frac{1}{\rho_0}$$ if the null geodesic can be extended that far.

>

> *if $$\rho=\rho_0$$ at $$\nu=\nu_0$$ then $$\rho$$ is greater than or equal to $$\frac{1}{\rho^{-1} + \nu_0-\nu}$$. Thus there is a conjugate point before $$\nu=\nu_0 + \rho^{-1}$$.*

I dont understand many terms here. Firstly, what is affine parameter distance? And I am at loss as to how does one get the 2nd relation between $$\rho$$ and $$\frac{1}{\rho^{-1} + \nu_0-\nu}$$. How can you derive it? Frankly, I dont understand ANYTHING about how does thhis equation come, though I suspect it just the Frobenius theorem.

Please give me DETAILED asnwers, as I have mentioned before, I am not too comfortable with it. I dont understand anything in blockquotes other than the RNP equation.

Thanks in advance!!!

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# Geodesic conjugate points

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