General relativity without Differential geometry

In summary, the conversation discusses the role of differential geometry in learning General Relativity. While it is not absolutely necessary to learn General Relativity, it is still a crucial aspect and is present in the subject. The difficulty of differential geometry and the necessary background in math is also mentioned. However, it is recommended to focus on the physics aspect of GR rather than solely on the math.
  • #1
shounakbhatta
288
1
Hello,

I am learning General Relativity through some books like 'Gravity' by Hartle and through some other textbooks. 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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  • #2
shounakbhatta said:
(1) Is differential geometry absolutely necessary to learn General Relativity?

No.

shounakbhatta said:
(2) Can GR be learned without differential geometry?

Yes.

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

All of it.
 
  • #3
Well, if differential geometry is not necessary to learn GR then how all of GR related to differential geometry?
 
  • #4
shounakbhatta said:
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.
 
  • #5
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.
 
  • #6
shounakbhatta said:
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.
 
  • #7
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.
 

FAQ: General relativity without Differential geometry

1. What is general relativity without differential geometry?

General relativity without differential geometry is a theoretical framework that describes the relationship between space, time, and gravity without the use of differential geometry. In this approach, the concept of curved spacetime is replaced with that of a flat spacetime that is subject to distortions caused by the presence of massive objects.

2. How does general relativity without differential geometry differ from the traditional version?

The traditional version of general relativity uses differential geometry to mathematically describe the curvature of spacetime. In contrast, general relativity without differential geometry relies on a simpler mathematical framework, making it more accessible for those without advanced mathematical training.

3. What are the advantages of using general relativity without differential geometry?

One of the main advantages of this approach is that it allows for easier visualization and conceptual understanding of the theory. It also opens up the possibility for new insights and perspectives on the nature of gravity and the universe.

4. Are there any limitations to general relativity without differential geometry?

General relativity without differential geometry is a simplified version of the theory and therefore may not be able to fully explain all observed phenomena. It also has not been extensively tested and validated through experiments and observations, so its accuracy and applicability may be limited.

5. How is general relativity without differential geometry relevant to current scientific research?

General relativity without differential geometry is a relatively new and developing area of research that is being explored by scientists and theorists. It has the potential to offer new insights and perspectives on gravity and the universe, and may lead to advancements in our understanding of the fundamental laws of physics.

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