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 = 0
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 its correct form is given in UFT313. The Einstein field equation is completely wrong and no conclusion based on it can be correct.