I'm a mathematician with hardly any knowledge of physics. Before I start reading volumes of physics books, I have a few questions that have been bugging me and that will help me start reading physics.
Let's forget Quantum Mechanics for the purpose of the discussion, and focus on Relativity. I always fail to understand what the objects are and how to relate them intuitively to perceivable space and time. What are the objects?
Here's what I mean: we begin with spacetime being a pseudo-Riemannian manifold with some metric (is this metric assumed to be locally Lorentzian?). For each point in this manifold, there should be some parameters, right? Like whether there's a particle there, what particle is it (information that includes, for example, mass), is there a magnetic field there and so forth. Together, this pseudo-Riemannian manifold, with a pre-specified metric, and with a set of parameters for each point, is what described the universe in relativity, right? What precisely are those parameters in General Relativity?
Answer
General relativity is a mathematical theory that generalizes the special theory of relativity and mechanics. It turns out that one can treat the gravitational effects as non-inertial forces, thanks to one of the principles that governs this theory, that is also considered the most important one, namely the principle of equivalence postulated by Einstein. That the metric of space-time must be locally flat, i.e. Minkowskian, is a consequence of this principle. In fact, the principle of equivalence states that in a free falling frame of reference, i.e. a reference where there are no inertial forces, the laws of physics are those of special relativity. As you are a mathematician there is no need for me to recall that if a manifold has a null curvature tensor, then there are global coordinates on the manifold such that the components of the metric tensor, say $g$, are exactly those of the Minkowski metric $\eta = \operatorname{diag}(1,-1,-1,-1)$.
The metric $g$ on the manifold $M$ describing the space-time is not a-priori given, but it is determined by the distribution of matter. Now comes your question on the objects of general relativity. These objects are all sorts of geometrical object that you can construct on a (smooth) manifold. Thus you'll have tensors of any ranks, and even spinors of any rank. What makes them physical objects is just their interpretation. Einstein's field equations
$$\text{Ric}-\frac12\operatorname{Tr}(\text{Ric})g = \chi T,$$
where $\text{Ric}\in T^*M\otimes T^*M$ is the Ricci tensor, $g\in T^*M\otimes T^*M$ the metric on $M$, and $T\in T^*M\otimes T^*M$ the energy-stress tensor, gives you the metic $g$ in terms of the energy-matter distribution $T$ on $M$, that is just a tensor density on $M$.
The mathematical problem of studying the geodesics on a manifold $M$ described by a metric $g$ is the equivalent of the physical problem of determining the motion of a particle that is freely falling in the gravitational field generated by a matter distribution $T$ such that the resulting metric is $g$.
One of the most spectacular prediction of the general theory of relativity is the existence of so-called black holes. Again you find these object by studying the mathematical properties of the solutions of the Einstein's equations and then give them a physical interpretation. There are some tough situation in physics when you can't base your theories on observations and experiments. Therefore you must start with a mathematical theory and develop it as further as you can. Each result you obtain is then physically interpreted. This is quite the situation of general relativity (see, e.g. the problem of the detection of gravitaional waves, or the observation of naked singularities).
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