Sean M. Carroll is a theoretical physicist and cosmologist known for his work in areas such as cosmology, quantum mechanics, and the foundations of physics. He is a professor at the California Institute of Technology (Caltech) and has made significant contributions to our understanding of the universe, including the nature of time, the arrow of time, and the role of entropy.
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Sean M. Carroll develops and teaches the geometric account of gravity in general relativity. His 1997 lecture notes connect differential geometry to Einstein field equations and applications in gravitational radiation, black holes and cosmology.
A central example is the covariant derivative: differentiating the coordinate components of a vector field alone does not produce another tensor. An additional affine connection term accounts for the changing coordinate basis. This construction allows physical equations to retain their meaning when coordinates change, supplying the mathematical foundation for the later gravitational applications.[ref]
Lecture Notes on General Relativity by Sean M. Carroll. Chapters 2–3: manifolds, tensor transformations, connection and covariant differentiation
Portrait of Sean M. Carroll
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Sean's series The Biggest Ideas in the Universe has some merit, but it's just to math-light falling a bit below the missing link between basic and advanced.