I have done some books on GTR like by Sean Carroll's Spacetime and Geometry, read a little bit from Kip Thorne's Gravitation, and also did Short course on General Relativity. But I didn't understand much. The problem is not mathematics but notation and line of reasoning, like I don't understand why did we use tortoise coordinate in Schwarzschild solution to explain black holes I mean what were the hints that were asking us to use tortoise coordinate. And if I am working on some problem on GTR in future how would I know that which coordinate transformation would be better and how many sort of coordinate transformation are there. I just want some or more books to understand how Physicist did solve Einstein field equation.
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1$\begingroup$ I think this is too broad to be usefully answered. If you're asking about tortoise coordinates then we use them to avoid the coordinate singularity at the event horizon that we get in the Schwarzschild coordinates. All coordinate systems have their advantages and disadvantages and we choose the coordinates that are best for the specific problem we're looking at. $\endgroup$– John RennieCommented Aug 9, 2018 at 6:47
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$\begingroup$ Yeah, but how did they know that they should use this particular transformation, did they tried trail and error, I don't think so, I want some book, so I could understand how did they figure out. $\endgroup$– Chandra PrakashCommented Aug 9, 2018 at 6:52
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$\begingroup$ It is basically trial and error, though obviously it helps if (like Eddington) you have a deep understanding of the geometry involved. Informed trial and error you might say. $\endgroup$– John RennieCommented Aug 9, 2018 at 6:59
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$\begingroup$ Okay thanks, so any books to get my understanding of geometry involved better $\endgroup$– Chandra PrakashCommented Aug 9, 2018 at 7:00
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$\begingroup$ Have a look at this list $\endgroup$– John RennieCommented Aug 9, 2018 at 7:08
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If you're confused about the basics of the Schwarzschild solution, I think it's best to go back to a basic reference like Carroll and work everything out for yourself thoroughly. Many of the coordinate transformations here are not obvious; they took decades to find. You can instead motivate them by seeing what formal properties they reveal.
If you want to see these ideas applied more generally, what you need is a second course on general relativity that focuses on black holes. For example, there's a great set of lecture notes here. You could also get this background from more advanced books like Wald.