Occasional, MTW's use of language is a bit obscure. I've highlighted a section from p491 (attached) which I just can't make any sense of.
I know about the 20 components of Rαβγδ but what does "these twenty components are arbitrary to the extent of the six parameters of a local Lorentz transformation" actually mean? And are they actually referring to Rαβγδ, because it goes on to say "However, (4)R is the only one of these 14 quantities...." which seems to imply that we aren't talking about the 20 Rαβγδ. So what are the 20 components? And what are the 6 parameters of the local Lorentz transformation?
Also, in the context of a local Lorentz transformation, where gαβ,γ= 0, all the values of Rαβγδ are linear in the second derivatives because Rαβγδ = Γαβδ,γ - Γαβγ,δ because the ΓΓ terms are all zero.
The Attempt at a Solution