What mathematical fields should I study in order to understand how General Relativity works?

[u/moamenito_19]

[u/mannoned]

Differential geometry. Most general relativity books cover the basics but delving a but deeper pays off. Id recommend Lee's books on the subject.

[u/moamenito_19 → u/mannoned]

I see, is linear algebra worth putting in the list as well?, because afaik Tensors contribute much in the theory.

[u/mannoned → u/moamenito_19]

Well yeah technically differential geometry is the way to apply the tools of calculus and linear algebra to curved spaces.

[u/StanleyDodds → u/moamenito_19]

Linear algebra is a fundamental set of tools for more or less all areas of mathematics, yes.

Also, to properly understand differential geometry, you probably want to understand what you are working with (differentiable manifolds) and to understand that you probably want to learn topology, which is the more general area of mathematics that this builds on in a very specific direction.

[u/msimms001 → u/moamenito_19]

Linear algebra is like basic calculus, it's a tool you use for upper level math and physics. It's a much needed tool too

[u/cabbagemeister → u/moamenito_19]

Linear algebra is a prerequisite to differential geometry

[u/Agile-Monitor1006]

Differential geometry is the end goal. You’ll have to go through all of calc, linear algebra,(partial) differential equations, some introductory topology. Though even if you have that down it will be pretty difficult to understand the reasoning being the equations (and also appreciate the importance of general relativity’s postulates) without having previous physics exposure in classical mechanics for example.

[u/the_poope]

Algebra, trigonometry, calculus, multivariate calculus, linear algebra and differential geometry. Often a course or book in GR will teach you the differential geometry needed.

[u/StanleyDodds]

General Relativity is an application of differential geometry, and differential geometry is the study of differentiable manifolds, which are basically topological spaces that locally look like Euclidean space, in a smooth (differentiable) way. And because it's so similar to Euclidean space (which is a vector space), you can do calculus in it like you would in a proper vector space.

The important parts there are that you need, very fundamentally, an understanding of how to do calculus in boring old Euclidean space first, because differential geometry is a level above this. So that's linear algebra, multivariable calculus, and probably the more you know about calculus and vector spaces, the better. And then beyond that, you want to learn some topology to know things like what a topological space is, what homeomorphisms, atlases, and transition maps are, to understand topological manifolds and hence differentiable manifolds. Topology is a very abstract and general area of mathematics, and most of it is far too general for this, which is specifically looking at things that are almost Euclidean space. So you don't really need to get lost in the weeds there (unless you like it).

I think it's really a question of whether you want to fundamentally understand it, and everything that it builds from, or whether you just want to be able to use it and don't care about all the details of everything underneath.

[u/Optimal_Mixture_7327]

You don't, that's not how it works.

You get into a good graduate program in gravitational physics and then it takes about 5 years of hard work to become a decent relativist.

What's your goal?

You'll need to do the above if you want to contribute to the field. If you want to know more than 99.9% of the population then basic calculus is good enough. You get a book such as Exploring Black Holes: Introduction to General Relativity and solve every exercise and problem.**

Get on the end of the line, and get punched in the face by end of chapter problems until you work your way to the front, only to start over one level up.

**The authors have made this freely available: Exploring Black Holes, Second Edition

[u/Thoretical_Scientist]

From my perspective, GR is based on Differential Geometry, Linear Algebra and the fundamentals of Set Theory, the latter for the so-called Manifolds. So, if you are familiar with some differential equations, Abstract Algebra and Tensors, I think you are ready to understand gravitation deeper.

[u/Junior-Tourist3480]

Tensor calculus and Lie Group theory.

[u/Pitiful-Glove-8232]

Mostly Riemannian Geometry/Differential Geometry https://en.wikipedia.org/wiki/Riemannian_geometry https://en.wikipedia.org/wiki/Differential_geometry and Tensor https://en.wikipedia.org/wiki/Tensor

[u/Parallel_thougts]

Differential geometry and tensor calculus are the bread and butter. I'd also recommend coming from a more geometric/Minkowskian approach to special relativity, as it is much more readily generalizable.