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The mathematical realization of equivalence principle (EP) is the principle of general covariance. General relativity (GR) equations must be covariant with respect to general coordinate transformations. To go from special relativity (SR) to GR equations, one replaces ordinary by covariant derivatives. The SR equation of motion turns into the geodesic equation. The Einstein equation, as the relativistic gravitation field equation, relates the energy momentum tensor to the Einstein curvature tensor. The Einstein equation in the space exterior to a spherical source is solved to obtain the Schwarzschild solution. The solutions of Einstein equation that satisfy the cosmological principle is the Robertson-Walker spacetime. The relation of the cosmological Friedmann equations to the Einstein field equation is explicated. The compatibility of the cosmological-constant term with the mathematical structure of Einstein equation and the interpretation of this term as the vacuum energy tensor are discussed.

*Keywords: *
principle of general covariance;
general relativity;
geodesic equation;
Einstein equation;
Newtonian limit;
Schwarzschild solution;
Robertson-Walker spacetime;
cosmological constant

*Chapter.*
*9445 words.*

*Subjects: *
Astronomy and Astrophysics

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