Case

Case (1): vector gauge symmetry

Case (2): scalar gauge symmetry

Case (3): conservation

Conditions

$h^{\mu \nu }\to h^{\mu \nu }+{\partial }^{\nu }{\Lambda }^{\mu }+{\partial }^{\mu }{\Lambda }^{\nu }$

$h^{\mu \nu }\to h^{\mu \nu }+{{\eta }^{\prime }}^{\mu \nu }\Lambda$

${\partial }_{\mu }{T}^{\mu \nu }=0$

Relation between coefficients

$1+b=0$

$a+b=0$

$c+d=0$

$e+f=0$

$1+2c+d=0$

$a+b=0$

$e+2f=0$

$1+b=0$

$a+c=0$

$b+d=0$

$e{\partial }_{\mu }{h}^{\mu \nu }+f{\eta }^{\mu \nu }{\partial }_{\mu }h=0$

Unknown coefficients

$c\mathrm{,}d\mathrm{,}e\mathrm{,}f$

$a\mathrm{,}b\mathrm{,}c\mathrm{,}d\mathrm{,}e\mathrm{,}f$

$a\mathrm{,}c\mathrm{,}e\mathrm{,}f$