Geodesic Conjugate Points Explained

[cr7einstein]

Dear all,
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 don't 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 don't 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 don't understand anything in blockquotes other than the RNP equation.

Thanks in advance!

 

[WannabeNewton]

Presumably by RNP you mean the Raychaudhuri equation, although the notations you have used are very non-standard and you seem to have redefined the expansion scalar to be its negation. Anyways, there isn't much detail involved. Just use the strong energy condition and the positive norm of the shear to get an differential inequality for the scalar $\rho$ and integrate to get the result. There should be a factor of $1/3$ that you seem to be missing, it follows from the Raychaudhuri equation. By the way this is a caustic not a pair of conjugate points.

 

[cr7einstein]

I have typed the exact same lines as that of the book.I know that the notation is different from the standard Raychauduri Equation, but this is because thiis the unified Raychaudhari-Newman -penrose(RNP) equation...as hawking puts it. the $$\frac{1}{3}$$ you are talking about is $$n=3$$ fot timelike curves. Here's a link to the pdf of the book : http://www.benpadiah.com/otherstuff/elib/HawkingNatureOfSpaceTime.pdf And can you please help me get the desired result from the strong energy condition? I don't get the head or tail of how it can be done to get the positive norm of shear...

 

[WannabeNewton]

For null geodesic fields you only need the weak energy condition, you don't need the strong energy condition.
The weak energy condition clearly implies that $R_{\alpha\beta}l^{\alpha}l^{\beta} \geq 0$ for any null vector $l^a$ (see p.8 of the pdf you linked).

Furthermore the shear has positive norm $\sigma_{\alpha\beta}\sigma^{\alpha\beta} \geq 0$ because it is space-like in both indices.

Therefore $\frac{d\rho}{dv} \geq \rho^2 \Rightarrow \frac{d\rho}{\rho^2} \geq dv \Rightarrow \rho^{-1}|_{\rho}^{\rho_0} \geq v - v_0 \Rightarrow \rho \geq \frac{1}{\rho_0^{-1} - v + v_0}$.

 

[sardar]

Dear reader,

Thank you for your interest in understanding geodesic conjugate points. I will do my best to explain the concepts mentioned in the content you provided.

Firstly, an affine parameter is a type of parameter used to describe the path of a geodesic (a curve that represents the shortest distance between two points in a curved space). It is used because it remains constant along the geodesic, making it a useful tool for describing the path. The affine parameter distance is simply the distance along the geodesic measured in terms of this parameter.

Moving on to the second relation between $$\rho$$ and $$\frac{1}{\rho^{-1} + \nu_0-\nu}$$, this is derived from the RNP equation mentioned in the content. The RNP equation is a mathematical expression that describes the behavior of a geodesic in a curved space. In this equation, $$\rho$$ represents the convergence of neighboring geodesics. In simpler terms, it measures how much the geodesics are bending towards each other.

Now, if we set $$\rho=\rho_0$$ at $$\nu=\nu_0$$, the RNP equation tells us that as we move along the geodesic, the convergence will increase and become infinite at a point $$q$$ within an affine parameter distance of $$\frac{1}{\rho_0}$$. In other words, at this point, the geodesics will be so close to each other that they will actually intersect.

To understand the second relation better, let's look at the denominator of $$\frac{1}{\rho^{-1} + \nu_0-\nu}$$, which is $$\rho^{-1} + \nu_0-\nu$$. As the affine parameter distance increases, the value of $$\nu$$ will also increase. Therefore, as we approach the conjugate point, the value of $$\nu_0-\nu$$ will decrease, making the denominator smaller. As a result, the overall value of $$\frac{1}{\rho^{-1} + \nu_0-\nu}$$ will increase, indicating that the convergence is getting larger and approaching infinity.

I hope this explanation helps you in understanding the concepts mentioned in the content. If you have any further questions, please do not hesitate to ask. I would be happy to provide more detailed explanations. Additionally, I suggest consulting other resources on