Penrose Diagram for Minkowski Space-Time: Step-by-Step Guide

[petecc]

TL;DR Summary

Question about calculating curves for constant time and radial coordinates.

I'm working through Ray d'Inverno's book "Introducing Einstein's Relativity" and I've got to the section that introduces Penrose diagrams. The first example is just Minkowski space-time. The construction goes from Schwarzschild coordinates $t$ and $r$, to define null coordinates $v = t + r$ and $w = t - r$, then $p = \tan^{-1} v$ and $q = \tan^{-1} w$, and finally to $t' = p + q$ and $r' = p - q$.

One of the questions is then to calculate the paths of constant $t$ and $r$ in terms of $t'$ and $r'$ and draw a Penrose diagram with those curves marked on it.

So, my solution was to take $t+r = \tan p$ and $t-r = \tan q$, so
$$ 2t = \tan p + \tan q $$ and $$2r = \tan p - \tan q,$$
so for some constant value of $t=t_c$, we get
$$q = \tan^{-1}(2t_c - \tan p).$$
I can then, for a given $t_c$, use $p$ as a parameter and plot $t' = p + q$ and $r' = p - q$. Using the same approach for constant $r$, I put this into a Tikz diagram, and got this:

The problem is that the curves don't look like the answer given in the book, which looks much more like (half of) the image on wiki :

Ignoring that they've drawn both halves, the curves don't seem to be 'bunch up' as they approach the infinity points, like mine. I have found a few sources online that have diagrams that look like mine, so my question is: have I made a (quite possibly silly) mistake?

 

[Ambitwistor]

After reviewing your calculations and comparing them to the given answer in the book, I believe you have made a mistake in your calculations. The key error appears to be in your expression for $q$. Instead of $q = \tan^{-1}(2t_c - \tan p)$, it should be $q = \tan^{-1}(2t_c - \tan p) - p$. This makes a significant difference in the resulting curves, as it accounts for the "bunching up" effect seen in the book's answer.

I would recommend double checking your calculations and making sure to account for any necessary adjustments or transformations in the expressions for $q$ and $p$. I also suggest referencing other sources or consulting with a colleague or mentor to confirm your calculations and approach.

Overall, it is important to carefully check your work and make sure it aligns with the given answer and any known principles or equations. Mistakes can happen, but it's important to identify and correct them in order to accurately represent the concepts and principles being studied. Good luck with your studies!