Second Bianchi Identity of 1902 and the Einstein Field Equation

These are described in Eq. (3.87) of Carroll’s online lecture notes to “Spacetime and Geometry: an Introduction to General Relativity”. Two index contractions give the 1902 second Bianchi identity as: D sup mu G sub mu nu = 0, where G sub mu nu is the Einstein tensor G sub mu nu = R sub mu nu – (1/2) R g sub mu nu. Conservation of energy / momentum is expressed as D sup mu T sub mu nu = 0 in terms of the canonical energy momentum tensor. So the Einstein field equation is

D sup mu G sub mu nu = D sup mu T sub mu nu =…

i..e
G sub mu nu = k T sub mu nu

where k is the Einstein constant. The second Bianchi identity is now known to be completely wrong (UFT88, UFT99, UFT109, UFT255. UFT313, UFT354 and many other papers) and…