General Relativity is often discussed in terms of the geometry of spacetime. But one can also think of it as just saying that gravity is associated with a field that has a certain strength or value at every point. This idea of a field is basically just like in electromagnetism, with its electric and magnetic fields. It’s also like in fluid mechanics, where there’s a velocity field that gives the velocity of the fluid at every point (like a wind velocity map for the weather).
What Einstein did in 1915 was to suggest particular equations that should be satisfied by the gravitational field. Mathematically, they’re partial differential equations, which means that they say how values of the field relate to rates of change (partial derivatives) of these values. They’re the same general kind of equations that we know work for electromagnetic fields, or for the velocity field in a fluid.
So what does one do with these equations? Well, one solves them to find out what the field is in any particular case. It turns out that for electromagnetism, the structure of the equations makes this in principle straightforward. But for fluid mechanics, it’s considerably more complicated—and for Einstein’s equations it’s much more complicated still.
In electromagnetism, one can just think of charges and currents as being sources of electromagnetic field, and there’s no “internal effect” of the field on itself (unless one considers quantum effects). But for fluid mechanics and Einstein’s equations, it’s a different story. In a first approximation, the velocity of a fluid is determined by whatever pressure is applied to it. But what complicates things greatly is that within the fluid there’s an internal effect of each part of the velocity field on others. And it’s similar with the gravitational field: In a first approximation, the field is just determined by whatever configuration of masses exists. But there’s also an “internal effect” of the field on itself. Physically, this is because the gravitational field can be thought of as having energy and momentum, which behave like mass in effectively being a source of the field. (The electromagnetic field has energy and momentum too, but it doesn’t itself have charge, so doesn’t act as a source for itself. In QCD, the color field itself has color, so it has the same general kind of nonlinear character as fluid mechanics or Einstein’s equations.)
In electromagnetism, with its simpler structure, one can’t have any region of static nonzero field unless one has charges or currents explicitly producing it. But when fields can act on themselves it’s a different story, and there can be structures that exist purely in the field, without any external sources being present. For example, in a fluid there can be a vortex that just exists within the fluid—because this happens to be a possible solution to the pure equations for the velocity field of the fluid, without any external forces.
What about the Einstein equations? Well, it’s somewhat the same story, though the details are considerably more complicated. There are nontrivial solutions to the Einstein equations even in the case of “pure gravity”, without any matter or external configuration of masses being present. And that’s exactly what black holes are. They’re examples of solutions to the Einstein equations that correspond to structures that can just exist independently in a gravitational field, a bit like vortices can just exist in the velocity field of a fluid.
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