It is the third-order tensor i j k k ij k k x T x e e e e T T grad Gradient of a Tensor Field (1.14.10) This differs from the quantity i j k i jk jk j k i i x T T x T e e e e e e. The gradient of a second order tensor field T is defined in a manner analogous to that of the gradient of a vector, Eqn. Einstein's original 1928 manuscripts translated into English are included. As an example, consider a space curve parameterised by s, with unit tangent vector. Using the vierbein field theory, presented is a derivation of the the Einstein equation and then the Dirac equation in curved space. Thus, the vierbein field theory is the most natural way to represent a relativistic quantum field theory in curved space. To date, one of the most important applications of the vierbein representation is for the derivation of the correction to a 4-spinor quantum field transported in curved space, yielding the correct form of the covariant derivative. Einstein discovered the spin connection in terms of the vierbein fields to take the place of the conventional affine connection. Einstein's vierbein theory is a gauge field theory for gravity the vierbein field playing the role of a gauge field but not exactly like the vector potential field does in Yang-Mills theory-the correction to the derivative (the covariant derivative) is not proportional to the vierbein field as it would be if gravity were strictly a Yang-Mills theory. It is based on the vierbein field taken as the "square root" of the metric tensor field. We further describe geodesics as special curves in a curved spacetime. Locally flat (Cartesian) coordinate systems are described in which the laws of special relativity are valid. General Relativity theory is reviewed following the vierbein field theory approach proposed in 1928 by Einstein. We then extend our analysis to obtain the Riemann Christoffel curvature tensor and hence characterise curved spaces or curved manifolds.
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