I am currently trying to study Petrov Classification so I may start my research in the summer. I have completed a year of GR course at the level of Sean Carroll.
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$\begingroup$ I would suggest first to study spinor approach in GR. With it Petrov classification is extremely simple and elegant. $\endgroup$– A.V.S.Commented Mar 25, 2020 at 6:23
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$\begingroup$ Can you refer me to a resource dealing with discussion of Spinors approach? $\endgroup$– jay121Commented Mar 25, 2020 at 14:15
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2$\begingroup$ There is a book by P. O'Donnell “Introduction to 2-spinors in GR”. There is also a classic 2-volume monograph by Penrose & Rindler, though this might be harder. $\endgroup$– A.V.S.Commented Mar 25, 2020 at 20:54
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I recommend you to first have a good handle of Spinors in GR. I personally took this route in getting a grip on Spinors:
A Child's Guide to Spinors : http://www.weylmann.com/spinor.pdf
An Introduction to Spinors : https://arxiv.org/abs/1312.3824
Penrose, Roger (1960). "A spinor approach to general relativity"
The third one is going to be a smooth if you have gone through the first two documents which introduces to spinors in a wonderful way.
Then you can directly jump into these for further learning about Petrov Classification:
Petrov, A.Z. (2000). "Classification of spaces defined by gravitational fields". General Relativity and Gravitation. 32 (8): 1665–1685 (This is an English translation)
Hall, Graham (2004). Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics)