# 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.

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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
How difficult is differential geometry? Can it be self taught? What background of maths is required?
Well it depends on how far you want to go. With regards to GR it's very easy so don't worry. With regards to background math, calculus 3 and linear algebra are a must. Topology and/or real analysis would certainly help if you're learning from a pure math text on differential geometry but for GR you don't need to know any topology or real analysis until you're far down the road (also a lot of differential geometry texts will assume you've seen topology and real analysis before but again those are higher level than you would need for GR until you're at an advanced stage of GR). Honestly I wouldn't worry too much about learning differential geometry separately before learning GR if your main goal is just to learn GR. There are a myriad of excellent GR books that teach you the necessary differential geometry so that you can jump straight into the physics without wasting time on the extra fluff. As far as GR is concerned, the math is easy but the physics is arguably not so you should focus much more on the physics-that's what books like Hartle are for.

Because Gravity does not formally introduce differential geometry, does not mean there is no differential geometry in it. There's quite a lot in it, actually.