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The study of electromagnetism (E&M) centers around the electric and magnetic fields, both their static configurations and dynamic, e.g waves carried by them. The Maxwell equations are essentially field equations.

I've never studied General Relativity so I am ignorant of it other than I know that gravitation is its subject. But is that essentially the same content of a course of study? You study the static and dynamic properties of the gravitational field through a set of field equations?

If true, would that make the study of General Relativity somewhat similar, in terms of the logic, to that of E&M?

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There are a couple of differences, for gravity is somewhat more complicated than electromagnetism. For example, while electromagnetism is a field theory defined on spacetime, while gravity is a field theory of spacetime. In other words, electromagnetism takes spacetime as given, while gravity involves solving for the very structure of spacetime.

This immediately means that it is not as easy to split in static and dynamic problems (although it can be done). Dealing with gravity doesn't mean simply figuring out the $\vec{E}$ and $\vec{B}$ fields as defined on space and time, but rather involves figuring out how spacetime actually behaves. There are situations in which we can split spacetime into space $+$ time and there are situations in which we can interpret the gravitational field as being stationary in the sense that it does not depend on time, but it gets a bit more subtle.

Furthermore, solving the Einstein Field Equations is considerably difficult, because they are non-linear and usually the sources involve the gravitational field as well. When dealing with E&M, one often (although not always) takes the sources to be given as independent of the fields themselves and then simply solves a couple of differential equations to obtain the fields. However, for a gravitational problem, it is not very common for the sources (which are any form of matter you are considering) to be independent of the field, because the field is spacetime itself, and matter is distributed along spacetime. Furthermore, the non-linearity of the equations makes them considerably harder to deal with.

A consequence of that is that solving the EFE often is taken to be a more advanced topic in General Relativity. Wald's book, which is one of the main references in the subject, delays the discussion of how to tackle solving the field equations to just a later moment. This contrasts quite a lot with E&M books, which often contain a lot of techniques to deal with Maxwell's equations.

Furthermore, the math of GR is way more complicated, especially due to this fact that the theory describes spacetime itself. While E&M deals mainly with vector calculus and related topics, GR requires you to learn Differential Geometry, which takes some time.

In general, an introductory course in General Relativity will follow the following general structure, which I list based on my undergraduate GR course, Wald's book, Frederic Schuller's lectures and some other sources. Naturally, each individual course will be different.

  1. Review of Special Relativity. Sometimes this goes deep dealing with tensors and stuff and the relativistic formulation of E&M, sometimes its just conceptual.
  2. Lots of Differential Geometry. How much often depends on the instructor, but you often will learn about what do we mean by a curved spacetime, what is a metric, what are the Christoffel symbols (they play a similar role to that of the electromagnetic potentials), what is curvature, what are geodesics (the paths that free particles make on spacetime), etc. Some undergraduate courses won't go so deep in this stage (see, e.g., Wald's account of how he teaches GR for further detail). I'd say this is analogous to how E&M courses sometimes start with a review of vector calculus (see, e.g., the book by Griffiths).
  3. The Einstein Field Equations. This is pretty much saying what GR is and uses all of that differential geometry just built. If there wasn't so much differential geometry in this stage, the discussion will likely be quite short, since the EFEs do employ the notions of curvature I mentioned and so forth (which are pretty much the most advanced topic in differential geometry you'll have seem so far). This is similar to presenting the Maxwell Equations in an E&M course. Since the equations describe spacetime itself, there isn't really a way of splitting them in static vs dynamic at this stage, we can only do it later (as far as I'm aware at least).
  4. Applications. There are three main ones which most courses will at least try to cover: gravitational waves (which are sort of similar to E&M waves), Friedmann–Lemaître–Robertson–Walker Cosmology (can't think of an analogy), and the Schwarzschild solution, which describes the vacuum region of spherically symmetric spacetimes with a central mass (think of a star of a black hole that is not charged and doesn't spin; it's also analogous to the Coulomb solution in E&M). Some courses might derive these solutions from scratch by making some symmetry assumptions, other courses might just give you the solution (and perhaps check or have you check that they do solve the EFE). These three solutions provide the tools for the most usual experimental verifications of GR, so those experimental confirmations are discussed in this topic as well. In the courses I took, this topic alone often takes half the time of the whole course.

You see that here and there there are a couple of similarities. Both theories need some math in the beginning, both theories have a spherically symmetric static solution (the Coulomb solution for E&M, the Schwarzschild solution for GR), and so on. However, GR is far more difficult when it comes to solving the equations, so it is more usual to work with just a few solutions (or family of solutions, as is the case of FLRW cosmology). Common exercises will be to prove some property of these solutions, to analyze how objects move in these spacetimes (for example, to solve the relativistic version of the Kepler problem), and stuff like that.

In short, albeit the theories do have some similarities, the equations of GR are much harder to solve, and that is reflected in how it is studied.

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There is a similarity in that both electromagnetism and gravitation are studied with their respective field equations.

Apart from the differing complexity of their field equations [number of equations, nonlinearity, techniques to solving, etc], there is one important distinction.

The electromagnetic field (as it is usually is studied) is a "matter field" on a fixed non-dynamical "background" Minkowski spacetime.

The "gravitational field" on the other hand is a "field" that, along with "matter fields", also determines the geometry of the [now] dynamical spacetime.

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