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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]