Bianchi identity

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

Group generator Bianchi identity Feynman Jacobi... 

The electromagnetic field equations on the 0(3) level can be obtained from this purely geometrical theory by using Eq. (631) in the Bianchi identity... 

Equation (643) is also a Bianchi identity in the theory of gravitation because G v is derived from the antisymmetric part of the Riemann tensor, whose symmetric part can be contracted to the Einstein tensor. 

In standard classical electrodynamics, the Maxwell equation d = 0 becomes a Bianchi identity by using the electromagnetic potential s J, defined as = ds J. The dynamical equation for this field in empty space is d 0. 

But the Minkowski spacetime R4 has trivial cohomology. This means that the Maxwell equation implies that. is a closed 2-form, so it is also an exact form and we can write. = d d, where ( is another potential 1-form in the Minkowski space. Now the dynamical equation becomes another Bianchi identity. This simple idea is a consequence of the electromagnetic duality, which is an exact symmetry in vacuum. In tensor components, with sJ = A dx and ((i = C(1dxt we have b iV = c, /tv — and b iV = SMCV - SvC or, in vector components... 

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