Gauge transformation
Bianchi identity
Bianchi identity /Maxwell’s Equation
from
x ′ α = x α + ε ξ α
with ε → 0 →
g ′ j k ( x ′ ) ≅ g j k ( x ) − ε ( ∇ k ξ j + ∇ j ξ k )
R ′ i j k l + R ′ i k l j + R ′ i l j k = 0 →
R i j k l + R i k l j + R i l j k = 0
(2-1)
∇ k F l i + ∇ i F k l + ∇ l F i k = 0
(2-12)
∇ j ( R ′ j k − 1 2 g j k R ′ ) = 0
or →
R ′ j k − 1 2 g j k R ′ = k T j k , ∇ j T j k = 0
R j k − 1 2 g j k R = k T j k
(3-2)
J i = def ∇ j F i j
(3-16)