General relativity without Differential geometry

Hello,

I am learning General Relativity through some books like 'Gravity' by Hartle and through some other text books. All those books, do not speak of general relativity from the context of differential geometry. I have a fair amount of knowledge of calculus as well as set theory. My understanding why differential geometry is required in GR: (Please correct me if I am wrong)

As it deals with curvature and topology hence it requires differential geometry to study curved surfaces.

My question:

(1) Is differential geometry absolutely necessary to learn General Relativity?
(2) Can GR be learned without differential geometry?
(3) How much of the GR is related to differential geometry?
(4) Can anybody please guide me, a step by step guide to differential geometry?

Thanks.

WannabeNewton
(1) Is differential geometry absolutely necessary to learn General Relativity?

No.

(2) Can GR be learned without differential geometry?

Yes.

(3) How much of the GR is related to differential geometry?

All of it.

Well, if differential geometry is not necessary to learn GR then how all of GR related to differential geometry?

WannabeNewton
Well, if differential geometry is not necessary to learn GR then how all of GR related to differential geometry?

Well first off you can learn a subject without properly learning it. Your original post didn't add the qualifier "proper" it simply asked if it's possible to learn GR without differential geometry. It's possible but that doesn't mean you're learning it the right way. Secondly, just because you learn GR taking the pedestrian route doesn't mean the contents of the pedestrian route define GR itself. People learn QM through wave-mechanics all the time. Does that mean there's no functional analysis in QM? Obviously not. Functional analysis is the core of QM but you can easily learn QM without ever knowing what functional analysis even is. It's much harder to learn GR without ever seeing any differential geometry but you can surely pull it off. Just don't expect the promised land at the end of the pedestrian road.

Hello WannabeNewton,

Thank you very much for this wonderful answer. So, while taking the pedestrian road, does not give a proper understanding of GR.

How difficult is differential geometry? Can it be self taught? What background of maths is required?

Thanks.

WannabeNewton