Gauge transformation

Bianchi identity

Bianchi identity /Maxwell’s Equation

from

${x^{\prime }}^{\alpha }=x^{\alpha }+\epsilon {\xi }^{\alpha }$

with $\epsilon \to 0$ →

${g^{\prime }}_{jk}\left(x^{\prime }\right)\cong g_{jk}\left(x\right)-\epsilon \left({\nabla }_k{\xi }_j+{\nabla }_j{\xi }_k\right)$

${R^{\prime }}_{ijkl}+{R^{\prime }}_{iklj}+{R^{\prime }}_{iljk}=0$ →

$R_{ijkl}+R_{iklj}+R_{iljk}=0$

(2-1)

${\nabla }_k{F}_{li}+{\nabla }_i{F}_{kl}+{\nabla }_l{F}_{ik}=0$

(2-12)

${\nabla }^j\left({R^{\prime }}_{jk}-\frac{1}{2}{g}_{jk}{R}^{\prime }\right)=0$

or →

${R^{\prime }}_{jk}-\frac{1}{2}{g}_{jk}{R}^{\prime }=k{T}_{jk},{\nabla }^j{T}_{jk}=0$

$R_{jk}-\frac{1}{2}{g}_{jk}R=k{T}_{jk}$

(3-2)

$J^i\stackrel{\text{def}}{=}{\nabla }_j{F}^{ij}$

(3-16)