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What spacetimes do you consider to be important or interesting? Important or interesting as solutions to the Einstein field equation, or important or interesting geometrically. I'll start the list.
Minkowski
Schwarzschild (including extended)
Kerr(-Newman)
Friedmann-Lemaitre-Robertson-Walker
Schwarzschild constant density spherical solution
Vaidya
Morris-Thorne wormholes (and generalizations)
Alcubierre warp drive
The first five spacetimes on the list are now usually treated even in introductory books. Off the top of my head (not with my books right now), I can only only think of two of my books (that aren't compilations of solutions) that cover Vaidya spacetimes, Poisson's A Relativist's Toolkit: The Mathematics of Black Hole Mechanics and Padmanabhan's new, advanced, comprehensive Gravitation: Foundations and Frontiers. Morris-Thorne wormholes and Alcubierre warp drive are covered in some introductory books.
Rindler spacetime and the Milne universe are already included in Minkowski. Similarly, de Sitter and anti-de Sitter are included in the Friedmann-Lemaitre-Robertson-Walker family. I take a spacetime to be a four-dimensional Lorentzian (differentiable) manifold, so each spacetime includes all compatible coordinate charts. Rindler spacetime and the Milne universe are particular coordinate charts for Minkowski spacetime.
If you add a (family of) spacetimes, and you know of a book that isn't a compilation of spacetimes which treats the spacetime, also identify the book.
The two standard compilations are Exact Space-Times in Einstein's General Relativity by Griffiths (Jerry, not David) and Podolsky (which I find it to be endlessly fascinating), and the second edition of Exact Solutions of Einstein's Field Equations by Stephani, Kramer, MacCullum, Hoenselaers, and Herlt.
Minkowski
Schwarzschild (including extended)
Kerr(-Newman)
Friedmann-Lemaitre-Robertson-Walker
Schwarzschild constant density spherical solution
Vaidya
Morris-Thorne wormholes (and generalizations)
Alcubierre warp drive
The first five spacetimes on the list are now usually treated even in introductory books. Off the top of my head (not with my books right now), I can only only think of two of my books (that aren't compilations of solutions) that cover Vaidya spacetimes, Poisson's A Relativist's Toolkit: The Mathematics of Black Hole Mechanics and Padmanabhan's new, advanced, comprehensive Gravitation: Foundations and Frontiers. Morris-Thorne wormholes and Alcubierre warp drive are covered in some introductory books.
Rindler spacetime and the Milne universe are already included in Minkowski. Similarly, de Sitter and anti-de Sitter are included in the Friedmann-Lemaitre-Robertson-Walker family. I take a spacetime to be a four-dimensional Lorentzian (differentiable) manifold, so each spacetime includes all compatible coordinate charts. Rindler spacetime and the Milne universe are particular coordinate charts for Minkowski spacetime.
If you add a (family of) spacetimes, and you know of a book that isn't a compilation of spacetimes which treats the spacetime, also identify the book.
The two standard compilations are Exact Space-Times in Einstein's General Relativity by Griffiths (Jerry, not David) and Podolsky (which I find it to be endlessly fascinating), and the second edition of Exact Solutions of Einstein's Field Equations by Stephani, Kramer, MacCullum, Hoenselaers, and Herlt.